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

theorem Th35:
  a <= b & r <= s implies R2Homeomorphism | the carrier of [:
  Closed-Interval-TSpace(a,b),Closed-Interval-TSpace(r,s):] is Function of [:
Closed-Interval-TSpace(a,b),Closed-Interval-TSpace(r,s):], Trectangle(a,b,r,s)
proof
  set C1 = Closed-Interval-TSpace(a,b);
  set C2 = Closed-Interval-TSpace(r,s);
  set TR = Trectangle(a,b,r,s);
  set h = R2Homeomorphism | the carrier of [:C1,C2:];
  assume a <= b & r <= s;
  then
A1: the carrier of [:C1,C2:] = [: [.a,b.], [.r,s.] :] by Th27;
  dom R2Homeomorphism = [:R,R:] by Lm1,FUNCT_2:def 1;
  then
A2: dom h = the carrier of [:C1,C2:] by A1,RELAT_1:62,TOPMETR:17,ZFMISC_1:96;
  rng h c= the carrier of TR
  proof
    let y be object;
A3: the carrier of TR = closed_inside_of_rectangle(a,b,r,s) &
closed_inside_of_rectangle(a,b,r,s) = {p where p is Point of TOP-REAL 2: a <= p
    `1 & p`1 <= b & r <= p`2 & p`2 <= s} by JGRAPH_6:def 2,PRE_TOPC:8;
    assume y in rng h;
    then consider x being object such that
A4: x in dom h and
A5: h.x = y by FUNCT_1:def 3;
    reconsider x as Point of [:R^1,R^1:] by A4;
    dom h c= [:R,R:] by A1,A2,TOPMETR:17,ZFMISC_1:96;
    then consider x1, x2 being Element of R such that
A6: x = [x1,x2] by A4,DOMAIN_1:1;
A7: x2 in [.r,s.] by A1,A2,A4,A6,ZFMISC_1:87;
A8: h.x = R2Homeomorphism.x by A4,FUNCT_1:47;
    then reconsider p = h.x as Point of TOP-REAL 2;
A9: h.x = <*x1,x2*> by A8,A6,Def2;
    then x2 = p`2;
    then
A10: r <= p`2 & p`2 <= s by A7,XXREAL_1:1;
A11: x1 in [.a,b.] by A1,A2,A4,A6,ZFMISC_1:87;
    x1 = p`1 by A9;
    then a <= p`1 & p`1 <= b by A11,XXREAL_1:1;
    hence thesis by A5,A3,A10;
  end;
  hence thesis by A2,FUNCT_2:2;
end;
