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 Th23:
  for X, S, F, f, A st for n holds F.n = A /\ great_dom(f,n)
  holds A /\ eq_dom(f,+infty) = meet 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 /\ eq_dom(f,+infty) holds x in meet rng F
  proof
    let x being object;
    assume
A2: x in A /\ eq_dom(f,+infty);
then A3: x in A by XBOOLE_0:def 4;
A4: x in eq_dom(f,+infty) by A2,XBOOLE_0:def 4;
 for Y being set holds Y in rng F implies x in Y
    proof
      let Y be set;
   Y in rng F implies x in Y
      proof
        assume Y in rng F;
        then consider m being Element of NAT such that
        m in dom F and
A5:     Y = F.m by PARTFUN1:3;
A6:    Y = A /\ great_dom(f,m) by A1,A5;
        reconsider x as Element of X by A2;
A7:    f.x = +infty by A4,Def15;
       m in REAL by XREAL_0:def 1;
    then x in dom f & not +infty <= m by A4,Def15,XXREAL_0:9;
then     x in great_dom(f,m) by A7,Def13;
        hence thesis by A3,A6,XBOOLE_0:def 4;
      end;
      hence thesis;
    end;
    hence thesis by SETFAM_1:def 1;
  end;
then A8: A /\ eq_dom(f,+infty) c= meet rng F;
 for x being object st x in meet rng F holds x in A /\ eq_dom(f,+infty)
  proof
    let x be object;
     reconsider xx=x as set by TARSKI:1;
    assume
A9: x in meet rng F;
A10: for m holds x in A & x in dom f & ex y being R_eal st y=f.x & y = +infty
    proof
      let m;
  m in NAT;
then   m in dom F by FUNCT_2:def 1;
then   F.m in rng F by FUNCT_1:def 3;
then   x in F.m by A9,SETFAM_1:def 1;
then A11:  x in A /\ great_dom(f,m) by A1;
then A12:  x in great_dom(f,m) by XBOOLE_0:def 4;
  for r holds r < f.xx
      proof
        let r;
        consider n such that
A13:    r <= n by Th8;
    n in NAT;
then     n in dom F by FUNCT_2:def 1;
then     F.n in rng F by FUNCT_1:def 3;
then     x in F.n by A9,SETFAM_1:def 1;
then     x in A /\ great_dom(f,n) by A1;
then     x in great_dom(f,n) by XBOOLE_0:def 4;
then     n < f.x by Def13;
        hence thesis by A13,XXREAL_0:2;
      end;
then   f.x = +infty by Th12;
      hence thesis by A11,A12,Def13,XBOOLE_0:def 4;
    end;
 1 in NAT;
then  1 in dom F by FUNCT_2:def 1;
then  F.1 in rng F by FUNCT_1:def 3;
then  x in F.1 by A9,SETFAM_1:def 1;
then  x in A /\ great_dom(f,1) by A1;
    then reconsider x as Element of X;
 x in eq_dom(f,+infty) by A10,Def15;
    hence thesis by A10,XBOOLE_0:def 4;
  end;
then  meet rng F c= A /\ eq_dom(f,+infty);
  hence thesis by A8,XBOOLE_0:def 10;
end;
