let f1, f2 be Function of L,L; :: thesis: ( ( for x being Element of L holds f1 . x = "/\" ((uparrow x) /\ the carrier of S),L ) & ( for x being Element of L holds f2 . x = "/\" ((uparrow x) /\ the carrier of S),L ) implies f1 = f2 )
assume that
A1: for x being Element of L holds f1 . x = "/\" ((uparrow x) /\ the carrier of S),L and
A2: for x being Element of L holds f2 . x = "/\" ((uparrow x) /\ the carrier of S),L ; :: thesis: f1 = f2

now

let x be Element of L; :: thesis: f1 . x = f2 . x
thus f1 . x = "/\" ((uparrow x) /\ the carrier of S),L by A1
.= f2 . x by A2 ; :: thesis: verum

end;

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