 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
  Th13: B <> C implies (A, B, C are_collinear iff A in Line(B, C))
proof
  assume A1: B <> C;
  hereby
    assume A, B, C are_collinear;
    then per cases by TOPREAL9:67;
    suppose A in LSeg(B, C);
      hence A in Line(B, C) by Th12;
    end;
    suppose B in LSeg(C, A);
      then A2: B in Line(C, A) by Th12;
      A3: C in Line(C, A) & A in Line(C, A) by EUCLID_4:41;
      then Line(C, A) c= Line(B, C) by A2,A1,EUCLID_4:43;
      hence A in Line(B, C) by A3;
    end;
    suppose C in LSeg(A, B);
      then A4: C in Line(A, B) by Th12;
      A5: A in Line(A, B) & B in Line(A, B) by EUCLID_4:41;
      then Line(A, B) c= Line(B, C) by A4,A1,EUCLID_4:43;
      hence A in Line(B, C) by A5;
    end;
  end;
  assume A in Line(B, C);
  then consider lambda such that
  A6: A = (1 - lambda) * B + lambda * C;
  the_area_of_polygon3(A, B, C) =
  (1 - lambda) * the_area_of_polygon3(B, B, C) +
  lambda * the_area_of_polygon3(C, B, C) by Th7,A6;
  hence thesis by Th9;
end;
