let MS be non empty satisfying_Real satisfying_equiv satisfying_interval satisfying_commutativity satisfying_Nat satisfying_harmonic_closed satisfying_fifth_constructible classical_fifth satisfying_octave_constructible classical_octave satisfying_octave_descendent_constructible MusicStruct ; :: thesis: for fondamentale being Element of MS holds (spiral_of_fifths (MS,fondamentale,fondamentale)) . 4 = (81 / 64) * (@ fondamentale)
let fondamentale be Element of MS; :: thesis: (spiral_of_fifths (MS,fondamentale,fondamentale)) . 4 = (81 / 64) * (@ fondamentale)
reconsider n4 = 4, n3 = 3, n2 = 2, n1 = 1, n0 = 0 as Nat ;
(spiral_of_fifths (MS,fondamentale,fondamentale)) . n3 is Element of MS ;
then reconsider r32 = (27 / 16) * (@ fondamentale) as Element of MS by Th60;
A1: (spiral_of_fifths (MS,fondamentale,fondamentale)) . 4 = (spiral_of_fifths (MS,fondamentale,fondamentale)) . (n3 + 1)
.= Fifth_reduct (MS,fondamentale,((spiral_of_fifths (MS,fondamentale,fondamentale)) . n3)) by Def19
.= Fifth_reduct (MS,fondamentale,r32) by Th60 ;
consider r, s being positive Real such that
A2: ( r = r32 & s = (3 / 2) * r & Fifth (MS,r32) = s ) by Th54;
A3: Fifth (MS,r32) = (81 / 32) * (@ fondamentale) by A2;
A4: ex fr being positive Real st
( fondamentale = fr & Octave (MS,fondamentale) = 2 * fr ) by Def15;
A5: ex r being positive Real st
( Fifth (MS,r32) = r & Octave_descendent (MS,(Fifth (MS,r32))) = r / 2 ) by Th51;
not Fifth (MS,r32) is_Between fondamentale, Octave (MS,fondamentale) by A3, A4, XREAL_1:68;
hence (spiral_of_fifths (MS,fondamentale,fondamentale)) . 4 = (81 / 64) * (@ fondamentale) by A1, A2, A5, Def18; :: thesis: verum