let f1, f2 be Function of REAL ,REAL ; :: thesis: ( ( for d being Real holds f1 . d = Re (Sum ((d * <i> ) ExpSeq )) ) & ( for d being Real holds f2 . d = Re (Sum ((d * <i> ) ExpSeq )) ) implies f1 = f2 )
assume A6: for d being Real holds f1 . d = Re (Sum ((d * <i> ) ExpSeq )) ; :: thesis: ( ex d being Real st not f2 . d = Re (Sum ((d * <i> ) ExpSeq )) or f1 = f2 )
assume A7: for d being Real holds f2 . d = Re (Sum ((d * <i> ) ExpSeq )) ; :: thesis: f1 = f2
for d being Real holds f1 . d = f2 . d

proof

let d be Real; :: thesis: f1 . d = f2 . d
thus f1 . d = Re (Sum ((d * <i> ) ExpSeq )) by A6
.= f2 . d by A7 ; :: thesis: verum

end;

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