
theorem
  sigma Family_of_Intervals = Borel_Sets &
  sigma(Field_generated_by Family_of_Intervals) = Borel_Sets
proof
   Family_of_Intervals c= sigma(Family_of_Intervals) by PROB_1:def 9; then
A1:Family_of_halflines c= sigma(Family_of_Intervals) by Th02;
   now let x be object;
    assume x in Family_of_Intervals; then
    ex I be Interval st x = I;
    hence x in Borel_Sets by Th03;
   end; then
   Family_of_Intervals c= Borel_Sets;
   hence sigma Family_of_Intervals = Borel_Sets by A1,PROB_1:def 9,def 12;
   hence thesis by SRINGS_3:23;
end;
