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 :: thesis: for x being Element of L holds f1 . x = f2 . x

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_2:63; :: thesis: verum