reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  f,g for PartFunc of X,COMPLEX,
  r for Real,
  c for Complex,
  E,A,B for Element of S;
reserve f for PartFunc of X,REAL,
  a for Real;

theorem
  A /\ eq_dom(f,a) = A /\ great_eq_dom(f,a) /\ less_eq_dom(f,a)
proof
  now
    let x be object;
    assume
A1: x in A /\ great_eq_dom(f,a) /\ less_eq_dom(f,a);
    then
A2: x in less_eq_dom(f,a) by XBOOLE_0:def 4;
    then
A3: x in dom f by MESFUNC6:4;
A4: x in A /\ great_eq_dom(f,a) by A1,XBOOLE_0:def 4;
    then x in great_eq_dom(f,a) by XBOOLE_0:def 4;
    then
A5: ex y1 be Real st y1 = f.x & a <= y1 by MESFUNC6:6;
    ex y2 be Real st y2 = f.x & y2 <= a by A2,MESFUNC6:4;
    then a = f.x by A5,XXREAL_0:1;
    then
A6: x in eq_dom(f,a) by A3,MESFUNC6:7;
    x in A by A4,XBOOLE_0:def 4;
    hence x in A /\ eq_dom(f,a) by A6,XBOOLE_0:def 4;
  end;
  then
A7: A /\ great_eq_dom(f,a) /\ less_eq_dom(f,a) c= A /\ eq_dom(f,a) by
TARSKI:def 3;
  now
    let x be object;
    assume
A8: x in A /\ eq_dom(f,a);
    then
A9: x in A by XBOOLE_0:def 4;
A10: x in eq_dom(f,a) by A8,XBOOLE_0:def 4;
    then
A11: ex y be Real st y = f.x & a = y by MESFUNC6:7;
A12: x in dom f by A10,MESFUNC6:7;
    then x in great_eq_dom(f,a) by A11,MESFUNC6:6;
    then
A13: x in A /\ great_eq_dom(f,a) by A9,XBOOLE_0:def 4;
    x in less_eq_dom(f,a) by A11,A12,MESFUNC6:4;
    hence x in A /\ great_eq_dom(f,a) /\ less_eq_dom(f,a) by A13,XBOOLE_0:def 4
;
  end;
  then A /\ eq_dom(f,a) c= A /\ great_eq_dom(f,a) /\ less_eq_dom(f,a) by
TARSKI:def 3;
  hence thesis by A7,XBOOLE_0:def 10;
end;
