set P = InclPoset (Ids S);
let f, g be Function of (InclPoset (Ids S)),(InclPoset (Ids L)); :: thesis: ( ( for I being Ideal of S ex J being Subset of L st
( I = J & f . I = downarrow J ) ) & ( for I being Ideal of S ex J being Subset of L st
( I = J & g . I = downarrow J ) ) implies f = g )
assume that
A6: for I being Ideal of S ex J being Subset of L st
( I = J & f . I = downarrow J ) and
A7: for I being Ideal of S ex J being Subset of L st
( I = J & g . I = downarrow J ) ; :: thesis: f = g
A8: the carrier of (InclPoset (Ids S)) = the carrier of RelStr(# (Ids S),(RelIncl (Ids S)) #) by YELLOW_1:def 1
.= Ids S ;

A9: 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 A8, WAYBEL_0:def 23;
then ex I being Ideal of S st x = I ;
then reconsider I = x as Ideal of S ;
A10: ex J2 being Subset of L st
( I = J2 & g . I = downarrow J2 ) by A7;
ex J1 being Subset of L st
( I = J1 & f . I = downarrow J1 ) by A6;
hence f . x = g . x by A10; :: thesis: verum

end;

A11: 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 A11, A9, FUNCT_1:2; :: thesis: verum