 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
  Th19: (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 &
  A, A1, B2, C2 are_collinear & B, B1, A2, C2 are_collinear &
  C, C1, A2, B2 are_collinear)
  implies (((1 - mu) + lambda * mu) * ((1 - lambda) +
  nu * lambda) * ((1 - nu) + mu * nu)) <> 0 & the_area_of_polygon3(A2, B2, C2)
  = ((mu * nu * lambda - (1 - mu) * (1 - nu) * (1 - lambda)) ^2 / (((1 - mu) +
  lambda * mu) * ((1 - lambda) + nu * lambda) * ((1 - nu) + mu * nu))) *
  the_area_of_polygon3(A, B, C)
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 and
A8: A, A1, B2, C2 are_collinear and
A9: B, B1, A2, C2 are_collinear and
A10: C, C1, A2, B2 are_collinear;
A11: A, A1, C2 are_collinear by A8;
A12: B, B1, C2 are_collinear by A9;
A13: B, B1, A2 are_collinear by A9;
A14: C, C1, A2 are_collinear by A10;
A15: A, A1, B2 are_collinear by A8;
A16: C, C1, B2 are_collinear by A10;
  A17: C, A, B is_a_triangle by A1;
  A18: B, C, A is_a_triangle by A1;
  set q1 = ((1 - mu) + lambda * mu);
  set q2 = ((1 - lambda) + nu * lambda);
  set q3 = ((1 - nu) + mu * nu);
  consider alpha such that A19: C2 = (1 - alpha) * A + alpha * A1 and
  A20: alpha = (1 - mu) / q1 by Lm2,A1,A2,A3,A6,A11,A12;
  A21: q1 <> 0 by Lm2,A1,A2,A3,A6,A11,A12;
  consider beta such that A22: B2 = (1 - beta) * C + beta * C1 and
  A23: beta = (1 - lambda) / q2 by Lm2,A17,A4,A2,A5,A16,A15;
  A24: q2 <> 0 by Lm2,A17,A4,A2,A5,A16,A15;
  consider gamma such that A25: A2 = (1 - gamma) * B + gamma * B1 and
  A26: gamma = (1 - nu) / q3 by Lm2,A18,A3,A4,A7,A13,A14;
  A27: q3 <> 0 by Lm2,A18,A3,A4,A7,A13,A14;
  1 - alpha = (1 * q1 - (1 - mu)) /q1 by A20,A21,XCMPLX_1:127;
  then A28: 1 - alpha = lambda * mu * (1 / q1) by XCMPLX_1:99;
  1 - beta = (1 * q2 - (1 - lambda)) /q2 by A23,A24,XCMPLX_1:127;
  then A29: 1 - beta = nu * lambda * (1 / q2) by XCMPLX_1:99;
  1 - gamma = (1 * q3 - (1 - nu)) /q3 by A26,A27,XCMPLX_1:127;
  then A30: 1 - gamma = mu * nu * (1 / q3) by XCMPLX_1:99;
  A31: alpha = (1 - mu) * (1 / q1) by A20,XCMPLX_1:99;
  A32: beta = (1 - lambda) * (1 / q2) by A23,XCMPLX_1:99;
  A33: gamma = (1 - nu) * (1 / q3) by A26,XCMPLX_1:99;
  thus (q1 * q2 * q3) <> 0 by A21,A24,A27;
  set S = the_area_of_polygon3(A, B, C);
  the_area_of_polygon3(A2, B2, C2)
  = (1 - gamma) * the_area_of_polygon3(B, B2, C2) +
  gamma * the_area_of_polygon3((1 - mu) * C + mu * A, B2, C2) by A3,Th7,A25
  .= (1 - gamma) * the_area_of_polygon3(B, B2, C2) + gamma * ((1 - mu) *
  the_area_of_polygon3(C, B2, C2) + mu * the_area_of_polygon3(A, B2, C2))
  by Th7
  .= - (1 - gamma) * the_area_of_polygon3(C2, B2, B) - gamma * ((1 - mu) *
  the_area_of_polygon3((1 - beta) * C + beta * C1, C, C2) + mu *
  the_area_of_polygon3(C2, B2, A)) by A22
  .= - (1 - gamma) * the_area_of_polygon3(C2, B2, B) - gamma * ((1 - mu) *
  ((1 - beta) * the_area_of_polygon3(C, C, C2) + beta *
  the_area_of_polygon3(C1, C, C2)) + mu * the_area_of_polygon3(C2, B2, A))
  by Th7
  .= - (1 - gamma) * the_area_of_polygon3((1 - alpha) * A + alpha * A1, B2,
  B) - gamma * (1 - mu) * beta * the_area_of_polygon3(C1, C, C2) - gamma *
  mu * the_area_of_polygon3(C2, B2, A) by A19
  .= - (1 - gamma) * ((1 - alpha) * the_area_of_polygon3(A, B2, B) + alpha *
  the_area_of_polygon3((1 - lambda) * B + lambda * C, B2, B)) - gamma *
  (1 - mu) * beta * the_area_of_polygon3(C1, C, C2) - gamma * mu *
  the_area_of_polygon3(C2, B2, A) by A2,Th7
  .= - (1 - gamma) * ((1 - alpha) * the_area_of_polygon3(A, B2, B) + alpha *
  ((1 - lambda) * the_area_of_polygon3(B, B2, B) + lambda *
  the_area_of_polygon3(C, B2, B))) - gamma * (1 - mu) * beta *
  the_area_of_polygon3(C1, C, C2) - gamma * mu *
  the_area_of_polygon3(C2, B2, A) by Th7
  .= - (1 - gamma) * ((1 - alpha) * the_area_of_polygon3(A, B2, B) - alpha *
  lambda * the_area_of_polygon3((1 - beta) * C + beta * C1, C, B)) - gamma *
  (1 - mu) * beta * the_area_of_polygon3(C1, C, C2) - gamma * mu *
  the_area_of_polygon3(C2, B2, A) by A22
  .= - (1 - gamma) * ((1 - alpha) * the_area_of_polygon3(A, B2, B) -
  alpha * lambda * ((1 - beta) * the_area_of_polygon3(C, C, B) +
  beta * the_area_of_polygon3(C1, C, B)))
  - gamma * (1 - mu) * beta * the_area_of_polygon3(C1, C, C2)
  - gamma * mu * the_area_of_polygon3(C2, B2, A) by Th7
  .= - (1 - gamma) * ((1 - alpha) * the_area_of_polygon3(A, B2, B) -
  alpha * lambda * beta * the_area_of_polygon3((1 - nu) * A + nu * B, C, B))
  - gamma * (1 - mu) * beta * the_area_of_polygon3(C1, C, C2)
  - gamma * mu * the_area_of_polygon3(C2, B2, A) by A4
  .= - (1 - gamma) * ((1 - alpha) * the_area_of_polygon3(A, B2, B) -
  alpha * lambda * beta * ((1 - nu) * the_area_of_polygon3(A, C, B) +
  nu * the_area_of_polygon3(B, C, B)))
  - gamma * (1 - mu) * beta * the_area_of_polygon3(C1, C, C2)
  - gamma * mu * the_area_of_polygon3(C2, B2, A) by Th7
  .= (1 - gamma) * ((1 - alpha) * the_area_of_polygon3((1 - beta) * C +
  beta * C1, A, B) - alpha * lambda * beta * (1 - nu) * S)
  - gamma * (1 - mu) * beta * the_area_of_polygon3(C1, C, C2)
  - gamma * mu * the_area_of_polygon3(C2, B2, A) by A22
  .= (1 - gamma) * ((1 - alpha) * ((1 - beta) * the_area_of_polygon3(C, A, B)
  + beta * the_area_of_polygon3(C1, A, B)) - alpha * lambda * beta * (1 - nu)
  * S) - gamma * (1 - mu) * beta * the_area_of_polygon3(C1, C, C2) - gamma *
  mu * the_area_of_polygon3(C2, B2, A) by Th7
  .= (1 - gamma) * ((1 - alpha) * ((1 - beta) * S
  + beta * ((1 - nu) * the_area_of_polygon3(A, A, B) +
  nu * the_area_of_polygon3(B, A, B))) -
  alpha * lambda * beta * (1 - nu) * S)
  - gamma * (1 - mu) * beta * the_area_of_polygon3(C1, C, C2)
  - gamma * mu * the_area_of_polygon3(C2, B2, A) by Th7,A4
  .= ((1 - gamma) * (1 - alpha) * (1 - beta) * S -
  (1 - gamma) * alpha * lambda * beta * (1 - nu) * S)
  - gamma * (1 - mu) * beta * ((1 - nu) * the_area_of_polygon3(A, C, C2) +
  nu * the_area_of_polygon3(B, C, C2))
  - gamma * mu * the_area_of_polygon3(C2, B2, A) by Th7,A4
  .= ((1 - gamma) * (1 - alpha) * (1 - beta) * S -
  (1 - gamma) * alpha * lambda * beta * (1 - nu) * S)
  - gamma * (1 - mu) * beta * ((1 - nu) * the_area_of_polygon3((1 - alpha) *
  A + alpha * A1, A, C) + nu * the_area_of_polygon3(C2, B, C))
  - gamma * mu * the_area_of_polygon3(C2, B2, A) by A19
  .= ((1 - gamma) * (1 - alpha) * (1 - beta) * S -
  (1 - gamma) * alpha * lambda * beta * (1 - nu) * S)
  - gamma * (1 - mu) * beta * ((1 - nu) * ((1 - alpha) *
  the_area_of_polygon3(A, A, C) + alpha * the_area_of_polygon3(A1, A, C)) +
  nu * the_area_of_polygon3(C2, B, C))
  - gamma * mu * the_area_of_polygon3(C2, B2, A) by Th7
  .= ((1 - gamma) * (1 - alpha) * (1 - beta) * S -
  (1 - gamma) * alpha * lambda * beta * (1 - nu) * S)
  - gamma * (1 - mu) * beta * ((1 - nu) * alpha *
  the_area_of_polygon3((1 - lambda) * B + lambda * C, A, C) +
  nu * the_area_of_polygon3(C2, B, C))
  - gamma * mu * the_area_of_polygon3(C2, B2, A) by A2
  .= ((1 - gamma) * (1 - alpha) * (1 - beta) * S -
  (1 - gamma) * alpha * lambda * beta * (1 - nu) * S)
  - gamma * (1 - mu) * beta * ((1 - nu) * alpha * (
  (1 - lambda) * the_area_of_polygon3(B, A, C) +
  lambda * the_area_of_polygon3(C, A, C)) +
  nu * the_area_of_polygon3(C2, B, C))
  - gamma * mu * the_area_of_polygon3(C2, B2, A) by Th7
  .= ((1 - gamma) * (1 - alpha) * (1 - beta) * S -
  (1 - gamma) * alpha * lambda * beta * (1 - nu) * S)
  - gamma * (1 - mu) * beta * (- (1 - nu) * alpha * (1 - lambda) * S +
  nu * ((1 - alpha) * S +
  alpha * the_area_of_polygon3((1 - lambda) * B + lambda * C, B, C)))
  - gamma * mu * the_area_of_polygon3(C2, B2, A) by A2,Th7,A19
  .= ((1 - gamma) * (1 - alpha) * (1 - beta) * S -
  (1 - gamma) * alpha * lambda * beta * (1 - nu) * S)
  - gamma * (1 - mu) * beta * (- (1 - nu) * alpha * (1 - lambda) * S +
  nu * ((1 - alpha) * S +
  alpha * ((1 - lambda) * the_area_of_polygon3(B, B, C) +
  lambda * the_area_of_polygon3(C, B, C))))
  - gamma * mu * the_area_of_polygon3(C2, B2, A) by Th7
  .= ((1 - gamma) * (1 - alpha) * (1 - beta) * S -
  (1 - gamma) * alpha * lambda * beta * (1 - nu) * S)
  + (gamma * (1 - mu) * beta * (1 - nu) * alpha * (1 - lambda) * S -
  gamma * (1 - mu) * beta * nu * (1 - alpha) * S) - gamma * mu * (
  (1 - alpha) * the_area_of_polygon3(A, B2, A) +
  alpha * the_area_of_polygon3(A1, B2, A)) by Th7,A19
  .= ((1 - gamma) * (1 - alpha) * (1 - beta) * S -
  (1 - gamma) * alpha * lambda * beta * (1 - nu) * S)
  + (gamma * (1 - mu) * beta * (1 - nu) * alpha * (1 - lambda) * S -
  gamma * (1 - mu) * beta * nu * (1 - alpha) * S) - gamma * mu * alpha *
  the_area_of_polygon3((1 - lambda) * B + lambda * C, B2, A) by A2
  .= ((1 - gamma) * (1 - alpha) * (1 - beta) * S -
  (1 - gamma) * alpha * lambda * beta * (1 - nu) * S)
  + (gamma * (1 - mu) * beta * (1 - nu) * alpha * (1 - lambda) * S -
  gamma * (1 - mu) * beta * nu * (1 - alpha) * S) - gamma * mu * alpha * (
  (1 - lambda) * the_area_of_polygon3(B, B2, A) + lambda *
  the_area_of_polygon3(C, B2, A)) by Th7
  .= ((1 - gamma) * (1 - alpha) * (1 - beta) * S -
  (1 - gamma) * alpha * lambda * beta * (1 - nu) * S)
  + (gamma * (1 - mu) * beta * (1 - nu) * alpha * (1 - lambda) * S -
  gamma * (1 - mu) * beta * nu * (1 - alpha) * S) - gamma * mu * alpha * (
  (1 - lambda) * the_area_of_polygon3((1 - beta) * C + beta * C1, A, B)
  + lambda * the_area_of_polygon3(B2, A, C)) by A22
  .= ((1 - gamma) * (1 - alpha) * (1 - beta) * S -
  (1 - gamma) * alpha * lambda * beta * (1 - nu) * S)
  + (gamma * (1 - mu) * beta * (1 - nu) * alpha * (1 - lambda) * S -
  gamma * (1 - mu) * beta * nu * (1 - alpha) * S) - gamma * mu * alpha * (
  (1 - lambda) * ((1 - beta) * the_area_of_polygon3(C, A, B) +
  beta * the_area_of_polygon3(C1, A, B))
  + lambda * the_area_of_polygon3(B2, A, C)) by Th7
  .= ((1 - gamma) * (1 - alpha) * (1 - beta) * S -
  (1 - gamma) * alpha * lambda * beta * (1 - nu) * S)
  + (gamma * (1 - mu) * beta * (1 - nu) * alpha * (1 - lambda) * S -
  gamma * (1 - mu) * beta * nu * (1 - alpha) * S) - gamma * mu * alpha * (
  (1 - lambda) * ((1 - beta) * S + beta * ((1 - nu) *
  the_area_of_polygon3(A, A, B) + nu * the_area_of_polygon3(B, A, B))) +
  lambda * the_area_of_polygon3(B2, A, C)) by Th7,A4
  .= ((1 - gamma) * (1 - alpha) * (1 - beta) * S - (1 - gamma) * alpha *
  lambda * beta * (1 - nu) * S) + (gamma * (1 - mu) * beta * (1 - nu) * alpha
  * (1 - lambda) * S - gamma * (1 - mu) * beta * nu * (1 - alpha) * S) -
  gamma * mu * alpha * ((1 - lambda) * (1 - beta) * S + lambda * (
  (1 - beta) * the_area_of_polygon3(C, A, C) + beta *
  the_area_of_polygon3(C1, A, C))) by Th7,A22
  .= ((1 - gamma) * (1 - alpha) * (1 - beta) * S - (1 - gamma) * alpha *
  lambda * beta * (1 - nu) * S) + (gamma * (1 - mu) * beta * (1 - nu) * alpha
  * (1 - lambda) * S - gamma * (1 - mu) * beta * nu * (1 - alpha) * S) -
  gamma * mu * alpha * ((1 - lambda) * (1 - beta) * S + lambda * beta * (
  the_area_of_polygon3((1 - nu) * A + nu * B, A, C))) by A4
  .= ((1 - gamma) * (1 - alpha) * (1 - beta) * S - (1 - gamma) * alpha *
  lambda * beta * (1 - nu) * S) + (gamma * (1 - mu) * beta * (1 - nu) * alpha
  * (1 - lambda) * S - gamma * (1 - mu) * beta * nu * (1 - alpha) * S) -
  gamma * mu * alpha * ((1 - lambda) * (1 - beta) * S + lambda * beta * ((1 -
  nu) * the_area_of_polygon3(A, A, C) + nu * the_area_of_polygon3(B, A, C)))
  by Th7
  .= (((1 - gamma) * (1 - alpha) * (1 - beta) - (1 - gamma) * alpha * lambda
  * beta * (1 - nu)) + (gamma * (1 - mu) * beta * (1 - nu) * alpha * (1 -
  lambda) - gamma * (1 - mu) * beta * nu * (1 - alpha)) + (gamma * mu * alpha
  * lambda * beta * nu - gamma * mu * alpha * (1 - lambda) * (1 - beta))) *
  S
  .= (((mu * nu * (1 / q3)) * (lambda * mu * (1 / q1)) * (nu * lambda * (1 /
  q2)) - (mu * nu * (1 / q3)) * ((1 - mu) * (1 / q1)) * lambda * ((1 - lambda
  ) * (1 / q2)) * (1 - nu)) + (((1 - nu) * (1 / q3)) * (1 - mu) * ((1 -
  lambda) * (1 / q2)) * (1 - nu) * ((1 - mu) * (1 / q1)) * (1 - lambda) - ((1
  - nu) * (1 / q3)) * (1 - mu) * ((1 - lambda) * (1 / q2)) * nu * (lambda *
  mu * (1 / q1))) + (((1 - nu) * (1 / q3)) * mu * ((1 - mu) * (1 / q1)) *
  lambda * ((1 - lambda) * (1 / q2)) * nu - ((1 - nu) * (1 / q3)) * mu * ((1
  - mu) * (1 / q1)) * (1 - lambda) * (nu * lambda * (1 / q2)))) * S
  by A28,A31,A29,A32,A30,A33
  .= (mu * nu * lambda - (1 - mu) * (1 - nu) * (1 - lambda)) ^2 *
  ((1 / q1) * (1 / q2) * (1 / q3)) * S
  .= (mu * nu * lambda - (1 - mu) * (1 - nu) * (1 - lambda)) ^2 *
  ((1 / (q1 * q2)) * (1 / q3)) * S by XCMPLX_1:102
  .= (mu * nu * lambda - (1 - mu) * (1 - nu) * (1 - lambda)) ^2 *
  (1 / (q1 * q2 * q3)) * S by XCMPLX_1:102;
  hence thesis by XCMPLX_1:99;
end;
