reserve T for non empty TopSpace,
  a, b, c, d for Point of T;

theorem Th42:
  a,b are_connected & b,c are_connected implies a,c are_connected
proof
  assume that
A1: a,b are_connected and
A2: b,c are_connected;
  set P = the Path of a,b,R = the Path of b,c;
A3: P is continuous & P.0 = a by A1,BORSUK_2:def 2;
  take P+R;
A4: R.0 = b & R.1 = c by A2,BORSUK_2:def 2;
  P.1 = b & R is continuous by A1,A2,BORSUK_2:def 2;
  hence thesis by A3,A4,BORSUK_2:14;
end;
