 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
  Th17: (A1 = (1 - lambda) * B + lambda * C &
  B1 = (1 - mu) * C + mu * A & C1 = (1 - nu) * A + nu * B) implies
  the_area_of_polygon3(A1, B1, C1) = ((1 - lambda) * (1 - mu) * (1 - nu) +
  lambda * mu * nu) * the_area_of_polygon3(A, B, C)
proof
  assume that A1: A1 = (1 - lambda) * B + lambda * C
  and A2: B1 = (1 - mu) * C + mu * A and A3: C1 = (1 - nu) * A + nu * B;
  the_area_of_polygon3(A1, B1, C1) =
  (1 - lambda) * the_area_of_polygon3(B, B1, C1) +
  lambda * the_area_of_polygon3(C, B1, C1) by Th7,A1
  .= - (1 - lambda) * the_area_of_polygon3((1 - mu) * C + mu * A, B, C1) -
  lambda * the_area_of_polygon3((1 - mu) * C + mu * A, C, C1) by A2
  .= - (1 - lambda) * ((1 - mu) * the_area_of_polygon3(C, B, C1) +
  mu * the_area_of_polygon3(A, B, C1)) -
  lambda * the_area_of_polygon3((1 - mu) * C + mu * A, C, C1) by Th7
  .= - (1 - lambda) * ((1 - mu) * the_area_of_polygon3(C, B, C1) +
  mu * the_area_of_polygon3(A, B, C1)) -
  lambda * ((1 - mu) * the_area_of_polygon3(C, C, C1) +
  mu * the_area_of_polygon3(A, C, C1)) by Th7
  .= (1 - lambda) * (1 - mu) * the_area_of_polygon3((1 - nu) * A + nu * B, B,
  C) + (1 - lambda) * mu * the_area_of_polygon3((1 - nu) * A +
  nu * B, B, A) + lambda * mu * the_area_of_polygon3((1 - nu) * A +
  nu * B, C, A) by A3
  .= (1 - lambda) * (1 - mu) * ((1 - nu) * the_area_of_polygon3(A, B, C) +
  nu * the_area_of_polygon3(B, B, C))
  + (1 - lambda) * mu * the_area_of_polygon3((1 - nu) * A +
  nu * B, B, A) + lambda * mu * the_area_of_polygon3((1 - nu) * A +
  nu * B, C, A) by Th7
  .= (1 - lambda) * (1 - mu) * (1 - nu) * the_area_of_polygon3(A, B, C)
  + (1 - lambda) * mu * ((1 - nu) * the_area_of_polygon3(A, B, A)
  + nu * the_area_of_polygon3(B, B, A)) +
  lambda * mu * the_area_of_polygon3((1 - nu) * A + nu * B, C, A) by Th7
  .= (1 - lambda) * (1 - mu) * (1 - nu) * the_area_of_polygon3(A, B, C) +
  lambda * mu * ((1 - nu) * the_area_of_polygon3(A, C, A) +
  nu * the_area_of_polygon3(B, C, A)) by Th7;
  hence thesis;
end;
