reserve T,T1,T2,S for non empty TopSpace;
reserve GY for non empty TopSpace,
  r,s for Real;

theorem
  for T being non empty TopSpace,a,b,c being Point of T, G1 being Path
of a,b, G2 being Path of b,c st G1 is continuous & G2 is continuous & G1.0=a &
G1.1=b & G2.0=b & G2.1=c holds G1+G2 is continuous & (G1+G2).0=a & (G1+G2).1=c
proof
  let T be non empty TopSpace,a,b,c be Point of T, G1 be Path of a,b, G2 be
  Path of b,c;
  assume
  G1 is continuous & G2 is continuous & G1.0=a & G1.1=b & G2.0=b & G2. 1= c;
  then
  ex h being Function of I[01],T st h is continuous & h.0=a & h.1=c & rng
  h c= (rng G1) \/ (rng G2) by Lm3;
  then a,c are_connected;
  hence thesis by Def2;
end;
