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 Th43:
  a,b are_connected & b,c are_connected & c,d are_connected & d,e
are_connected & a,f are_connected implies for A being Path of a,b, B being Path
of b,c, C being Path of c,d, D being Path of d,e, E being Path of f,c holds A+(
  B+C)+D, A+B+-E+(E+C+D) are_homotopic
proof
  assume that
A1: a,b are_connected & b,c are_connected and
A2: c,d are_connected & d,e are_connected and
A3: a,f are_connected;
  let A be Path of a,b, B be Path of b,c, C be Path of c,d, D be Path of d,e,
  E be Path of f,c;
A4: A+B+-E, A+B+-E are_homotopic by A3,BORSUK_2:12;
A5: a,c are_connected by A1,BORSUK_6:42;
  then
A6: f,c are_connected by A3,BORSUK_6:42;
  then
A7: E+(C+D), E+C+D are_homotopic by A2,BORSUK_6:73;
A8: c,e are_connected by A2,BORSUK_6:42;
  then
A9: A+B+-E+(E+(C+D)), A+B+-E+E+(C+D) are_homotopic by A3,A6,BORSUK_6:73;
A10: A+B+(C+D), A+(B+C)+D are_homotopic by A1,A2,Th35;
  f,e are_connected by A8,A6,BORSUK_6:42;
  then A+B+-E+(E+(C+D)), A+B+-E+(E+C+D) are_homotopic by A3,A7,A4,BORSUK_6:75;
  then
A11: A+B+-E+E+(C+D), A+B+-E+(E+C+D) are_homotopic by A9,BORSUK_6:79;
  A+B+-E+E+(C+D), A+B+(C+D) are_homotopic by A5,A8,A6,Th37;
  then A+(B+C)+D, A+B+-E+E+(C+D) are_homotopic by A10,BORSUK_6:79;
  hence thesis by A11,BORSUK_6:79;

end;
