theorem
  Th7: the_area_of_polygon3((1 - lambda) * A + lambda * A1, B, C) =
  (1 - lambda) * the_area_of_polygon3(A, B, C) +
  lambda * the_area_of_polygon3(A1, B, C)
proof
  the_area_of_polygon3((1 - lambda) * A + lambda * A1, B, C) =
  ((((1- lambda) * A `1 + lambda * A1 `1) * B `2
  - B `1 * ((1- lambda) * A + lambda * A1) `2)
  + (B `1 * C `2 - C `1 * B `2) + (C `1 * ((1- lambda) * A + lambda * A1) `2
  - ((1- lambda) * A + lambda * A1) `1 * C `2)) / 2 by Th4
  .= ((((1- lambda) * A `1 + lambda * A1 `1) * B `2 -
  B `1 * ((1- lambda) * A `2 + lambda * A1 `2)) + (B `1 * C `2 - C `1 * B `2)
  + (C `1 * ((1- lambda) * A + lambda * A1) `2
  - ((1- lambda) * A + lambda * A1) `1 * C `2)) / 2 by Th4
  .= ((((1- lambda) * A `1 + lambda * A1 `1) * B `2 -
  B `1 * ((1- lambda) * A `2 + lambda * A1 `2)) + (B `1 * C `2 - C `1 * B `2)
  + (C `1 * ((1- lambda) * A `2 + lambda * A1 `2)
  - ((1- lambda) * A + lambda * A1) `1 * C `2)) / 2 by Th4
  .= ((((1- lambda) * A `1 + lambda * A1 `1) * B `2 -
  B `1 * ((1- lambda) * A `2 + lambda * A1 `2)) + (B `1 * C `2 - C `1 * B `2)
  + (C `1 * ((1- lambda) * A `2 + lambda * A1 `2)
  - ((1- lambda) * A `1 + lambda * A1 `1) * C `2)) / 2 by Th4;
  hence thesis;
end;
