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 Th20:
for r being Real st for n holds F.n = A /\ less_dom(f,(r+1/(n+1)))
  holds A /\ less_eq_dom(f,r) = meet rng F
proof
  let r be Real;
  assume
A1: for n holds F.n = A /\ less_dom(f,(r+1/(n+1)));
   for x being object st x in A /\ less_eq_dom(f,r) holds x in meet rng F
  proof
    let x be object;
    assume
A2: x in A /\ less_eq_dom(f,r);
then A3: x in A by XBOOLE_0:def 4;
A4: x in less_eq_dom(f,r) 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,(r+1/(m+1))) by A1,A5;
A7:    x in dom f by A4,Def12;
        reconsider x as Element of X by A2;
A8:    f.x <= r by A4,Def12;
    (m+1)" > 0;
then     1/(m+1) > 0 by XCMPLX_1:215;
then     r < (r+1/(m+1)) by XREAL_1:29;
then     f.x < (r+1/(m+1)) by A8,XXREAL_0:2;
then     x in less_dom(f,(r+1/(m+1))) 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 A9: A /\ less_eq_dom(f,r) c= meet rng F;
   for
 x being object st x in meet rng F holds x in A /\ less_eq_dom(f,r)
  proof
    let x be object;
    assume
A10: x in meet rng F;
A11: for m holds x in A & x in dom f & f.x < (r+1/(m+1))
    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 A10,SETFAM_1:def 1;
then A12:  x in A /\ less_dom(f,(r+1/(m+1))) by A1;
then   x in less_dom(f,(r+1/(m+1))) by XBOOLE_0:def 4;
      hence thesis by A12,Def11,XBOOLE_0:def 4;
    end;
    reconsider y=f.x as R_eal by XXREAL_0:def 1;
 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 A10,SETFAM_1:def 1;
then  x in A /\ less_dom(f,(r+1/(1+1))) by A1;
    then reconsider x as Element of X;
 y <= r
    proof
  now per cases;
        suppose
      y=-infty;
          hence thesis by XXREAL_0:5;
        end;
        suppose
      not y=-infty;
then A13:      not y <= -infty by XXREAL_0:6;
      y < (r+1/(1+1)) by A11;
          then reconsider y1=y as Element of REAL by A13,XXREAL_0:48;
      for m holds y1-1/(m+1) <= r
          proof
            let m;
         y1 < (r+1/(m+1)) by A11;
            hence thesis by XREAL_1:20;
          end;
          hence thesis by Th11;
        end;
      end;
      hence thesis;
    end;
then  x in less_eq_dom(f,r) by A11,Def12;
    hence thesis by A11,XBOOLE_0:def 4;
  end;
then  meet rng F c= A /\ less_eq_dom(f,r);
  hence thesis by A9,XBOOLE_0:def 10;
end;
