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
  A /\ eq_dom(f,a) = A /\ great_eq_dom(f,a) /\ less_eq_dom(f,a)
proof
 for x being object st x in A /\ eq_dom(f,a) holds
  x in A /\ great_eq_dom(f,a) /\ less_eq_dom(f,a)
  proof
    let x be object;
    assume
A1: x in A /\ eq_dom(f,a);
then A2: x in A by XBOOLE_0:def 4;
A3: x in eq_dom(f,a) by A1,XBOOLE_0:def 4;
then A4: a = f.x by Def15;
    reconsider x as Element of X by A1;
A5: x in dom f by A3,Def15;
then  x in great_eq_dom(f,a) by A4,Def14;
then A6: x in A /\ great_eq_dom(f,a) by A2,XBOOLE_0:def 4;
 x in less_eq_dom(f,a) by A4,A5,Def12;
    hence thesis by A6,XBOOLE_0:def 4;
  end;
then A7: A /\ eq_dom(f,a) c= A /\ great_eq_dom(f,a) /\ less_eq_dom(f,a);
   for x being object st x in A /\ great_eq_dom(f,a) /\ less_eq_dom(f,a)
holds
  x in A /\ eq_dom(f,a)
  proof
    let x being object;
    assume
A8: x in A /\ great_eq_dom(f,a) /\ less_eq_dom(f,a);
then A9: x in A /\ great_eq_dom(f,a) by XBOOLE_0:def 4;
A10: x in less_eq_dom(f,a) by A8,XBOOLE_0:def 4;
A11: x in A by A9,XBOOLE_0:def 4;
 x in great_eq_dom(f,a) by A9,XBOOLE_0:def 4;
then A12: a <= f.x by Def14;
A13: f.x <= a by A10,Def12;
    reconsider x as Element of X by A8;
A14: x in dom f by A10,Def12;
 a = f.x by A12,A13,XXREAL_0:1;
then  x in eq_dom(f,a) by A14,Def15;
    hence thesis by A11,XBOOLE_0:def 4;
  end;
then
 A /\ great_eq_dom(f,a) /\ less_eq_dom(f,a) c= A /\ eq_dom(f,a);
  hence thesis by A7,XBOOLE_0:def 10;
end;
