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)) . 1 = (3 / 2) * (@ fondamentale)
let fondamentale be Element of MS; :: thesis: (spiral_of_fifths (MS,fondamentale,fondamentale)) . 1 = (3 / 2) * (@ fondamentale)
reconsider n1 = 1, n0 = 0 as Nat ;
A1: (spiral_of_fifths (MS,fondamentale,fondamentale)) . 1 = (spiral_of_fifths (MS,fondamentale,fondamentale)) . (n0 + 1)
.= Fifth_reduct (MS,fondamentale,((spiral_of_fifths (MS,fondamentale,fondamentale)) . n0)) by Def19
.= Fifth_reduct (MS,fondamentale,fondamentale) by Def19 ;
consider r, s being positive Real such that
A2: ( r = fondamentale & s = (3 / 2) * r & Fifth (MS,fondamentale) = s ) by Th54;
A3: ex fr being positive Real st
( fondamentale = fr & Octave (MS,fondamentale) = 2 * fr ) by Def15;
( 1 * r <= (3 / 2) * r & (3 / 2) * r < 2 * r ) by XREAL_1:68;
then Fifth (MS,fondamentale) is_Between fondamentale, Octave (MS,fondamentale) by A2, A3;
hence (spiral_of_fifths (MS,fondamentale,fondamentale)) . 1 = (3 / 2) * (@ fondamentale) by A1, A2, Def18; :: thesis: verum