reserve a,b,c,d for Real;

theorem
  a <= c & d <= b & c <= d implies
  Closed-Interval-TSpace(c,d) is closed SubSpace of Closed-Interval-TSpace(a,b)
proof
  assume that
A1: a <= c and
A2: d <= b and
A3: c <= d;
  [.c,d.] c= [.a,b.] by A1,A2,XXREAL_1:34;
  then
A4: the carrier of Closed-Interval-TSpace(c,d) c= [.a,b.] by A3,TOPMETR:18;
A5: Closed-Interval-TSpace(c,d) is closed SubSpace of R^1 by A3,Th2;
  a <= d by A1,A3,XXREAL_0:2;
  then the carrier of Closed-Interval-TSpace(c,d) c= the carrier of
  Closed-Interval-TSpace(a,b) by A2,A4,TOPMETR:18,XXREAL_0:2;
  hence thesis by A5,TSEP_1:14;
end;
