set P = InclPoset (Ids S);
let f, g be Function of (InclPoset (Ids S)),L; :: thesis: ( ( for I being Ideal of S holds f . I = "\/" (I,L) ) & ( for I being Ideal of S holds g . I = "\/" (I,L) ) implies f = g )
assume that
A3: for I being Ideal of S holds f . I = "\/" (I,L) and
A4: for I being Ideal of S holds g . I = "\/" (I,L) ; :: thesis: f = g
A5: the carrier of (InclPoset (Ids S)) = the carrier of RelStr(# (Ids S),(RelIncl (Ids S)) #) by YELLOW_1:def 1
.= Ids S ;

A6: now :: thesis: for x being object st x in the carrier of (InclPoset (Ids S)) holds
f . x = g . x

let x be object ; :: thesis: ( x in the carrier of (InclPoset (Ids S)) implies f . x = g . x )
assume x in the carrier of (InclPoset (Ids S)) ; :: thesis: f . x = g . x
then x in { X where X is Ideal of S : verum } by A5, WAYBEL_0:def 23;
then ex I being Ideal of S st x = I ;
then reconsider I = x as Ideal of S ;
f . I = "\/" (I,L) by A3;
hence f . x = g . x by A4; :: thesis: verum

end;

A7: dom g = the carrier of (InclPoset (Ids S)) by FUNCT_2:def 1;
dom f = the carrier of (InclPoset (Ids S)) by FUNCT_2:def 1;
hence f = g by A7, A6, FUNCT_1:2; :: thesis: verum