reserve a,b,c,d for Real;

theorem
  a <= b implies Closed-Interval-TSpace(a,b) is connected
proof
  assume
A1: a <= b;
  now
    per cases by A1,XXREAL_0:1;
    suppose
      a < b;
      then L[01]((#)(a,b),(a,b)(#)) is being_homeomorphism by Th17;
      then
A2:   rng L[01]((#)(a,b),(a,b)(#) ) = [#](Closed-Interval-TSpace(a,b )) &
      L[01]( (#)(a,b),(a,b)(#)) is continuous by TOPS_2:def 5;
      set A = the carrier of Closed-Interval-TSpace(0,1);
      A = [#](Closed-Interval-TSpace(0,1)) & L[01]((#)(a,b),(a,b)(#)).:(A)
      = rng L[01]((#)(a,b),(a,b)(#) ) by RELSET_1:22;
      hence thesis by A2,Th19,CONNSP_1:14,TOPMETR:20;
    end;
    suppose
A3:   a = b;
      then [.a,b.] = {a} & a = (#)(a,b) by Def1,XXREAL_1:17;
      then [#] Closed-Interval-TSpace(a,b) = {(#)(a,b)} by A3,TOPMETR:18;
      hence thesis by CONNSP_1:27;
    end;
  end;
  hence thesis;
end;
