let f1, f2 be Function of REAL,REAL; :: thesis: ( ( for x being real number holds f1 . x = x #Z q ) & ( for x being real number holds f2 . x = x #Z q ) implies f1 = f2 )
assume that
A2: for d being real number holds f1 . d = d #Z q and
A3: for d being real number holds f2 . d = d #Z q ; :: 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 = d #Z q by A2
.= f2 . d by A3 ; :: thesis: verum

end;

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