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 Th25:
  for X, S, F, f, A st for n being Nat holds F.n = A /\ less_dom(f,(-n))
  holds A /\ eq_dom(f,-infty) = meet rng F
proof
  let X,S,F,f,A;
  assume
A1: for n being Nat holds F.n = A /\ less_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 /\ less_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 (-m) <= -infty by A4,Def15,XXREAL_0:12;
then     x in less_dom(f,(-m)) by A7,Def11;
        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 /\ less_dom(f,(-m)) by A1;
then A12:  x in less_dom(f,(-m)) by XBOOLE_0:def 4;
  for r holds f.xx < r
      proof
        let r;
        consider n being Nat such that
A13:    -n <= r by Th9;
n in NAT by ORDINAL1:def 12;
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 /\ less_dom(f,(-n)) by A1;
then     x in less_dom(f,(-n)) by XBOOLE_0:def 4;
then     f.x < (-n) by Def11;
        hence thesis by A13,XXREAL_0:2;
      end;
then   f.x = -infty by Th13;
      hence thesis by A11,A12,Def11,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 /\ less_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;
