let f1, f2 be Function of REAL ,REAL ; :: thesis: ( ( for d being Element of REAL holds f1 . d = Im (Sum ((d * <i> ) ExpSeq )) ) & ( for d being Element of REAL holds f2 . d = Im (Sum ((d * <i> ) ExpSeq )) ) implies f1 = f2 )
assume A2: for d being Element of REAL holds f1 . d = Im (Sum ((d * <i> ) ExpSeq )) ; :: thesis: ( ex d being Element of REAL st not f2 . d = Im (Sum ((d * <i> ) ExpSeq )) or f1 = f2 )
assume A3: for d being Element of REAL holds f2 . d = Im (Sum ((d * <i> ) ExpSeq )) ; :: thesis: f1 = f2
A4: 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 = Im (Sum ((d * <i> ) ExpSeq )) by A2
.= f2 . d by A3 ; :: thesis: verum

end;

thus f1 = f2 by A4, FUNCT_2:113; :: thesis: verum