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 Th29:
  a,b are_connected & a,c are_connected implies for A, B being
  Path of a,b, C being Path of c,a st C+A, C+B are_homotopic holds A, B
  are_homotopic
proof
  assume that
A1: a,b are_connected and
A2: a,c are_connected;
  let A, B be Path of a,b, C be Path of c,a;
A3: -C+C+A, -C+(C+A) are_homotopic by A1,A2,BORSUK_6:73;
  assume
A4: C+A, C+B are_homotopic;
  b,c are_connected & -C, -C are_homotopic by A1,A2,BORSUK_2:12,BORSUK_6:42;
  then -C+(C+A), -C+(C+B) are_homotopic by A2,A4,BORSUK_6:75;
  then
A5: -C+C+A, -C+(C+B) are_homotopic by A3,BORSUK_6:79;
  -C+C+B, -C+(C+B) are_homotopic by A1,A2,BORSUK_6:73;
  then
A6: -C+C+A, -C+C+B are_homotopic by A5,BORSUK_6:79;
  -C+C+A, A are_homotopic by A1,A2,Th23,BORSUK_2:12;
  then
A7: A, -C+C+B are_homotopic by A6,BORSUK_6:79;
  -C+C+B, B are_homotopic by A1,A2,Th23,BORSUK_2:12;
  hence thesis by A7,BORSUK_6:79;

end;
