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 ;

dom f = the carrier of (InclPoset (Ids S)) by FUNCT_2:def 1;

hence f = g by A7, A6, FUNCT_1:2; :: thesis: verum

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

A7:
dom g = the carrier of (InclPoset (Ids S))
by FUNCT_2:def 1;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;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

dom f = the carrier of (InclPoset (Ids S)) by FUNCT_2:def 1;

hence f = g by A7, A6, FUNCT_1:2; :: thesis: verum