reserve k for Element of NAT;
reserve r,r1 for Real;
reserve i for Integer;
reserve q for Rational;
reserve X for set;
reserve f for PartFunc of X,ExtREAL;
reserve S for SigmaField of X;
reserve F for sequence of S;
reserve A for set;
reserve a for ExtReal;
reserve r,s for Real;
reserve n,m for Element of NAT;

theorem Th15:
  A c= dom f implies A /\ great_dom(f,a) = A\(A /\ less_eq_dom(f,a))
proof
  assume
A1: A c= dom f;
 dom f c= X by RELAT_1:def 18;
then A2: A c= X by A1;
 for x being object st x in A /\ great_dom(f,a) holds x in
  A\(A /\ less_eq_dom(f,a))
  proof
    let x be object;
    assume
A3: x in A /\ great_dom(f,a);
then A4: x in A by XBOOLE_0:def 4;
 x in great_dom(f,a) by A3,XBOOLE_0:def 4;
then  a < f.x by Def13;
then  not x in less_eq_dom(f,a) by Def12;
then  not x in (A /\ less_eq_dom(f,a)) by XBOOLE_0:def 4;
    hence thesis by A4,XBOOLE_0:def 5;
  end;
then A5: A /\ great_dom(f,a) c= A\(A /\ less_eq_dom(f,a));
 for x being object st x in A\(A /\ less_eq_dom(f,a)) holds x in
  A /\ great_dom(f,a)
  proof
    let x be object;
    assume
A6: x in A\(A /\ less_eq_dom(f,a));
then A7: x in A;
 not x in A /\ less_eq_dom(f,a) by A6,XBOOLE_0:def 5;
then A8: not x in less_eq_dom(f,a) by A6,XBOOLE_0:def 4;
    reconsider x as Element of X by A2,A7;
    reconsider y=f.x as R_eal;
 not y <= a by A1,A7,A8,Def12;
then  x in great_dom(f,a) by A1,A7,Def13;
    hence thesis by A6,XBOOLE_0:def 4;
  end;
then  A\(A /\ less_eq_dom(f,a)) c= A /\ great_dom(f,a);
  hence thesis by A5,XBOOLE_0:def 10;
end;
