let HPS be Heptatonic_Pythagorean_Score; :: thesis: for frequency being Element of HPS holds heptatonic_pythagorean_scale (HPS,frequency) is heptatonic
let frequency be Element of HPS; :: thesis: heptatonic_pythagorean_scale (HPS,frequency) is heptatonic
set MS = HPS;
set gamme = heptatonic_pythagorean_scale (HPS,frequency);

now :: thesis: ( ex frequency being Element of HPS ex r1, r2, r3, r4, r5, r6, r7, r8 being positive Real st
( (heptatonic_pythagorean_scale (HPS,frequency)) . 1 = r1 & (heptatonic_pythagorean_scale (HPS,frequency)) . 2 = r2 & (heptatonic_pythagorean_scale (HPS,frequency)) . 3 = r3 & (heptatonic_pythagorean_scale (HPS,frequency)) . 4 = r4 & (heptatonic_pythagorean_scale (HPS,frequency)) . 5 = r5 & (heptatonic_pythagorean_scale (HPS,frequency)) . 6 = r6 & (heptatonic_pythagorean_scale (HPS,frequency)) . 7 = r7 & (heptatonic_pythagorean_scale (HPS,frequency)) . 8 = r8 & r1 < r2 & r2 < r3 & r3 < r4 & r4 < r5 & r5 < r6 & r6 < r7 & r7 < r8 ) & (heptatonic_pythagorean_scale (HPS,frequency)) . 8 = Octave (HPS,frequency) & (heptatonic_pythagorean_scale (HPS,frequency)) . 1 = frequency )

now :: thesis: ex r1, r2, r3, r4, r5, r6, r7, r8 being positive Real st
( (heptatonic_pythagorean_scale (HPS,frequency)) . 1 = frequency & (heptatonic_pythagorean_scale (HPS,frequency)) . 1 = r1 & (heptatonic_pythagorean_scale (HPS,frequency)) . 2 = r2 & (heptatonic_pythagorean_scale (HPS,frequency)) . 3 = r3 & (heptatonic_pythagorean_scale (HPS,frequency)) . 4 = r4 & (heptatonic_pythagorean_scale (HPS,frequency)) . 5 = r5 & (heptatonic_pythagorean_scale (HPS,frequency)) . 6 = r6 & (heptatonic_pythagorean_scale (HPS,frequency)) . 7 = r7 & (heptatonic_pythagorean_scale (HPS,frequency)) . 8 = r8 & r1 < r2 & r2 < r3 & r3 < r4 & r4 < r5 & r5 < r6 & r6 < r7 & r7 < r8 )

reconsider r1 = 1 * (@ frequency), r2 = (9 / 8) * (@ frequency), r3 = (81 / 64) * (@ frequency), r4 = (4 / 3) * (@ frequency), r5 = (3 / 2) * (@ frequency), r6 = (27 / 16) * (@ frequency), r7 = (243 / 128) * (@ frequency), r8 = 2 * (@ frequency) as positive Real ;
take r1 = r1; :: thesis: ex r2, r3, r4, r5, r6, r7, r8 being positive Real st
( (heptatonic_pythagorean_scale (HPS,frequency)) . 1 = frequency & (heptatonic_pythagorean_scale (HPS,frequency)) . 1 = r1 & (heptatonic_pythagorean_scale (HPS,frequency)) . 2 = r2 & (heptatonic_pythagorean_scale (HPS,frequency)) . 3 = r3 & (heptatonic_pythagorean_scale (HPS,frequency)) . 4 = r4 & (heptatonic_pythagorean_scale (HPS,frequency)) . 5 = r5 & (heptatonic_pythagorean_scale (HPS,frequency)) . 6 = r6 & (heptatonic_pythagorean_scale (HPS,frequency)) . 7 = r7 & (heptatonic_pythagorean_scale (HPS,frequency)) . 8 = r8 & r1 < r2 & r2 < r3 & r3 < r4 & r4 < r5 & r5 < r6 & r6 < r7 & r7 < r8 )
take r2 = r2; :: thesis: ex r3, r4, r5, r6, r7, r8 being positive Real st
( (heptatonic_pythagorean_scale (HPS,frequency)) . 1 = frequency & (heptatonic_pythagorean_scale (HPS,frequency)) . 1 = r1 & (heptatonic_pythagorean_scale (HPS,frequency)) . 2 = r2 & (heptatonic_pythagorean_scale (HPS,frequency)) . 3 = r3 & (heptatonic_pythagorean_scale (HPS,frequency)) . 4 = r4 & (heptatonic_pythagorean_scale (HPS,frequency)) . 5 = r5 & (heptatonic_pythagorean_scale (HPS,frequency)) . 6 = r6 & (heptatonic_pythagorean_scale (HPS,frequency)) . 7 = r7 & (heptatonic_pythagorean_scale (HPS,frequency)) . 8 = r8 & r1 < r2 & r2 < r3 & r3 < r4 & r4 < r5 & r5 < r6 & r6 < r7 & r7 < r8 )
take r3 = r3; :: thesis: ex r4, r5, r6, r7, r8 being positive Real st
( (heptatonic_pythagorean_scale (HPS,frequency)) . 1 = frequency & (heptatonic_pythagorean_scale (HPS,frequency)) . 1 = r1 & (heptatonic_pythagorean_scale (HPS,frequency)) . 2 = r2 & (heptatonic_pythagorean_scale (HPS,frequency)) . 3 = r3 & (heptatonic_pythagorean_scale (HPS,frequency)) . 4 = r4 & (heptatonic_pythagorean_scale (HPS,frequency)) . 5 = r5 & (heptatonic_pythagorean_scale (HPS,frequency)) . 6 = r6 & (heptatonic_pythagorean_scale (HPS,frequency)) . 7 = r7 & (heptatonic_pythagorean_scale (HPS,frequency)) . 8 = r8 & r1 < r2 & r2 < r3 & r3 < r4 & r4 < r5 & r5 < r6 & r6 < r7 & r7 < r8 )
take r4 = r4; :: thesis: ex r5, r6, r7, r8 being positive Real st
( (heptatonic_pythagorean_scale (HPS,frequency)) . 1 = frequency & (heptatonic_pythagorean_scale (HPS,frequency)) . 1 = r1 & (heptatonic_pythagorean_scale (HPS,frequency)) . 2 = r2 & (heptatonic_pythagorean_scale (HPS,frequency)) . 3 = r3 & (heptatonic_pythagorean_scale (HPS,frequency)) . 4 = r4 & (heptatonic_pythagorean_scale (HPS,frequency)) . 5 = r5 & (heptatonic_pythagorean_scale (HPS,frequency)) . 6 = r6 & (heptatonic_pythagorean_scale (HPS,frequency)) . 7 = r7 & (heptatonic_pythagorean_scale (HPS,frequency)) . 8 = r8 & r1 < r2 & r2 < r3 & r3 < r4 & r4 < r5 & r5 < r6 & r6 < r7 & r7 < r8 )
take r5 = r5; :: thesis: ex r6, r7, r8 being positive Real st
( (heptatonic_pythagorean_scale (HPS,frequency)) . 1 = frequency & (heptatonic_pythagorean_scale (HPS,frequency)) . 1 = r1 & (heptatonic_pythagorean_scale (HPS,frequency)) . 2 = r2 & (heptatonic_pythagorean_scale (HPS,frequency)) . 3 = r3 & (heptatonic_pythagorean_scale (HPS,frequency)) . 4 = r4 & (heptatonic_pythagorean_scale (HPS,frequency)) . 5 = r5 & (heptatonic_pythagorean_scale (HPS,frequency)) . 6 = r6 & (heptatonic_pythagorean_scale (HPS,frequency)) . 7 = r7 & (heptatonic_pythagorean_scale (HPS,frequency)) . 8 = r8 & r1 < r2 & r2 < r3 & r3 < r4 & r4 < r5 & r5 < r6 & r6 < r7 & r7 < r8 )
take r6 = r6; :: thesis: ex r7, r8 being positive Real st
( (heptatonic_pythagorean_scale (HPS,frequency)) . 1 = frequency & (heptatonic_pythagorean_scale (HPS,frequency)) . 1 = r1 & (heptatonic_pythagorean_scale (HPS,frequency)) . 2 = r2 & (heptatonic_pythagorean_scale (HPS,frequency)) . 3 = r3 & (heptatonic_pythagorean_scale (HPS,frequency)) . 4 = r4 & (heptatonic_pythagorean_scale (HPS,frequency)) . 5 = r5 & (heptatonic_pythagorean_scale (HPS,frequency)) . 6 = r6 & (heptatonic_pythagorean_scale (HPS,frequency)) . 7 = r7 & (heptatonic_pythagorean_scale (HPS,frequency)) . 8 = r8 & r1 < r2 & r2 < r3 & r3 < r4 & r4 < r5 & r5 < r6 & r6 < r7 & r7 < r8 )
take r7 = r7; :: thesis: ex r8 being positive Real st
( (heptatonic_pythagorean_scale (HPS,frequency)) . 1 = frequency & (heptatonic_pythagorean_scale (HPS,frequency)) . 1 = r1 & (heptatonic_pythagorean_scale (HPS,frequency)) . 2 = r2 & (heptatonic_pythagorean_scale (HPS,frequency)) . 3 = r3 & (heptatonic_pythagorean_scale (HPS,frequency)) . 4 = r4 & (heptatonic_pythagorean_scale (HPS,frequency)) . 5 = r5 & (heptatonic_pythagorean_scale (HPS,frequency)) . 6 = r6 & (heptatonic_pythagorean_scale (HPS,frequency)) . 7 = r7 & (heptatonic_pythagorean_scale (HPS,frequency)) . 8 = r8 & r1 < r2 & r2 < r3 & r3 < r4 & r4 < r5 & r5 < r6 & r6 < r7 & r7 < r8 )
take r8 = r8; :: thesis: ( (heptatonic_pythagorean_scale (HPS,frequency)) . 1 = frequency & (heptatonic_pythagorean_scale (HPS,frequency)) . 1 = r1 & (heptatonic_pythagorean_scale (HPS,frequency)) . 2 = r2 & (heptatonic_pythagorean_scale (HPS,frequency)) . 3 = r3 & (heptatonic_pythagorean_scale (HPS,frequency)) . 4 = r4 & (heptatonic_pythagorean_scale (HPS,frequency)) . 5 = r5 & (heptatonic_pythagorean_scale (HPS,frequency)) . 6 = r6 & (heptatonic_pythagorean_scale (HPS,frequency)) . 7 = r7 & (heptatonic_pythagorean_scale (HPS,frequency)) . 8 = r8 & r1 < r2 & r2 < r3 & r3 < r4 & r4 < r5 & r5 < r6 & r6 < r7 & r7 < r8 )
( (heptatonic_pythagorean_scale (HPS,frequency)) . 1 = 1 * (@ frequency) & (heptatonic_pythagorean_scale (HPS,frequency)) . 2 = (9 / 8) * (@ frequency) & (heptatonic_pythagorean_scale (HPS,frequency)) . 3 = (81 / 64) * (@ frequency) & (heptatonic_pythagorean_scale (HPS,frequency)) . 4 = (4 / 3) * (@ frequency) & (heptatonic_pythagorean_scale (HPS,frequency)) . 5 = (3 / 2) * (@ frequency) & (heptatonic_pythagorean_scale (HPS,frequency)) . 6 = (27 / 16) * (@ frequency) & (heptatonic_pythagorean_scale (HPS,frequency)) . 7 = (243 / 128) * (@ frequency) & (heptatonic_pythagorean_scale (HPS,frequency)) . 8 = 2 * (@ frequency) ) by Th88;
hence ( (heptatonic_pythagorean_scale (HPS,frequency)) . 1 = frequency & (heptatonic_pythagorean_scale (HPS,frequency)) . 1 = r1 & (heptatonic_pythagorean_scale (HPS,frequency)) . 2 = r2 & (heptatonic_pythagorean_scale (HPS,frequency)) . 3 = r3 & (heptatonic_pythagorean_scale (HPS,frequency)) . 4 = r4 & (heptatonic_pythagorean_scale (HPS,frequency)) . 5 = r5 & (heptatonic_pythagorean_scale (HPS,frequency)) . 6 = r6 & (heptatonic_pythagorean_scale (HPS,frequency)) . 7 = r7 & (heptatonic_pythagorean_scale (HPS,frequency)) . 8 = r8 ) ; :: thesis: ( r1 < r2 & r2 < r3 & r3 < r4 & r4 < r5 & r5 < r6 & r6 < r7 & r7 < r8 )
thus ( r1 < r2 & r2 < r3 & r3 < r4 & r4 < r5 & r5 < r6 & r6 < r7 & r7 < r8 ) by XREAL_1:98; :: thesis: verum

end;

hence ex frequency being Element of HPS ex r1, r2, r3, r4, r5, r6, r7, r8 being positive Real st
( (heptatonic_pythagorean_scale (HPS,frequency)) . 1 = r1 & (heptatonic_pythagorean_scale (HPS,frequency)) . 2 = r2 & (heptatonic_pythagorean_scale (HPS,frequency)) . 3 = r3 & (heptatonic_pythagorean_scale (HPS,frequency)) . 4 = r4 & (heptatonic_pythagorean_scale (HPS,frequency)) . 5 = r5 & (heptatonic_pythagorean_scale (HPS,frequency)) . 6 = r6 & (heptatonic_pythagorean_scale (HPS,frequency)) . 7 = r7 & (heptatonic_pythagorean_scale (HPS,frequency)) . 8 = r8 & r1 < r2 & r2 < r3 & r3 < r4 & r4 < r5 & r5 < r6 & r6 < r7 & r7 < r8 ) ; :: thesis: ( (heptatonic_pythagorean_scale (HPS,frequency)) . 8 = Octave (HPS,frequency) & (heptatonic_pythagorean_scale (HPS,frequency)) . 1 = frequency )
A1: ex fr being positive Real st
( frequency = fr & Octave (HPS,frequency) = 2 * fr ) by Def15;
( (heptatonic_pythagorean_scale (HPS,frequency)) . 1 = 1 * (@ frequency) & (heptatonic_pythagorean_scale (HPS,frequency)) . 2 = (9 / 8) * (@ frequency) & (heptatonic_pythagorean_scale (HPS,frequency)) . 3 = (81 / 64) * (@ frequency) & (heptatonic_pythagorean_scale (HPS,frequency)) . 4 = (4 / 3) * (@ frequency) & (heptatonic_pythagorean_scale (HPS,frequency)) . 5 = (3 / 2) * (@ frequency) & (heptatonic_pythagorean_scale (HPS,frequency)) . 6 = (27 / 16) * (@ frequency) & (heptatonic_pythagorean_scale (HPS,frequency)) . 7 = (243 / 128) * (@ frequency) & (heptatonic_pythagorean_scale (HPS,frequency)) . 8 = 2 * (@ frequency) ) by Th88;
hence ( (heptatonic_pythagorean_scale (HPS,frequency)) . 8 = Octave (HPS,frequency) & (heptatonic_pythagorean_scale (HPS,frequency)) . 1 = frequency ) by A1; :: thesis: verum

end;

hence heptatonic_pythagorean_scale (HPS,frequency) is heptatonic ; :: thesis: verum