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_dom f,a) = A \ (A /\ (less_eq_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_dom f,a) = A \ (A /\ (less_eq_dom f,a))
let A be set ; :: thesis: for a being R_eal st A c= dom f holds
A /\ (great_dom f,a) = A \ (A /\ (less_eq_dom f,a))
let a be R_eal; :: thesis: ( A c= dom f implies A /\ (great_dom f,a) = A \ (A /\ (less_eq_dom f,a)) )
assume A1: A c= dom f ; :: thesis: A /\ (great_dom f,a) = A \ (A /\ (less_eq_dom f,a))
dom f c= X by RELAT_1:def 18;
then A3: A c= X by A1, XBOOLE_1:1;
for x being set st x in A /\ (great_dom f,a) holds
x in A \ (A /\ (less_eq_dom f,a))

proof

let x be set ; :: thesis: ( x in A /\ (great_dom f,a) implies x in A \ (A /\ (less_eq_dom f,a)) )
assume A5: x in A /\ (great_dom f,a) ; :: thesis: x in A \ (A /\ (less_eq_dom f,a))
then A6: x in A by XBOOLE_0:def 4;
x in great_dom f,a by A5, XBOOLE_0:def 4;
then a < f . x by Def14;
then not x in less_eq_dom f,a by Def13;
then not x in A /\ (less_eq_dom f,a) by XBOOLE_0:def 4;
hence x in A \ (A /\ (less_eq_dom f,a)) by A6, XBOOLE_0:def 5; :: thesis: verum

end;

then A11: A /\ (great_dom f,a) c= A \ (A /\ (less_eq_dom f,a)) by TARSKI:def 3;
for x being set st x in A \ (A /\ (less_eq_dom f,a)) holds
x in A /\ (great_dom f,a)

proof

let x be set ; :: thesis: ( x in A \ (A /\ (less_eq_dom f,a)) implies x in A /\ (great_dom f,a) )
assume A13: x in A \ (A /\ (less_eq_dom f,a)) ; :: thesis: x in A /\ (great_dom f,a)
then A14: x in A ;
not x in A /\ (less_eq_dom f,a) by A13, XBOOLE_0:def 5;
then A16: not x in less_eq_dom f,a by A13, XBOOLE_0:def 4;
reconsider x = x as Element of X by A3, A14;
reconsider y = f . x as R_eal ;
not y <= a by A1, A14, A16, Def13;
then x in great_dom f,a by A1, A14, Def14;
hence x in A /\ (great_dom f,a) by A13, XBOOLE_0:def 4; :: thesis: verum

end;

then A \ (A /\ (less_eq_dom f,a)) c= A /\ (great_dom f,a) by TARSKI:def 3;
hence A /\ (great_dom f,a) = A \ (A /\ (less_eq_dom f,a)) by A11, XBOOLE_0:def 10; :: thesis: verum