let f1, f2 be Function of the carrier of R,(bool the carrier of R); :: thesis: ( ( for x being Element of R holds f1 . x = Coim ( the InternalRel of R,x) ) & ( for x being Element of R holds f2 . x = Coim ( the InternalRel of R,x) ) implies f1 = f2 )
assume that
A1: for x being Element of R holds f1 . x = Coim ( the InternalRel of R,x) and
A2: for x being Element of R holds f2 . x = Coim ( the InternalRel of R,x) ; :: thesis: f1 = f2
for x being Element of R holds f1 . x = f2 . x

proof

let x be Element of R; :: thesis: f1 . x = f2 . x
f1 . x = Coim ( the InternalRel of R,x) by A1
.= f2 . x by A2 ;
hence f1 . x = f2 . x ; :: thesis: verum

end;

hence f1 = f2 ; :: thesis: verum