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
  Family_of_halflines c= Ext_Borel_Sets
proof
  let x be object;
  assume x in Family_of_halflines;
  then consider r being Element of REAL such that
   A1: x = halfline(r);
  reconsider Set1 = [.-infty,r.] as Element of Ext_Borel_Sets by Th3;
  reconsider Set2 = {r} as Element of Ext_Borel_Sets by Th71;
  reconsider Set3 = Set1 \ Set2 as Element of Ext_Borel_Sets;
  A3: Set3=[.-infty,r.[ by XXREAL_1:135,XXREAL_0:5;
  reconsider Set4 = {-infty} as Element of Ext_Borel_Sets by Th500,Th600;
  reconsider Set5 = Set3 \ Set4 as Element of Ext_Borel_Sets;
  Set5 = ].-infty,r.[ by XXREAL_1:136,A3,XXREAL_0:12;
  hence thesis by A1;
end;
