let X be set ; :: thesis: for f being PartFunc of X,ExtREAL
for A being set
for a being R_eal st A c= dom f holds
A /\ (great_eq_dom f,a) = A \ (A /\ (less_dom f,a))
let f be PartFunc of X,ExtREAL ; :: thesis: for A being set
for a being R_eal st A c= dom f holds
A /\ (great_eq_dom f,a) = A \ (A /\ (less_dom f,a))
let A be set ; :: thesis: for a being R_eal st A c= dom f holds
A /\ (great_eq_dom f,a) = A \ (A /\ (less_dom f,a))
let a be R_eal; :: thesis: ( A c= dom f implies A /\ (great_eq_dom f,a) = A \ (A /\ (less_dom f,a)) )
assume A1: A c= dom f ; :: thesis: A /\ (great_eq_dom f,a) = A \ (A /\ (less_dom f,a))
dom f c= X by RELAT_1:def 18;
then A2: A c= X by A1, XBOOLE_1:1;
A3: A /\ (great_eq_dom f,a) c= A \ (A /\ (less_dom f,a))

proof

for x being set st x in A /\ (great_eq_dom f,a) holds
x in A \ (A /\ (less_dom f,a))

proof

let x be set ; :: thesis: ( x in A /\ (great_eq_dom f,a) implies x in A \ (A /\ (less_dom f,a)) )
assume x in A /\ (great_eq_dom f,a) ; :: thesis: x in A \ (A /\ (less_dom f,a))
then A4: ( x in A & x in great_eq_dom f,a ) by XBOOLE_0:def 4;
then a <= f . x by Def15;
then not x in less_dom f,a by Def12;
then not x in A /\ (less_dom f,a) by XBOOLE_0:def 4;
hence x in A \ (A /\ (less_dom f,a)) by A4, XBOOLE_0:def 5; :: thesis: verum

end;

hence A /\ (great_eq_dom f,a) c= A \ (A /\ (less_dom f,a)) by TARSKI:def 3; :: thesis: verum

end;

A \ (A /\ (less_dom f,a)) c= A /\ (great_eq_dom f,a)

proof

for x being set st x in A \ (A /\ (less_dom f,a)) holds
x in A /\ (great_eq_dom f,a)

proof

let x be set ; :: thesis: ( x in A \ (A /\ (less_dom f,a)) implies x in A /\ (great_eq_dom f,a) )
assume A5: x in A \ (A /\ (less_dom f,a)) ; :: thesis: x in A /\ (great_eq_dom f,a)
then A6: ( x in A & not x in A /\ (less_dom f,a) ) by XBOOLE_0:def 5;
then A7: not x in less_dom f,a by XBOOLE_0:def 4;
reconsider x = x as Element of X by A2, A6;
reconsider y = f . x as R_eal ;
not y < a by A1, A6, A7, Def12;
then x in great_eq_dom f,a by A1, A6, Def15;
hence x in A /\ (great_eq_dom f,a) by A5, XBOOLE_0:def 4; :: thesis: verum

end;

hence A \ (A /\ (less_dom f,a)) c= A /\ (great_eq_dom f,a) by TARSKI:def 3; :: thesis: verum

end;

hence A /\ (great_eq_dom f,a) = A \ (A /\ (less_dom f,a)) by A3, XBOOLE_0:def 10; :: thesis: verum