
theorem Th6:
  for a,b,c,d be Real st a < b & b < c & c < d holds
    ['a,d'] \ ['b,c'] c= ['a,b'] \/ ['c,d']
proof
  let a,b,c,d be Real;
  assume A1: a < b & b < c & c < d; then
  ADD: a < c by XXREAL_0:2;
   let x be object;
   assume A2: x in ['a,d'] \ ['b,c']; then
   reconsider x as Real;
   not x in ['b,c'] by XBOOLE_0:def 5,A2; then
 A3:   not (x in [. b,c .]) by INTEGRA5:def 3,A1;
   x in ['a,d'] by XBOOLE_0:def 5,A2; then
   x in [.a,d.] by INTEGRA5:def 3,ADD,XXREAL_0:2,A1; then
   A4: a <= x & x <= d by XXREAL_1:1;
   per cases by A3;
    suppose x < b; then
     x in [. a,b .] by A4; then
     x in [' a,b '] by INTEGRA5:def 3,A1;
     hence thesis by XBOOLE_0:def 3;
     end;
    suppose x > c; then
     x in [. c,d .] by A4; then
     x in [' c,d '] by INTEGRA5:def 3,A1;
     hence thesis by XBOOLE_0:def 3;
    end;
end;
