let seq1, seq2 be sequence of MS; :: thesis: ( seq1 . 0 = frequency & ( for n being Nat holds seq1 . (n + 1) = Fifth_reduct (MS,fondamentale,(seq1 . n)) ) & seq2 . 0 = frequency & ( for n being Nat holds seq2 . (n + 1) = Fifth_reduct (MS,fondamentale,(seq2 . n)) ) implies seq1 = seq2 )
assume that
A1: ( seq1 . 0 = frequency & ( for n being Nat holds seq1 . (n + 1) = Fifth_reduct (MS,fondamentale,(seq1 . n)) ) ) and
A2: ( seq2 . 0 = frequency & ( for n being Nat holds seq2 . (n + 1) = Fifth_reduct (MS,fondamentale,(seq2 . n)) ) ) ; :: thesis: seq1 = seq2
defpred S₁[ Nat] means seq1 . $1 = seq2 . $1;
A3: S₁[ 0 ] by A1, A2;
A4: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A5: S₁[k] ; :: thesis: S₁[k + 1]
seq1 . (k + 1) = Fifth_reduct (MS,fondamentale,(seq1 . k)) by A1
.= seq2 . (k + 1) by A2, A5 ;
hence S₁[k + 1] ; :: thesis: verum

end;

A6: for k being Nat holds S₁[k] from NAT_1:sch 2(A3, A4);

now :: thesis: ( dom seq1 = dom seq2 & ( for x being object st x in dom seq1 holds
seq1 . x = seq2 . x ) )

dom seq1 = NAT by PARTFUN1:def 2;
hence dom seq1 = dom seq2 by PARTFUN1:def 2; :: thesis: for x being object st x in dom seq1 holds
seq1 . x = seq2 . x
thus for x being object st x in dom seq1 holds
seq1 . x = seq2 . x by A6; :: thesis: verum

end;

hence seq1 = seq2 by FUNCT_1:def 11; :: thesis: verum