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 /\ (less_dom (f,a)) = A \ (A /\ (great_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 /\ (less_dom (f,a)) = A \ (A /\ (great_eq_dom (f,a)))
let A be set ; :: thesis: for a being R_eal st A c= dom f holds
A /\ (less_dom (f,a)) = A \ (A /\ (great_eq_dom (f,a)))
let a be R_eal; :: thesis: ( A c= dom f implies A /\ (less_dom (f,a)) = A \ (A /\ (great_eq_dom (f,a))) )
assume A1: A c= dom f ; :: thesis: A /\ (less_dom (f,a)) = A \ (A /\ (great_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 /\ (less_dom (f,a)) holds
x in A \ (A /\ (great_eq_dom (f,a)))

proof

let x be set ; :: thesis: ( x in A /\ (less_dom (f,a)) implies x in A \ (A /\ (great_eq_dom (f,a))) )
assume A5: x in A /\ (less_dom (f,a)) ; :: thesis: x in A \ (A /\ (great_eq_dom (f,a)))
then A6: x in A by XBOOLE_0:def 4;
x in less_dom (f,a) by A5, XBOOLE_0:def 4;
then f . x < a by Def12;
then not x in great_eq_dom (f,a) by Def15;
then not x in A /\ (great_eq_dom (f,a)) by XBOOLE_0:def 4;
hence x in A \ (A /\ (great_eq_dom (f,a))) by A6, XBOOLE_0:def 5; :: thesis: verum

end;

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

proof

let x be set ; :: thesis: ( x in A \ (A /\ (great_eq_dom (f,a))) implies x in A /\ (less_dom (f,a)) )
assume A13: x in A \ (A /\ (great_eq_dom (f,a))) ; :: thesis: x in A /\ (less_dom (f,a))
then A14: x in A ;
not x in A /\ (great_eq_dom (f,a)) by A13, XBOOLE_0:def 5;
then A16: not x in great_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 a <= y by A1, A14, A16, Def15;
then x in less_dom (f,a) by A1, A14, Def12;
hence x in A /\ (less_dom (f,a)) by A13, XBOOLE_0:def 4; :: thesis: verum

end;

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