let f, g be Function; :: thesis: ( dom f = X & ( for x being object st x in X holds
f . x = Im (W,x) ) & dom g = X & ( for x being object st x in X holds
g . x = Im (W,x) ) implies f = g )
assume that
A1: dom f = X and
A2: for x being object st x in X holds
f . x = Im (W,x) and
A3: dom g = X and
A4: for x being object st x in X holds
g . x = Im (W,x) ; :: thesis: f = g

now :: thesis: for x being object st x in X holds
f . x = g . x

let x be object ; :: thesis: ( x in X implies f . x = g . x )
assume A5: x in X ; :: thesis: f . x = g . x
hence f . x = Im (W,x) by A2
.= g . x by A4, A5 ;
:: thesis: verum

end;

hence f = g by A1, A3, FUNCT_1:2; :: thesis: verum