reserve T for non empty TopSpace,
  a, b, c, d for Point of T;
reserve X for non empty pathwise_connected TopSpace,
  a1, b1, c1, d1 for Point of X;

theorem Th73:
  for P be Path of a, b, Q be Path of b, c, R be Path of c, d st a
,b are_connected & b,c are_connected & c,d are_connected holds (P + Q) + R, P +
  (Q + R) are_homotopic
proof
  let P be Path of a, b, Q be Path of b, c, R be Path of c, d such that
A1: a,b are_connected & b,c are_connected and
A2: c,d are_connected;
  a,c are_connected & RePar (P + Q + R, 3RP) = P + (Q + R) by A1,A2,Th42,Th52;
  hence thesis by A2,Th42,Th45,Th49;
end;
