reserve i for Integer,
  a, b, r, s for Real;

theorem
  a <= b & r <= s implies for A being closed Subset of
Closed-Interval-TSpace(a,b), B being closed Subset of Closed-Interval-TSpace(r,
  s) holds product ((1,2)-->(A,B)) is closed Subset of Trectangle(a,b,r,s)
proof
  set T = Closed-Interval-TSpace(a,b);
  set S = Closed-Interval-TSpace(r,s);
  assume
A1: a <= b & r <= s;
  then reconsider
  h = R2Homeomorphism | the carrier of [:Closed-Interval-TSpace(a,b
  ),Closed-Interval-TSpace(r,s):] as Function of [:Closed-Interval-TSpace(a,b),
  Closed-Interval-TSpace(r,s):], Trectangle(a,b,r,s) by Th35;
  let A be closed Subset of T, B be closed Subset of S;
  reconsider P = product ((1,2)-->(A,B)) as Subset of Trectangle(a,b,r,s) by A1
,Th38;
A2: [:A,B:] is closed by TOPALG_3:15;
  the carrier of S is Subset of R^1 by TSEP_1:1;
  then
A3: B is Subset of REAL by TOPMETR:17,XBOOLE_1:1;
  the carrier of T is Subset of R^1 by TSEP_1:1;
  then
A4: A is Subset of REAL by TOPMETR:17,XBOOLE_1:1;
A5: h.:[:A,B:] = R2Homeomorphism.:[:A,B:] by RELAT_1:129
    .= P by A4,A3,Th33;
  h is being_homeomorphism & Trectangle(a,b,r,s) is non empty by A1,Th32,Th36;
  hence thesis by A5,A2,TOPS_2:58;
end;
