let f1, f2 be Function of the carrier of X, the carrier of X; :: thesis: ( ( for z being Element of X holds f1 . z = Sum (z ExpSeq) ) & ( for z being Element of X holds f2 . z = Sum (z ExpSeq) ) implies f1 = f2 )
assume that
A1: for z being Element of X holds f1 . z = Sum (z ExpSeq) and
A2: for z being Element of X holds f2 . z = Sum (z ExpSeq) ; :: thesis: f1 = f2
for z being Element of X holds f1 . z = f2 . z

proof

let z be Element of X; :: thesis: f1 . z = f2 . z
thus f1 . z = Sum (z ExpSeq) by A1
.= f2 . z by A2 ; :: thesis: verum

end;

hence f1 = f2 by FUNCT_2:63; :: thesis: verum