let seq1, seq2 be sequence of X; :: thesis: ( seq1 . 0 = seq . 0 & ( for n being Nat holds seq1 . (n + 1) = (seq1 . n) + (seq . (n + 1)) ) & seq2 . 0 = seq . 0 & ( for n being Nat holds seq2 . (n + 1) = (seq2 . n) + (seq . (n + 1)) ) implies seq1 = seq2 )
assume that
A2: seq1 . 0 = seq . 0 and
A3: for n being Nat holds seq1 . (n + 1) = (seq1 . n) + (seq . (n + 1)) and
A4: seq2 . 0 = seq . 0 and
A5: for n being Nat holds seq2 . (n + 1) = (seq2 . n) + (seq . (n + 1)) ; :: thesis: seq1 = seq2
defpred S₁[ Nat] means seq1 . $1 = seq2 . $1;
A6: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume seq1 . k = seq2 . k ; :: thesis: S₁[k + 1]
hence seq1 . (k + 1) = (seq2 . k) + (seq . (k + 1)) by A3
.= seq2 . (k + 1) by A5 ;
:: thesis: verum

end;

A7: S₁[ 0 ] by A2, A4;
for n being Nat holds S₁[n] from NAT_1:sch 2(A7, A6);
then for n being Element of NAT holds S₁[n] ;
hence seq1 = seq2 ; :: thesis: verum