reserve GX,GY for non empty TopSpace,
  x,y for Point of GX,
  r,s for Real;

theorem Th1:   ::Arcwise connectedness derives connectedness
  for GX being non empty TopSpace st (for x,y being Point of GX
  ex h being Function of I[01],GX st h is continuous & x=h.0 & y=h.1)
  holds GX is connected
proof
  let GX;
  assume
A1: for x,y being Point of GX
  ex h being Function of I[01],GX st h is continuous & x=h.0 & y=h.1;
  for x,y being Point of GX ex GY st (GY is connected &
  ex f being Function of GY,GX st f is continuous & x in rng(f)& y in rng(f))
  proof
    let x,y;
    now
      consider h being Function of I[01],GX such that
A2:   h is continuous and
A3:   x=h.0 and
A4:   y=h.1 by A1;
A5:   0 in dom h by Lm1,BORSUK_1:40,FUNCT_2:def 1;
A6:   1 in dom h by Lm1,BORSUK_1:40,FUNCT_2:def 1;
A7:   x in rng h by A3,A5,FUNCT_1:def 3;
      y in rng h by A4,A6,FUNCT_1:def 3;
      hence thesis by A2,A7,TREAL_1:19;
    end;
    hence thesis;
  end;
  hence thesis by TOPS_2:63;
end;
