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 Th21:
  a,b are_connected & c,b are_connected implies for A1,A2 being
Path of a,b, B being Path of c,b holds A1,A2 are_homotopic implies A1, A2+-B+B
  are_homotopic
proof
  assume that
A1: a,b are_connected and
A2: c,b are_connected;
  set X = the constant Path of b,b;
  let A1,A2 be Path of a,b, B be Path of c,b;
A3: A1, A1+X are_homotopic by A1,BORSUK_6:80;
  assume
A4: A1, A2 are_homotopic;
  -B+B, X are_homotopic by A2,BORSUK_6:86;
  then A2+(-B+B), A1+X are_homotopic by A1,A4,BORSUK_6:75;
  then
A5: A2+(-B+B), A1 are_homotopic by A3,BORSUK_6:79;
  A2+(-B+B), A2+-B+B are_homotopic by A1,A2,BORSUK_6:73;
  hence thesis by A5,BORSUK_6:79;
end;
