let f1, f2 be Function of REAL,REAL; :: thesis: ( ( for x being Real holds f1 . x = x #Z q ) & ( for x being Real holds f2 . x = x #Z q ) implies f1 = f2 )

assume that

A2: for d being Real holds f1 . d = d #Z q and

A3: for d being Real holds f2 . d = d #Z q ; :: thesis: f1 = f2

for d being Element of REAL holds f1 . d = f2 . d

assume that

A2: for d being Real holds f1 . d = d #Z q and

A3: for d being Real holds f2 . d = d #Z q ; :: thesis: f1 = f2

for d being Element of REAL holds f1 . d = f2 . d

proof

hence
f1 = f2
; :: thesis: verum
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;thus f1 . d = d #Z q by A2

.= f2 . d by A3 ; :: thesis: verum