let f1, f2 be Function of REAL ,REAL ; :: thesis: ( ( for d being real number holds f1 . d = Sum (d rExpSeq ) ) & ( for d being real number holds f2 . d = Sum (d rExpSeq ) ) implies f1 = f2 )
assume A3: for d being real number holds f1 . d = Sum (d rExpSeq ) ; :: thesis: ( ex d being real number st not f2 . d = Sum (d rExpSeq ) or f1 = f2 )
assume A4: for d being real number holds f2 . d = Sum (d rExpSeq ) ; :: thesis: f1 = f2
for d being Element of REAL holds f1 . d = f2 . d

proof

let d be Element of REAL ; :: thesis: f1 . d = f2 . d
thus f1 . d = Sum (d rExpSeq ) by A3
.= f2 . d by A4 ; :: thesis: verum

end;

hence f1 = f2 by FUNCT_2:113; :: thesis: verum