reserve Omega for non empty set;
reserve Sigma for SigmaField of Omega;
reserve S for non empty Subset of REAL;
reserve r for Real;
reserve T for Nat;
reserve I for TheEvent of r;

theorem Th6000:
  Intersection ext_left_closed_sets(0) = {+infty}
proof
 for c being object holds c in Intersection ext_left_closed_sets(0) iff
  c in {+infty}
 proof
  let c be object;
  thus c in Intersection ext_left_closed_sets(0) implies c in {+infty}
proof
 assume
Y: c in Intersection ext_left_closed_sets(0);
 assume not c in {+infty};
 then WW: c <> +infty by TARSKI:def 1;
 reconsider c as Element of ExtREAL by Y;
 per cases by XXREAL_0:14,WW;
  suppose c = -infty; then
   not c in [.0+0,+infty.] by XXREAL_1:1; then
   not c in (ext_left_closed_sets(0)).0 by Def4000;
  hence thesis by Y,PROB_1:13;
  end;
  suppose c in REAL;
   then reconsider c as Element of REAL;
   per cases;
   suppose c < 0; then
    not c in [.0+1,+infty.] by XXREAL_1:1; then
    not c in (ext_left_closed_sets(0)).1 by Def4000;
    hence thesis by Y,PROB_1:13;
   end;
   suppose S1: c >= 0;
    set finerg=[\ c+2*1 /];
     WQ1: (c+2*1)-1 < finerg by INT_1:def 6;
     WQ: finerg > 0
     proof
      c+(1+1-1)>0 by S1;
      hence thesis by WQ1;
     end;
    finerg is Nat
    proof
     finerg in INT by INT_1:def 2;
     then consider k being Nat such that
      ZZ: finerg = k or finerg = -k by INT_1:def 1;
    thus thesis by ZZ,WQ;
    end;
    then reconsider finerg as Nat;
W0: (c+2*1)-1 < finerg by INT_1:def 6;
    (c+2*1)-1 = c+1; then
W1: c < (c+2*1)-1 by XREAL_1:29;
W2: finerg < 0+(finerg+1) by XREAL_1:29;
    c < finerg by W0,W1,XXREAL_0:2; then
    c < 0+(finerg+1) by W2,XXREAL_0:2; then
    not c in [.0 + (finerg+1),+infty.] by XXREAL_1:1; then
    not c in (ext_left_closed_sets(0)).(finerg+1) by Def4000;
    hence thesis by Y,PROB_1:13;
   end;
  end;
end;
   assume A12: c in {+infty};
   for n being Nat holds c in (ext_left_closed_sets(0)).n
   proof
    let n be Nat;
s2:  (ext_left_closed_sets(0)).n=[.0+n,+infty.] by Def4000;
     0+n<=+infty by XXREAL_0:3; then
     {+infty} c= [.0+n,+infty.] by ZFMISC_1:31,XXREAL_1:1;
     hence thesis by A12,s2;
   end;
  hence thesis by PROB_1:13;
  end;
 hence thesis;
end;
