let HPS be Heptatonic_Pythagorean_Score; :: thesis: for frequency being Element of HPS holds (spiral_of_fifths (HPS,frequency,(Fourth (HPS,frequency)))) . 2 = (3 / 2) * (@ frequency)
let frequency be Element of HPS; :: thesis: (spiral_of_fifths (HPS,frequency,(Fourth (HPS,frequency)))) . 2 = (3 / 2) * (@ frequency)
set MS = HPS;
set q = Fourth (HPS,frequency);
reconsider n1 = 1 as Nat ;
A1: (spiral_of_fifths (HPS,frequency,(Fourth (HPS,frequency)))) . 2 = (spiral_of_fifths (HPS,frequency,(Fourth (HPS,frequency)))) . (n1 + 1)
.= Fifth_reduct (HPS,frequency,((spiral_of_fifths (HPS,frequency,(Fourth (HPS,frequency)))) . n1)) by Def19
.= Fifth_reduct (HPS,frequency,frequency) by Th82 ;
consider r, s being positive Real such that
A2: ( r = frequency & s = (3 / 2) * r & Fifth (HPS,frequency) = s ) by Th54;
A3: ex fr being positive Real st
( frequency = fr & Octave (HPS,frequency) = 2 * fr ) by Def15;
( 1 * r <= (3 / 2) * r & (3 / 2) * r < 2 * r ) by XREAL_1:68;
then Fifth (HPS,frequency) is_Between frequency, Octave (HPS,frequency) by A2, A3;
hence (spiral_of_fifths (HPS,frequency,(Fourth (HPS,frequency)))) . 2 = (3 / 2) * (@ frequency) by A1, A2, Def18; :: thesis: verum