reserve a,b,c,d for Real;

theorem
  a <= c & b <= d & c <= b implies Closed-Interval-TSpace(a,d) =
  Closed-Interval-TSpace(a,b) union Closed-Interval-TSpace(c,d) &
  Closed-Interval-TSpace(c,b) = Closed-Interval-TSpace(a,b) meet
  Closed-Interval-TSpace(c,d)
proof
  assume that
A1: a <= c and
A2: b <= d and
A3: c <= b;
A4: the carrier of Closed-Interval-TSpace(a,b) = [.a,b.] & the carrier of
  Closed-Interval-TSpace(c,d) = [.c,d.] by A1,A2,A3,TOPMETR:18,XXREAL_0:2;
  a <= b by A1,A3,XXREAL_0:2;
  then
A5: the carrier of Closed-Interval-TSpace(a,d) = [.a,d.] by A2,TOPMETR:18
,XXREAL_0:2;
A6: the carrier of Closed-Interval-TSpace(c,b) = [.c,b.] by A3,TOPMETR:18;
  [.a,b.] \/ [.c,d.] = [.a,d.] by A1,A2,A3,XXREAL_1:174;
  hence Closed-Interval-TSpace(a,d) = Closed-Interval-TSpace(a,b) union
  Closed-Interval-TSpace(c,d) by A4,A5,TSEP_1:def 2;
A7: [.a,b.] /\ [.c,d.] = [.c,b.] by A1,A2,XXREAL_1:143;
  then (the carrier of Closed-Interval-TSpace(a,b)) /\ (the carrier of
  Closed-Interval-TSpace(c,d)) <> {} by A3,A4,XXREAL_1:1;
  then (the carrier of Closed-Interval-TSpace(a,b)) meets (the carrier of
  Closed-Interval-TSpace(c,d)) by XBOOLE_0:def 7;
  then Closed-Interval-TSpace(a,b) meets Closed-Interval-TSpace(c,d) by
TSEP_1:def 3;
  hence thesis by A4,A6,A7,TSEP_1:def 4;
end;
