 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
  (A, B, C is_a_triangle & A1 = (1 - lambda) * B + lambda * C & B1 = (1 - mu)
  * C + mu * A & C1 = (1 - nu) * A + nu * B & lambda <> 1 & mu <> 1 & nu <> 1)
  implies ((lambda / (1 - lambda)) * (mu / (1 - mu)) * (nu / (1 - nu)) = 1 iff
  Line(A, A1), Line(B, B1), Line(C, C1) are_concurrent)
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: C1 = (1 - nu) * A +
  nu * B and A5: lambda <> 1 and A6: mu <> 1 and A7: nu <> 1;
  A8: A <> A1 by Th14,A1,A2;
  A9: B, C, A is_a_triangle by A1;
  then A10: B <> B1 by Th14,A3;
  A11: C, A, B is_a_triangle by A1;
  then A12: C <> C1 by Th14,A4;
  hereby
    assume A13:
    (lambda / (1 - lambda)) * (mu / (1 - mu)) * (nu / (1 - nu)) = 1;
    per cases;
    suppose 1 - mu + lambda * mu <> 0 & 1 - lambda + nu * lambda <> 0 &
      1 - nu + mu * nu <> 0;
      then consider A2 such that A14: A, A1, A2 are_collinear & B, B1, A2
      are_collinear & C, C1, A2 are_collinear by A13,Th21,A1,A2,A3,A4,A5,A6,A7;
      A2, A, A1 are_collinear & A2, B, B1 are_collinear & A2, C, C1
      are_collinear by A14;
      then A2 in Line(A, A1) & A2 in Line(B, B1) & A2 in Line(C, C1) by Th13,A8
,A10,A12;
      hence Line(A, A1), Line(B, B1), Line(C, C1) are_concurrent;
    end;
    suppose not (1 - mu + lambda * mu <> 0 & 1 - lambda + nu * lambda <> 0 &
      1 - nu + mu * nu <> 0);
      then Line(A, A1) is_parallel_to Line(B, B1) & Line(B, B1)
      is_parallel_to Line(C, C1) & Line(C, C1) is_parallel_to Line(A, A1)
      by A13,Lm6,A1,A2,A3,A4,A5,A6,A7;
      hence Line(A, A1), Line(B, B1), Line(C, C1) are_concurrent;
    end;
  end;
  assume Line(A, A1), Line(B, B1), Line(C, C1) are_concurrent;
  then per cases;
  suppose A15: Line(A, A1) is_parallel_to Line(B, B1) & Line(B, B1)
    is_parallel_to Line(C, C1) & Line(C, C1) is_parallel_to Line(A, A1);
    then 1 - mu + lambda * mu = 0 & 1 - nu + mu * nu = 0 & 1 - lambda + nu *
    lambda = 0 by Th16,A1,A9,A11,A2,A3,A4,A5,A6,A7;
    hence thesis by Lm6,A15,A1,A2,A3,A4,A5,A6,A7;
  end;
  suppose ex C2 st (C2 in Line(A, A1) & C2 in Line(B, B1) &
    C2 in Line(C, C1));
    then consider C2 such that A16: C2 in Line(A, A1) & C2 in Line(B, B1) &
    C2 in Line(C, C1);
    not Line(A, A1) is_parallel_to Line(B, B1) & not Line(B, B1)
    is_parallel_to Line(C, C1) & not Line(C, C1) is_parallel_to Line(A, A1)
    by A16,XBOOLE_0:3;
    then A17: 1 - mu + lambda * mu <> 0 & 1 - nu + mu * nu <> 0 & 1 - lambda +
    nu * lambda <> 0 by Th16,A1,A2,A3,A4,A5,A6,A7,A9,A11;
    C2, A, A1 are_collinear & C2, B, B1 are_collinear & C2, C, C1 are_collinear
    by A16,Th13,A8,A10,A12;
    then A, A1, C2 are_collinear & B, B1, C2 are_collinear & C, C1, C2
    are_collinear;
    hence thesis by A17,Th21,A1,A2,A3,A4,A5,A6,A7;
  end;
end;
