let it1, it2 be kernel Function of L,L; :: thesis: ( ( for x being Element of L holds it1 . x = inf (Class (EqRel R),x) ) & ( for x being Element of L holds it2 . x = inf (Class (EqRel R),x) ) implies it1 = it2 )
assume that
A17: for x being Element of L holds it1 . x = inf (Class (EqRel R),x) and
A18: for x being Element of L holds it2 . x = inf (Class (EqRel R),x) ; :: thesis: it1 = it2

now

let x be set ; :: thesis: ( x in the carrier of L implies it1 . x = it2 . x )
assume x in the carrier of L ; :: thesis: it1 . x = it2 . x
then reconsider x' = x as Element of L ;
thus it1 . x = inf (Class (EqRel R),x') by A17
.= it2 . x by A18 ; :: thesis: verum

end;

hence it1 = it2 by FUNCT_2:18; :: thesis: verum