 reserve A, B, C, A1, B1, C1, A2, B2, C2, C3 for Point of TOP-REAL 2,
  lambda, mu, nu, alpha, beta, gamma for Real,
  X, Y, Z for Subset of TOP-REAL 2;

theorem
  Th16: (A, B, C is_a_triangle & A1 = (1 - lambda) * B + lambda * C &
  B1 = (1 - mu) * C + mu * A  & mu <> 1) implies ((1 - mu) + lambda * mu <> 0
  iff not Line(A, A1) is_parallel_to Line(B, B1))
proof
  assume that A1: A, B, C is_a_triangle and A2: A1 = (1 - lambda) * B +
  lambda * C and A3: B1 = (1 - mu) * C + mu * A and A4: mu <> 1;
  A5: A <> A1 by A1,A2,Th14;
  B, C, A is_a_triangle by A1;
  then A6: B <> B1 by A3,Th14;
  hereby
    assume (1 - mu) + lambda * mu <> 0;
    then consider C2 such that
    A7: A, A1, C2 are_collinear & B, B1, C2 are_collinear by Lm3,A1,A2,A3;
    C2, A, A1 are_collinear & C2, B, B1 are_collinear by A7;
    then C2 in Line(A, A1) & C2 in Line(B, B1) by A5,A6,Th13;
    hence not Line(A, A1) is_parallel_to Line(B, B1) by XBOOLE_0:3;
  end;
  assume not Line(A, A1) is_parallel_to Line(B, B1);
  then consider C2 being object such that
A8: C2 in Line(A, A1) & C2 in Line(B, B1)
  by XBOOLE_0:3;
   reconsider C2 as Point of TOP-REAL 2 by A8;
  C2, A, A1 are_collinear & C2, B, B1 are_collinear by A8,A5,A6,Th13;
  then A, A1, C2 are_collinear & B, B1, C2 are_collinear;
  hence thesis by Lm2,A1,A2,A3,A4;
end;
