let f1, f2 be Function of L,L; :: thesis: ( ( for x being Element of L holds f1 . x = eval p,x ) & ( for x being Element of L holds f2 . x = eval p,x ) implies f1 = f2 )
assume that
A2: for x being Element of L holds f1 . x = eval p,x and
A3: for x being Element of L holds f2 . x = eval p,x ; :: thesis: f1 = f2

now

let x be Element of L; :: thesis: f1 . x = f2 . x
thus f1 . x = eval p,x by A2
.= f2 . x by A3 ; :: thesis: verum

end;

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