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 Th26:
  for X, S, F, f, A st for n holds F.n = A /\ great_dom(f,(-n))
  holds A /\ great_dom(f,-infty) = union rng F
proof
  let X,S,F,f,A;
  assume
A1: for n holds F.n = A /\ great_dom(f,(-n));
   for x being object st x in A /\ great_dom(f,-infty) holds x in union rng F
  proof
    let x be object;
    assume
A2: x in A /\ great_dom(f,-infty);
then A3: x in A by XBOOLE_0:def 4;
A4: x in great_dom(f,-infty) by A2,XBOOLE_0:def 4;
then A5: x in dom f by Def13;
A6: -infty < f.x by A4,Def13;
 ex n being Element of NAT st (-n) < f.x
    proof
      per cases;
      suppose
A7:     f.x = +infty;
        take 0;
        thus thesis by A7;
      end;
      suppose
    not f.x = +infty;
then     not +infty <= f.x by XXREAL_0:4;
        then reconsider y1=f.x as Element of REAL by A6,XXREAL_0:48;
        consider n1 being Nat such that
A8:    -n1 <= y1 by Th9;
    n1 < n1+1 by NAT_1:13;
then A9:    -(n1+1) < -n1 by XREAL_1:24;
        reconsider m=n1+1 as Element of NAT;
        take m;
        thus thesis by A8,A9,XXREAL_0:2;
      end;
    end;
    then consider n being Element of NAT such that
A10: (-n) < f.x;
    reconsider x as Element of X by A2;
 x in great_dom(f,(-n)) by A5,A10,Def13;
then  x in A /\ great_dom(f,(-n)) by A3,XBOOLE_0:def 4;
then A11: x in F.n by A1;
 n in NAT;
then  n in dom F by FUNCT_2:def 1;
then  F.n in rng F by FUNCT_1:def 3;
    hence thesis by A11,TARSKI:def 4;
  end;
then A12: A /\ great_dom(f,-infty) c= union rng F;
 for x being object st x in union rng F holds x in A /\ great_dom(f,-infty )
  proof
    let x being 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 /\ great_dom(f,(-m)) by A1,A13,A15;
then A17: x in A by XBOOLE_0:def 4;
A18: x in great_dom(f,(-m)) by A16,XBOOLE_0:def 4;
then A19: x in dom f by Def13;
A20: (-m) < f.x by A18,Def13;
    reconsider x as Element of X by A13,A14;
   -m in REAL by XREAL_0:def 1;
 then -infty < f.x by A20,XXREAL_0:2,12;
then  x in great_dom(f,-infty) by A19,Def13;
    hence thesis by A17,XBOOLE_0:def 4;
  end;
then  union rng F c= A /\ great_dom(f,-infty);
  hence thesis by A12,XBOOLE_0:def 10;
end;
