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 Th21:
  for r being Real st for n
  holds F.n = A /\ less_eq_dom(f,(r-1/(n+1)))
  holds A /\ less_dom(f,r) = union rng F
proof
  let r be Real;
  assume
A1: for n holds F.n = A /\ less_eq_dom(f,(r-1/(n+1)));
   for x being object st x in A /\ less_dom(f,r) holds x in union rng F
  proof
    let x be object;
    assume
A2: x in A /\ less_dom(f,r);
then A3: x in A by XBOOLE_0:def 4;
A4: x in less_dom(f,r) by A2,XBOOLE_0:def 4;
 ex Y being set st x in Y & Y in rng F
    proof
      reconsider x as Element of X by A2;
A5:   x in dom f by A4,Def11;
A6:   f.x < r by A4,Def11;
  ex m st f.x <= (r-1/(m+1))
      proof
        per cases;
        suppose
A7:      f.x = -infty;
          take 1;
          thus thesis by A7,XXREAL_0:5;
        end;
        suppose
      not f.x =-infty;
then       not f.x <= -infty by XXREAL_0:6;
          then reconsider y1=f.x as Element of REAL by A6,XXREAL_0:48;
          consider m such that
A8:      1/(m+1) < r-y1 by A6,Th10;
      y1+1/(m+1) < r by A8,XREAL_1:20;
then       f.x <= (r-1/(m+1)) by XREAL_1:20;
          hence thesis;
        end;
      end;
      then consider m such that
A9:  f.x <= (r-1/(m+1));
  x in less_eq_dom(f,(r-1/(m+1))) by A5,A9,Def12;
then A10:  x in A /\ less_eq_dom(f,(r-1/(m+1))) by A3,XBOOLE_0:def 4;
  m in NAT;
then A11:  m in dom F by FUNCT_2:def 1;
      take F.m;
      thus thesis by A1,A10,A11,FUNCT_1:def 3;
    end;
    hence thesis by TARSKI:def 4;
  end;
then A12: A /\ less_dom(f,r) c= union rng F;
   for x being object st x in union rng F holds x in A /\ less_dom(f,r )
  proof
    let x be object;
    assume x in union rng F;
    then consider Y being set such that
A13: x in Y and
A14: Y in rng F by TARSKI:def 4;
    consider m such that
    m in dom F and
A15: F.m = Y by A14,PARTFUN1:3;
A16: x in A /\ less_eq_dom(f,(r-1/(m+1))) by A1,A13,A15;
then A17: x in A by XBOOLE_0:def 4;
A18: x in less_eq_dom(f,(r-1/(m+1))) by A16,XBOOLE_0:def 4;
then A19: x in dom f by Def12;
A20: f.x <= (r-1/(m+1)) by A18,Def12;
    reconsider x as Element of X by A13,A14;
 f.x < r
    proof
     now
      r < r+1/(m+1) by XREAL_1:29,139;
then       r-1/(m+1) < r by XREAL_1:19;
          hence thesis by A20,XXREAL_0:2;
      end;
      hence thesis;
    end;
then  x in less_dom(f,r) by A19,Def11;
    hence thesis by A17,XBOOLE_0:def 4;
  end;
then  union rng F c= A /\ less_dom(f,r);
  hence thesis by A12,XBOOLE_0:def 10;
end;
