let s1, s2 be complex-valued ManySortedSet of NAT ; :: thesis: ( s1 . 0 = s . 0 & ( for n being Nat holds s1 . (n + 1) = (s1 . n) + (s . (n + 1)) ) & s2 . 0 = s . 0 & ( for n being Nat holds s2 . (n + 1) = (s2 . n) + (s . (n + 1)) ) implies s1 = s2 )
assume that
A5: s1 . 0 = s . 0 and
A6: for n being Nat holds s1 . (n + 1) = (s1 . n) + (s . (n + 1)) and
A7: s2 . 0 = s . 0 and
A8: for n being Nat holds s2 . (n + 1) = (s2 . n) + (s . (n + 1)) ; :: thesis: s1 = s2
defpred S₁[ Nat] means s1 . $1 = s2 . $1;
A9: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume s1 . k = s2 . k ; :: thesis: S₁[k + 1]
hence s1 . (k + 1) = (s2 . k) + (s . (k + 1)) by A6
.= s2 . (k + 1) by A8 ;
:: thesis: verum

end;

A10: S₁[ 0 ] by A5, A7;
for n being Nat holds S₁[n] from NAT_1:sch 2(A10, A9);
hence s1 = s2 ; :: thesis: verum