
theorem Th10:
  for a,b,c,d be Real st b <= c holds [.a,b.] /\ [.c,d.] c= [.b,b.]
  proof
    let a,b,c,d be Real;
    assume
    A1: b <= c;
    per cases;
    suppose
      A2: a <= b;
      per cases;
      suppose c <= d; then
        b <= d by A1,XXREAL_0:2; then
        [.a,b.] /\ [.b,d.] = [.b,b.] by A2,XXREAL_1:143;
        hence [.a,b.] /\ [.c,d.] c= [.b,b.] by A1,XBOOLE_1:26,XXREAL_1:34;
      end;
      suppose d < c; then
        [.c,d.] = {} by XXREAL_1:29;
        hence [.a,b.] /\ [.c,d.] c= [.b,b.];
      end;
    end;
    suppose b < a; then
      [.a,b.] = {} by XXREAL_1:29;
      hence [.a,b.] /\ [.c,d.] c= [.b,b.];
    end;
  end;
