 reserve Omega for non empty set;
 reserve F for SigmaField of Omega;

theorem A1:
  ].0,+infty.[ is Element of Borel_Sets
 proof
  set Bor1=].0,+infty.[;
  reconsider Bor3 = {0} as Event of Borel_Sets by FINANCE2:5;
  reconsider Bor2 = [.0,+infty.[ as Event of Borel_Sets by FINANCE1:3;
  Bor1 = Bor2 \ Bor3
  proof
   for x being object holds x in Bor1 iff (x in Bor2 \ Bor3)
   proof
    let x be object;
    thus x in Bor1 implies x in Bor2\Bor3
    proof
     assume ASS0: x in Bor1;
     then reconsider x as Real;
d1:  0<x & x<+infty by ASS0,XXREAL_1:4;
     then D2: x in [.0,+infty.[ by XXREAL_1:3;
     not x in {0} by d1,TARSKI:def 1;
    hence thesis by XBOOLE_0:def 5,D2;
    end;
     assume ASS0: x in Bor2\Bor3;
     then reconsider x as Real;
     x in Bor2 & not x in Bor3 by ASS0,XBOOLE_0:def 5;
     then 0<=x & x<+infty & x<>0 by TARSKI:def 1,XXREAL_1:3;
    hence thesis by XXREAL_1:4;
   end;
  hence thesis by TARSKI:2;
  end;
  hence thesis;
 end;
