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 Th4:
  for X be non empty set, f being Functional_Sequence of X,REAL, n
being Nat holds dom((inferior_realsequence f).n) = dom(f.0) & for x
  be Element of X st x in dom((inferior_realsequence f).n) holds ((
  inferior_realsequence f).n).x=(inferior_realsequence R_EAL(f#x)).n
proof
  let X be non empty set;
  let f be Functional_Sequence of X,REAL;
  let n be Nat;
  set IF = inferior_realsequence f;
  dom(IF.n) = dom((R_EAL f).0) by MESFUNC8:def 5
    .= dom R_EAL(f.0);
  hence dom((inferior_realsequence f).n) = dom(f.0);
  hereby
    let x be Element of X;
    assume x in dom(IF.n);
    then
A1: (IF.n).x = (inferior_realsequence((R_EAL f)#x)).n by MESFUNC8:def 5
      .= inf( ((R_EAL f)#x)^\n ) by RINFSUP2:27;
    (R_EAL f)#x = f#x by Th1;
    hence (IF.n).x = (inferior_realsequence R_EAL(f#x)).n by A1,RINFSUP2:27;
  end;
end;
