
theorem Th27:
  for S, T being non empty TopSpace, f being continuous Function
of S,T, a, b being Point of S, P, Q being Path of a,b, P1, Q1 being Path of f.a
  ,f.b st P,Q are_homotopic & P1 = f*P & Q1 = f*Q holds P1,Q1 are_homotopic
proof
  let S, T be non empty TopSpace;
  let f be continuous Function of S,T;
  let a, b be Point of S;
  let P, Q be Path of a,b;
  let P1, Q1 be Path of f.a,f.b;
  assume that
A1: P,Q are_homotopic and
A2: P1 = f*P and
A3: Q1 = f*Q;
  set F = the Homotopy of P,Q;
  take G = f*F;
  F is continuous by A1,BORSUK_6:def 11;
  hence G is continuous;
  let s be Point of I[01];
  thus G.(s,0) = f.(F.(s,j0)) by Lm1,BINOP_1:18
    .= f.(P.s) by A1,BORSUK_6:def 11
    .= P1.s by A2,FUNCT_2:15;
  thus G.(s,1) = f.(F.(s,j1)) by Lm1,BINOP_1:18
    .= f.(Q.s) by A1,BORSUK_6:def 11
    .= Q1.s by A3,FUNCT_2:15;
  thus G.(0,s) = f.(F.(j0,s)) by Lm1,BINOP_1:18
    .= f.a by A1,BORSUK_6:def 11;
  thus G.(1,s) = f.(F.(j1,s)) by Lm1,BINOP_1:18
    .= f.b by A1,BORSUK_6:def 11;
end;
