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;

theorem
  for r being Real st r >= 0 holds [.0,+infty.] \ [.0,r.[ = [.r,+infty.]
  proof
   let r be Real;
   assume A0: r >= 0;
   for x being object holds x in [.0,+infty.] \ [.0,r.[ iff x in [.r,+infty.]
   proof
    let x be object;
    thus x in [.0,+infty.] \ [.0,r.[ implies x in [.r,+infty.]
    proof
     assume x in [.0,+infty.] \ [.0,r.[;
     then G1: x in [.0,+infty.] & (not x in [.0,r.[) by XBOOLE_0:def 5;
     then x in {w where w is Element of ExtREAL: 0<=w & w<=+infty}
      by XXREAL_1:def 1;
     then consider w being Element of ExtREAL such that
G2:  x=w & 0<=w & w<=+infty;
G3:  not w in {w where w is Element of ExtREAL: 0<=w & w<r}
      by XXREAL_1:def 2,G1,G2;
     0>w or w>=r by G3;
     hence thesis by XXREAL_1:1,G2;
    end;
     assume x in [.r,+infty.];
     then x in {w where w is Element of ExtREAL: r<=w & w<=+infty}
      by XXREAL_1:def 1;
     then consider w being Element of ExtREAL such that
      H1: w=x & w>=r & w<=+infty;
     reconsider x as Element of ExtREAL by H1;
     H2: x in [.0,+infty.] by A0,XXREAL_1:1,H1;
     not x in [.0,r.[ by XXREAL_1:3,H1;
    hence thesis by H2,XBOOLE_0:def 5;
   end;
  hence thesis;
  end;
