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,
  k for Real,
  n for Nat,
  E for Element of S;

theorem
  for f be with_the_same_dom Functional_Sequence of X,REAL, F be
SetSequence of S, r be Real st (for n be Nat holds F.n = dom(
  f.0) /\ great_dom(f.n,r)) holds union rng F = dom(f.0) /\ great_dom(sup f,r)
proof
  let f be with_the_same_dom Functional_Sequence of X,REAL, F be SetSequence
  of S, r be Real;
  set E = dom(f.0);
  assume
A1: for n be Nat holds F.n = E /\ great_dom(f.n,r);
  now
    let x be object;
    assume
A2: x in E /\ great_dom(sup f,r);
    then reconsider z=x as Element of X;
A3: x in E by A2,XBOOLE_0:def 4;
    x in great_dom(sup f,r) by A2,XBOOLE_0:def 4;
    then
A4: r < (sup f).z by MESFUNC1:def 13;
    ex n be Element of NAT st r < (f#z).n
    proof
      assume
A5:   for n be Element of NAT holds (f#z).n <= r;
      for p be ExtReal holds p in rng R_EAL(f#z) implies p <= r
      proof
        let p be ExtReal;
        assume p in rng R_EAL(f#z);
        then ex n be object st n in NAT & (R_EAL(f#z)).n = p by FUNCT_2:11;
        hence thesis by A5;
      end;
      then r is UpperBound of rng R_EAL(f#z) by XXREAL_2:def 1;
      then
A6:   sup rng R_EAL(f#z) <= r by XXREAL_2:def 3;
      x in dom sup f by A3,MESFUNC8:def 4;
      hence contradiction by A4,A6,Th3;
    end;
    then consider n be Element of NAT such that
A7: r < (f#z).n;
A8: x in dom (f.n) by A3,MESFUNC8:def 2;
    r < (f.n).z by A7,SEQFUNC:def 10;
    then
A9: x in great_dom(f.n,r) by A8,MESFUNC1:def 13;
A10: F.n in rng F by FUNCT_2:4;
    F.n = E /\ great_dom(f.n,r) by A1;
    then x in F.n by A3,A9,XBOOLE_0:def 4;
    hence x in union rng F by A10,TARSKI:def 4;
  end;
  then
A11: E /\ great_dom(sup f,r) c= union rng F;
  now
    let x be object;
    assume x in union rng F;
    then consider y be set such that
A12: x in y and
A13: y in rng(F qua SetSequence of X) by TARSKI:def 4;
    reconsider z=x as Element of X by A12,A13;
    consider n be object such that
A14: n in dom F and
A15: y=F.n by A13,FUNCT_1:def 3;
    reconsider n as Element of NAT by A14;
A16: F.n = E /\ great_dom(f.n,r) by A1;
    then x in great_dom(f.n,r) by A12,A15,XBOOLE_0:def 4;
    then
A17: r < (f.n).z by MESFUNC1:def 13;
    f#z = (R_EAL f)#z by Th1;
    then (f.n).z = ((R_EAL f)#z).n by SEQFUNC:def 10;
    then
A18: (f.n).z <= sup rng((R_EAL f)#z) by FUNCT_2:4,XXREAL_2:4;
A19: x in E by A12,A15,A16,XBOOLE_0:def 4;
    then
A20: x in dom sup f by MESFUNC8:def 4;
    then (sup f).z = sup((R_EALf)#z) by MESFUNC8:def 4;
    then r < (sup f).z by A17,A18,XXREAL_0:2;
    then x in great_dom(sup f,r) by A20,MESFUNC1:def 13;
    hence x in E /\ great_dom(sup f,r) by A19,XBOOLE_0:def 4;
  end;
  then union rng F c= E /\ great_dom(sup f,r);
  hence thesis by A11;
end;
