 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 = (2 / 3) * B + (1 / 3) * C &
  B1 = (2 / 3) * C + (1 / 3) * A & C1 = (2 / 3) * A + (1 / 3) * B &
  A, A1, B2, C2 are_collinear & B, B1, A2, C2 are_collinear &
  C, C1, A2, B2 are_collinear)
  implies the_area_of_polygon3(A2, B2, C2) = the_area_of_polygon3(A, B, C) / 7
proof
  assume that A1: A, B, C is_a_triangle and
  A2: A1 = (2 / 3) * B + (1 / 3) * C & B1 = (2 / 3) * C + (1 / 3) * A &
  C1 = (2 / 3) * A + (1 / 3) * B and
A3:  A, A1, B2, C2 are_collinear & B, B1, A2, C2 are_collinear &::
  C, C1, A2, B2  are_collinear;
  consider lambda, mu, nu such that
  A4: lambda = (1 / 3) & mu = (1 / 3) & nu = (1 / 3);
  A1 = (1 - lambda) * B + lambda * C & B1 = (1 - mu) * C + mu * A &
  C1 = (1 - nu) * A + nu * B & lambda <> 1 & mu <> 1 & nu <> 1 &
  1 - lambda = 2 / 3 & 1 - mu = 2 / 3 & 1 - nu = 2 / 3 by A2,A4;
  then the_area_of_polygon3(A2, B2, C2) = ((1 / 3) * (1 / 3) * (1 / 3) -
  (2 / 3) * (2 / 3) * (2 / 3)) ^2 / (((2 / 3) + (1 / 3) * (1 / 3)) * ((2 / 3)
  + (1 / 3) * (1 / 3)) * ((2 / 3) + (1 / 3) * (1 / 3))) *
  the_area_of_polygon3(A, B, C) by Th19,A1,A3;
  hence thesis;
end;
