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 Th41:
  a,b are_connected & a,c are_connected & d,c are_connected
implies for A being Path of a,b, B being Path of c,d, C being Path of a,c holds
  A+(-A+C+B)+-B, C are_homotopic
proof
  assume that
A1: a,b are_connected & a,c are_connected and
A2: d,c are_connected;
  let A be Path of a,b, B be Path of c,d, C be Path of a,c;
A3: A+-A+C+B+-B, C are_homotopic by A1,A2,Th39;
A4: -B, -B are_homotopic by A2,BORSUK_2:12;
  A+(-A+C+B), A+-A+C+B are_homotopic & a,d are_connected by A1,A2,Th33,
BORSUK_6:42;
  then A+(-A+C+B)+-B, A+-A+C+B+-B are_homotopic by A2,A4,BORSUK_6:75;
  hence thesis by A3,BORSUK_6:79;
end;
