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, frequency being Element of MS st frequency is_Between fondamentale, Octave (MS,fondamentale) holds
for n being Nat holds (spiral_of_fifths (MS,fondamentale,frequency)) . n is_Between fondamentale, Octave (MS,fondamentale)
let fondamentale, frequency be Element of MS; :: thesis: ( frequency is_Between fondamentale, Octave (MS,fondamentale) implies for n being Nat holds (spiral_of_fifths (MS,fondamentale,frequency)) . n is_Between fondamentale, Octave (MS,fondamentale) )
assume A1: frequency is_Between fondamentale, Octave (MS,fondamentale) ; :: thesis: for n being Nat holds (spiral_of_fifths (MS,fondamentale,frequency)) . n is_Between fondamentale, Octave (MS,fondamentale)
let n be Nat; :: thesis: (spiral_of_fifths (MS,fondamentale,frequency)) . n is_Between fondamentale, Octave (MS,fondamentale)
defpred S₁[ Nat] means (spiral_of_fifths (MS,fondamentale,frequency)) . $1 is_Between fondamentale, Octave (MS,fondamentale);
A2: S₁[ 0 ] by Def19, A1;
A3: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A4: S₁[k] ; :: thesis: S₁[k + 1]
(spiral_of_fifths (MS,fondamentale,frequency)) . (k + 1) = Fifth_reduct (MS,fondamentale,((spiral_of_fifths (MS,fondamentale,frequency)) . k)) by Def19;
hence S₁[k + 1] by A4, Th56; :: thesis: verum

end;

for k being Nat holds S₁[k] from NAT_1:sch 2(A2, A3);
hence (spiral_of_fifths (MS,fondamentale,frequency)) . n is_Between fondamentale, Octave (MS,fondamentale) ; :: thesis: verum