 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
  Th14: A, B, C is_a_triangle & A1 = (1 - lambda) * B + lambda * C implies
  A <> A1
proof
  assume that A1: A, B, C is_a_triangle and A2: A1 = (1 - lambda) * B + lambda
  * C;
  A, B, C are_mutually_distinct by A1,EUCLID_6:20;
  then A3: A <> B & A <> C & B <> C by ZFMISC_1:def 5;
  hereby
    assume A = A1;
    then A in Line(B, C) by A2;
    hence contradiction by A3,Th13,A1;
  end;
end;
