reserve p, q, x, y for Real,
  n for Nat;
reserve X for non empty TopSpace,
  a, b, c, d, e, f for Point of X,
  T for non empty pathwise_connected TopSpace,
  a1, b1, c1, d1, e1, f1 for Point of T;

theorem Th27:
  a,b are_connected & c,b are_connected implies for A, B being
  Path of a,b, C being Path of b,c st A+C, B+C are_homotopic holds A, B
  are_homotopic
proof
  assume that
A1: a,b are_connected and
A2: c,b are_connected;
  let A, B be Path of a,b, C be Path of b,c;
A3: A+C+-C, A are_homotopic by A1,A2,Th19,BORSUK_2:12;
  assume
A4: A+C, B+C are_homotopic;
  a,c are_connected & -C, -C are_homotopic by A1,A2,BORSUK_2:12,BORSUK_6:42;
  then A+C+-C, B+C+-C are_homotopic by A2,A4,BORSUK_6:75;
  then
A5: A, B+C+-C are_homotopic by A3,BORSUK_6:79;
  B+C+-C, B are_homotopic by A1,A2,Th19,BORSUK_2:12;
  hence thesis by A5,BORSUK_6:79;

end;
