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