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ԍhh@ Lح1Э1Ѣ 24.4 24Z}4`D4E` 2BJ k3LVRH` 2BD4EhK)I JLV333 BASIC languagi}e OBJ > MACHINE language DOC > DOCumentation PIC|MIC|FNT|MVM > Graphics AMS|MUS|MBD > Music LSTj} > BASIC LISTing -Some MACHINE language programs can only be run by booting the disk MENU "without" BASIC! -For sk}ome programs, XL/XE models may require a "TRANSLATOR" program to be booted first! [Also available from BELLCOM] -l}Most programs are relatively self-explanatory; however, some have separate DOCumentation files available. Some BASICm} programs may contain instructions in their program listings. -To use this great MENU program on your own disks, n}simply copy the file "AUTORUN.SYS" onto your disk. * MENU written by DAVID CASTELL * program on your own disks, ^ [This article originally appeared in the March 1986 issue of The Australian Atari Gazette, the newsletter of M.A.C.Ep}., P.O. Box 340, Rosanna, Victoria, Australia 3084.] Adventures with... ^^^^^^^^^^^^^^^^^^ THE MAq}NDELBROT SET USING AN ATARI 800 ^^^^^^^^^^^^^^^^^^ or... A SLOW ROAD TO INSANITY br}y Rita Plukss A TUTORIAL ========== Girolama Cardano was an astrol- oger and physician tokis}ngs, a compulsive gambler who spent his life on the verge of bankruptcy and prison as well as on the brink of atheismt} and heresy. He was also a scientific writer from whose quill books flowed in fountains. Over 300 years later he hau}s been responsible for my near compulsive behaviour and for grave doubts being cast upon my sanity. Because of his av}stounding concept of negative numbers, and fictitious or sophisticated quantities, my family have placed a sign on thw}e computer room door <> and refuse to come in close contact with me. All they hear through the crackx} in the door, or when I emerge for sustenance is "Why is it not there?", or "Where is it?". They just shake their heady}s and hope that there is a power failure and then I will return to human form and remember all those tasks that now rz}emain undone. What Cardano put forward is today called an "imaginary" number. This is the square root of a negative{} number. What? you say. Has she finally flipped her lid? We all agree that it is impossible to have a square root of|} a negative number, but, just suppose, if it could be imagined and we combined this imaginary number with a real numb}}er eg 1+2/-2, then, the result would be known as a "complex" number. In fact these "complex" numbers have all sorts ~}of applications in areas such as atomic physics, engineering etc. But who can understand such highly abstracted conc}epts? All I'm interested in is using them to generate pretty pictures. A "complex" number is one that is composed o}f an ordinary number plus some multiple of the imaginary unit (the square root of minus one). eg 2+5/-1. The custom}ary way of writing /-1 in such numbers is i, and any complex number as a+bi. In our example a is 2, b is 5 and i is }the square root of minus one. So we can write it as 2+5i. Ordinary numbers can all be thought of as lying along a s}ingle straight line without any gaps in it - a continuum. But a typical complex number a+bi has no place on the line o}f ordinary numbers. So what can you do with it? Where does it fit? Gauss proposed that the complex number could be} thought of as labelling a point on a two dimensional plane; where a would be the horizontal distance and b the verti}cal distance and that the full a+bi could determine the position of a point on the plane just as the x and y of a Car}tesian coordinate couplet determines the point on a graph. So where does all this lead to? It leads to the Mandelbrot} Set. The Mandelbrot set is named for Benoit B. Mandelbrot, a research fellow at the IBM Thomas J. Watson Research C}entre in Yorktown Heights New York. He developed the field of fractal geometry (the mathematical study of forms havi}ng a fractional dimension). This work was carried on by John H. Hubbard, a mathematician at Cornell University and H}ubbard was one of the first people to make computer generated images of the Mandelbrot set. According to Hubbard it is} "the most complicated object in mathematics". If you wish to gain a more mathematical insight into the Mandelbrot} set, then you will need to have some understanding of two mathematical concepts - iterative procedures and complex n}umbers. The first is fairly easy (just like FOR-NEXT loops) but the second is a bit more abstruse. The articles on th}e Mandelbrot set in both Scientific American (August 1985) and Your Computer (January 1986) will give you sufficient }background of both concepts, but better still, consult with a senior high school maths student. To reiterate, a comp}lex number takes the form of C=A+Bi where both A and B are real numbers (eg -5, 6, 8.996 etc) and i is defined as the} square root of -1. The complex number C consists of a real part (A) and an imaginary part (Bi). We can now describe }the iterative procedure that generates the Mandelbrot set. Start with the expression Z=Z*Z+C where both Z and C are co}mplex numbers and allow Z to vary. Set Z=0 to give us Z=0+C=C (C is now equal to Z); substituting this value back int}o the equation for the next iteration will give Z=C*C+C, then Z=(C*C+C)(C*C+C)+C and so on. The value of Z fluctuates} with successive iterations because we have introduced the value i*i=-1 into our calculations. The Mandelbrot set is} the set of all complex numbers for which the size of Z*Z+C remains finite even after an indefinitely large number of } iterations and is situated at the centre of a vast two dimensional plane. The boundary of the set is a fractal (and y}ou thought I'd given up on fractals)! But what a fractal! It must be the ultimate. Our interest lies on these edge}s of the set where Z falls outside the Mandelbrot set i.e. where Z does reach the value of 2 and will go off to infi}nity. The fractal area! The number 2 is the crucial factor in working with the Mandelbrot set. "A straight forwa}rd result in the theory of complex number iterations guarantees that the iterations will drive Z to infinity IF AND O}NLY IF at some stage Z reaches the value of 2 or greater." But, actually there are very few points where the value of }2 will not be reached after a small number of iterations. This situation becomes rarer as the iteration count increa}ses. The Mandelbrot set itself is the black (static) area within the plane, where Z does not reach the value of 2. (}That is, black on the screen, white on the printouts.) It is shaped like a squat, wart covered figure eight lying on} its side. The coloured and patterned areas around the set are where Z has reached the value of 2 and the colour of }the pixel should tell you how many iterations it took to reach that value. This is the area that is of greatest inte}rest, the areas that surround the set. These areas are a myriad of colours and patterns, no two areas being identical,} yet repeating themselves over and over again. There are riots of organic looking tendrils and circular sweeps, whir}ls and whorls, and the colour! You can look at the full set, look at the edges, or you can zoom in as with a microsc}ope and go deeper and deeper, the patterns and colours just keep happening. The program that searches for these valu}es has been called MANDELZOOM (Program listing 1). As the name implies it investigates the Mandelbrot set and also a}llows you to zoom into the set for closer investigation of any part. With the Atari 800 we do have a couple of minor} problems; no double precision therefore we have to limit our zooming, and the resolution we can actually achieve, on}ce again limiting our final screen output. But with these limitations recognised we can produce very acceptable disp}lays within those constraints. Our first programs produced immediate displays - if anyone is interested in these let m}e know and I will pass them on to you. The final versions (we think they are the final versions - but we have though}t that before!) of the programs that follow this article were put together by Dick. But there have been a small core o}f enthused people (or nuts?) working diligently through the various stages to come up with the final version. This s}ubset of the Mandelbrot set have worked and bolstered each other and they are Chris Ryan, Dick Kellett, Ron Collis, a}nd of course, myself. There have been a couple of enthusiastic late comers, Gary Fyfe and Ian Conner (both from Geel}ong). The main problem with immediate screen plotting was that we could not reallocate where the colour changes should} take place. Not having the memory available to read the data into an array, Dick had a stroke of genius - put it to} disk as a data file, then when you are ready to plot, read it back from the disk! Brilliant I said! Only one probl}em, each data file is 246 sectors long! Oh well, what else are disks for? It still amazes me that with only 48K of }memory (less BASIC of course), an 8 bit ATARI computer and a TV screen that we could produce screens that are quite a}cceptable considering that what you see elsewhere have been produced on computers much more powerful than ours. But, th}en of course, we have an Atari! Power without the price. Following is a brief explanation of the 4 programs. See} Dick's article (3. More background) on how each of the programs work. The file "KELLETT.DOC" contains Dick Kellett's} article. PROGRAM 1 --------- MANDELZOOM - MANDLCAL performs the calculations on that part o}f the Mandelbrot set under investigation and writes it as a data file to disk (15-30 hours), depending on how much of} the area is within the actual set (black). PROGRAM 2 --------- MANDPLOT - Reads the data fi}le obtained by program 1 and plots that data to screen. (15-20 minutes) PROGRAM 3 ---------} DATACHK - Scans the data disk, reads the number of iterations required to generate each piece of data (pixel) and c}ollates this information. This allows you to see at what levels of iteration to make your colour changes when runnin}g program 2. PROGRAM 4 --------- COLDUMP - This has nothing to do with the Mandelbrot set, b}ut is a screen dump to dump each colour register separately to the printer (7+ screen). By using this, colour carbon} paper and a good eye when you roll your paper back, you can produce good colour printouts. HAVE FUN AND GOOD }LUCK! ----------------------- For more information on the Mandelbrot set refer to Scientific American (August 19}85) or if really involved, try The Fractal Geometry of Nature by Benoit B.Mandelbrot (W.H.Freeman & Co. New York, 197}7) and Fractals - Form, Chance and Dimension by Benoit B.Mandelbrot (W.H.Freeman & Co. San Francisco, 1977). York, 197t ^^^^^^^^^^^^^^^^^^ THE MANDELBROT SET ^^^^^^^^^^^^^^^^^^ The disk "MANDELBROT SET #1" contai}ns the four programs listed within the Mandelbrot articles in the M.A.C.E newsletter. They allow you to create the Man}delbrot data file, plot the data file to screen, investigate where to make the required colour changes, and finally t}o dump the screen to printer, colour register by colour register (for colour printouts), or in a variety of grey shad}es. The disk "MANDELBROT SET #2" contains a slide show (using Fader 2) of graphics 7+ Mandelbrot screensaves. These} were generated by Dick and Rita and are really fantastic to view. INSTRUCTIONS ============ }To generate your own Mandelbrot use Program 1 and refer to Figure 1 in the article (figure not on this disk). This is }the complete set, and from here you can select the coordinates to investigate any part of that set. ACORNER (Real co}ordinate) is the horizontal axis, BCORNER (Imaginary coordinate) is the vertical axis. SIDE is the horizontal length} of the 'square' you wish to view. (The ratio is 1 horizontal to .64 vertical.) The smaller the SIDE value, the mor}e powerful the zooming function and you find yourself deeper within the set. I have gone to 8 decimal places (and fu}rther). The deeper you go, the more precise you need to be with your measurements to get the right coordinates to find} something of interest, and you do need some luck, otherwise you may find nothing but the blackness of the set itself}. Load in program 1. Insert a disk with 250 free sectors, this will be your data disk. Follow the prompts. Turn o}ff the screen and leave the computer and drive to work for the next 10-30 hours (yes hours!) while you do all those o}ther tasks that need doing. When all the computation has finished and the file has been completed, turn on the scree}n which will show FINISHED and the name of the file it was saved to. Load in Program 2. Type in the name of the pic}ture file, then the name of the data file to generate the picture. For the first run through select option 2 (MOD 3)} and watch the mystery of the selected area unfold before your eyes (15 minutes). Save this screen by pressing the S}ELECT KEY. Use Program 5 (Loadscrn) to retrieve this picture at a later time. Program 3. Follow the prompts. This} program shows you at what levels of iterations activity was occurring. This will help in the choice of where to set t}he colour changes for the best effects. After noting (or dumping) the information generated by this program use that} information to experiment where to place your colour changes in program 2. Continue experimenting until you have th}e effect you desire. Program 4. If you have a PX-80 printer this will dump your screen to the printer either in sha}des of grey, or in separate colour registers. Program 5. This is a rough and ready retrieval program. Follow the p}rompts (graphics 15 is graphics 7+). I added this as an afterthought, just incase you did not have the means to retrie}ve the picture files you had created. This small program will actually retrieve any type of saved screen except for} compressed screens. PROGRAM DIRECTORY ================= PROGRAM 1: --------- MANDELZOOM part 1 }- MANDLCAL.BAS PROGRAM 2: --------- MANDELZOOM part 2 - MANDPLOT.BAS PROGRAM 3: --------- DATACHK.BAS PRO}GRAM 4: --------- COLDUMP.BAS PROGRAM 5: --------- LOADSCRN.BAS MANDELBROT SET #2 ==========}======= MANDELBROT SET #2 disk is an autorun. (Press START to speed up the display of each screen.) Just slip it in}to the drive, turn on the computer, sit back and watch a slide show of what you can produce using the programs on the} disk MANDELBROT SET #1. The views are fantastic! tch a slide show of what you can produce using the programs on the7 [This article originally appeared in the March 1986 issue of The Australian Atari Gazette, the newsletter of M.A.C.E}., P.O. Box 340, Rosanna, Victoria, Australia 3084.] MANDELBROT SETS and the ATARI ===========================}== by DICK KELLETT Have you looked at those computer generated pictures in SCIENTIFIC AMERICAN AUGUST 19}85 and YOUR COMPUTER YEAR BOOK JANUARY 1986 and thought you would like to try the same type of graphics? The articl}es in these magazines suggest setting up arrays of up to 1000 by 1000 and using 1000 iterations to check if each point} is in the MANDELBROT set. This is fine if you have access to a super mini or a mainframe but not much help with only }32K available. While the ATARI cannot match the resolution of the published pictures, GRAPHICS 7+ (GRAPHICS 15+16 on} XL and XE models) will give sufficient resolution and three colours plus background colour to produce extremely int}eresting pictures. The pictures are produced by using the equation Z^2+C where Z and C are complex numbers and repea}ting the calculation with the answer replacing Z in the equation. Counting the number of iterations before Z=2 and a}ssigning a colour to this number generates the picture. If the number of iterations exceeds the selected maximum (in }our case 100 iterations) the area (pixel) is in the MANDELBROT set and is plotted in the background colour. After ma}ny frustrating hours plotting screens directly and having a single colour or a small area in one corner, I realized t}hat the number of iterations selected to change colours was very important and differed for each picture. My approac}h to the problem was to use two programs. The first one called MANDLCAL.BAS selects an area according to the formulae }in the SCIENTIFIC AMERICAN article, scales it to suit a GRAPHICS 7+ screen and stores the results on disk as a DATA f}ile of 246 sectors. I used one hundred iterations to approximate whether or not the point was in the MANDELBROT set.} The second program called MANDPLOT.BAS allows you to select the levels of iteration that colour changes will occur }at, and then plot the point in the selected colour. Points in the MANDELBROT set are always plotted in the backgroun}d colour. The colour changes may be selected to change at preset counts, plot through the colours in MOD.3 or plot c}olours below the first change in MOD.3 The same colour is used for the lowest and highest counts as there is suffic}ient difference in the position of the points plotted to avoid running the areas together. It almost gives the effec}t of having an extra colour. Both of these programs run very slowly. MANDLCAL takes between ten and twenty hours to} calculate the data file. The worst case involves 100*159*191 separate calculations for an area completely in the MA}NDELBROT set. Obviously, an area with a lot of background points takes longer to calculate than one with a lot of col}our. MANDPLOT will set up a graphics 7+ screen and plot the picture from the data file in approximately twenty minut}es. Before the picture is plotted you will be asked for a file name to save the picture to. If you do not select a }name, the picture will be saved using the default name of PICTURE. The pictures are saved as 62 sector files. I pre}fer to save my pictures with individual names and use the DOS copy file option to create a new file called PICTURE, w}hich can then be loaded into ATARI ARTIST (MICRO ILLUSTRATOR) by pressing the CLEAR key after ATARI ARTIST has been l}oaded. The colours can then be adjusted and pattern fills added as required. The picture can then be saved in the }normal way and used with FADER 11 as a slide show. The pictures could also be put through Rapid Graphics Converter and} be used as a background file for MOVIE MAKER etc. NOTE: View the completed pictures displayed on the disk "MANDELBR}OT SET #2". Breath taking! Both these programs will compile with the MMG compiler; or, try running them using the Pu}blic Domain Turbo Basic XL. MANDLCAL does not show a significant increase in speed and I let the program run overnigh }t and while at work the next day. When compiled MANDPLOT will plot the picture in approximately ten minutes instead
}of twentyfive minutes. If you try these programs I would suggest that you obtain a copy of the photograph of the ful}l MANDELBROT set from the SCIENTIFIC AMERICAN article, as it shows the co-ordinates for the set. This will allow you} to select the starting points (ACORNER, BCORNER) and the length of the side for the area to be plotted (SIDE). If y
}ou have trouble locating this article see RITA or DICK at the next meeting or come to the screen art group. To get s}tarted try the following points. The first set of figures should give you the complete MANDELBROT set (a colour versio}n of Fig.1). Try... ACORNER=-2.5 BCORNER=-1.25 SIDE=3.5 and... ACORNER=.2665 BCORNER=-.0049 SIDE=.00}2 and... ACORNER=-.9 BCORNER=.263 SIDE=.005 Happy plotting... You will develope lots and lots of patienc}e! and... ACORNER=-.9 BCORNER=.263 SIDE=.005 Happy plotting... You will develope lots and lots of patienc67MANFILEHVIACNBCNSSCGAVGABAABCOUN}
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+>;@,;@,;@,;@,HAP%;}dV(>:A%,R(=ENTER FILE NAME OF DATA FILE.DO NOT USE DEVICE NAME OR EXT.VnE67@<@,.D:,67@%<},.E67B:,%@,..DATx
A (>:A%,NN(FENTER TO DISPLAY PICTURE WITH USER SELECTED COUNTS FOR C%=}OLOR CHANGE::(2ENTER TO DISPLAY PICTURE WITH COLOR CHANGE MOD 3MM(EENTER TO PLOT THE COUNTS LOWER THAN THE FIRST %>}CHANGE POINT IN MOD 3%% @)!@A !
B!"@APp(>:A%,p([THE COUNTS FO%?}R CHANGE ARE BETWEEN 1 AND 100. ENTER THE CHANGE NUMBERS IN ASCENDING ORDER(
A-@@(( _-%@}@@=(#ENTER COUNT FOR FIRST COLOR CHANGE A_ @)!AA(
A-@@(( %A} Z-@@>($ENTER COUNT FOR SECOND COLOR CHANGE BZ)!AA(
A0-@@ %B}(( Y-@@=(#ENTER COUNT FOR THIRD COLOR CHANGE AY)!AA0
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A`Р%H}ӠΛb+@%@76-F:A`,%F:Aa,$AVK%@@xb6-F:@,%@6-%@%I}%A*F:,"@6@F:,"@y@x @@$̠Π%J}k(>:A%,(k(SOnce design is complete, press SELECT to save to disk. The default name is PICTURE]AdAU%K}](CPress START to plot a new picture without saving current screenS(((Enter Filename to save designO("DO NOT U%L}SE DEVICE NAME OR EXTENDERS26-B:,"(6. D:PICTURE2
AW67@<@,.D:3-@%%M}@S67<,.7&@<&@,W ##67%@<%@,..PIC$!!---------------------------%N}ŠϠˠŠ+@+@@&;%6-F:@,%AV$F:@,36-%Av;6-&0z%O}6-P:'AV,&6-&+$AV,8AP@MARF:@,bASF:@,nAVzAW:*6%P}-?:C:hhhLV,<@,*@DX-@@4-@6?0P2@T X N%Q}
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+AA1A@$4(E(ΠWAH@'('( GRAPHICS 7+ TO PX80 P)U}RINTER,(,(!by Jerry White & Fernando Herrera(S(*(Altered to PX80 and print each S($register separately by Dick )V}Kellett.-kF(>WHEN PICTURE IS LOADED PRESS TO PRINT REGISTRATION LINEk( PRESS TO COMMENCE PRINTING2+(+()W} PLEASE WAIT. SCREEN DUMP LOADING3-@A <
A^***SCREEN DUMP****_*BY JERRY WHITE &*`)X}*FERNANDO HERREA *a* ANTIC MAGAZINE *b* JANUARY, 1984 *c******************dРiRR;A)Y},;A,;@,;@,;@B,;@3,;Ap,n
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#A@!-@@B%"367<,.>:,7 LL)^}104,104,104,10,10,10,10,170,104,104,157,66,3,104,157,69,3,104,157,68,3KK104,157,73,3,104,157,72,3,32,86,228,169,0,133,21)_}3,189,67,3,133,212,96٠Ԡś7
#A!-@@3%"367<,.>:,7 ==104,173,48,2,1)`}33,203,173,49,2,133,204,160,1,200,177,203"@@201,15,240,4,201,79,208,4,233,1,145,203,192,200,208,237,96,Ҡ)a}Λ-p
A(>:A%,,-@@p(?WHEN PICTURE IS LOADED PRESS TO PRINT REGISTRATION MARKS.)b}(PRESS TO PRINT/-@@0b(#(ENTER REGISTER TO PRINT 9(0. DEF COL ORANGEN(1. DEF COL GREEN)c}b(2. DEF COL BLUE1**("3.BACKGROUND (REG 4).DEF COL BLACK2[*("4. PRINT ALL REG IN SHADES OF GREY9( INPUT REG=[)d} )!@A5#$@%A06%-A1AF"!% I%%ҠǮԠ̠ś)e}J))0,0,0,0,15,15,15,15,0,0,0,0,0,0,0,0S**ҠǮԠ̠ԠΛT))0,0,0,0,0,0,0,0,15,15,15,15,0,0,0,0])f}##ҠǮԠ̠ś^))0,0,0,0,0,0,0,0,0,0,0,0,15,15,15,15g))ĠǮԠ̠˛h))15,1)g}5,15,15,0,0,0,0,0,0,0,0,0,0,0,0q&&Ԡ̠ǮΠӠƠٛr**0,0,0,0,1,0,4,0,3,0,12,0,15,15,15,15!!Р)q}6?b%DOS SYSb*)DUP SYSbSAUTORUN SYSb_HELP DOCbVoMANDLBRTDOCb!INSTRUCTDOCb,KELLETT DOCbMANDLCALBASb+DATACHK BASb6MANDPLOTBASb-RCOLDUMP BASbLOADSCRNBASà٠ԛ;
+@$/6-F:@,%F:@,$AV;6-?:C:,,Q@+@@Q6)r}-?:C:,<@<@<<Av,ԠΠӛ F:B2y,"@A F:B2y,"@A)s}
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A 0@.@@P:: @w(@>:@',>:@d, )t} COLOR REGISTER (@q"N(@1+- - - - - - - - - - - - +b-@)u}@m(@q q"@N(@1+ - - - - - - - - - - - - +b-@@m()v}@q q"@N(@1+ - - - - - - - - - - - -+b-@@m(@q )w}q"@N(@1+ - - - - - - - - - - - +b-@@m(@q q"@)x}N(@1+--------ALL REGISTERS--SHADES OF GREY----------+b-@@m(@q F:B2y,"@)y}A F:B2y,"@A F:B2y,@A РҠҠӛ РҠ)z}Ҡӛ@.@@P:(@>:@',3>:@,>:@',L>:@,>:@){},>:@',>:@V,6-C:,6-C:,ΠϠҠЛb-@9/6-A$@@%%D)|}6-?:A6<<<,Q(@^(@b W(@(@:(@>:@',>:@d,E@WAd)}}AUz
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#A !-A6A0%"+/ 33104,104,133,204,104,133,203,104,141,192,6,104//141,191,6)},104,141,194,6,104,141,193,6,169111,133,207,169,191,133,208,160,0,177,203,14100190,6,165,207,240,28,169,0,133,207,173)},193,,6,24,105,4,141,193,6,133,205,173,194,6++105,0,141,194,6,133,206,76,95,6,169,1..133,207,173,191,6,24,105,4,1)}41,191,6,133..205,173,192,6,105,0,141,192,6,133,206,32))137,6,32,146,6,32,160,6,32,137,6,32//169,6,32,160,6,165,2)}07,240,177,56,165,20322233,40,176,2,198,204,133,203,198,208,165,208--201,255,208,151,96,173,190,6,41,3,10,1000170)},96,160,0,189,195,6,145,205,232,200,192,,4,208,245,96,173,190,6,74,74,141,190,6++96,160,0,189,195,6,10,10,10,10,24,11)}3"44205,145,205,232,200,192,4,208,238,96,0,0,0,0,0,$U-@A32@@U2)}^
+AA1A@$C-@@^(ҠԠ7-@)}@7(CHECK PRINTER AND PRESS F:B2y,@A @
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-}@SCRENLOD.LSTREF COMPUTE OCT 85 P10MY VERSION... THERE -}ISA SAVE PROG TO GO WITH THIS....
SCREEN LOAD ROUTINE SETUP
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