theorem Th51:
  A,B,C is_a_triangle & |(C-A,B-C)| is zero implies
  the_foot_of_the_altitude(A,B,C) = C & the_foot_of_the_altitude(B,C,A) = C
  proof
    assume that
A1: A,B,C is_a_triangle and
A2: |(C-A,B-C)| is zero;
A3: A,B,C are_mutually_distinct by A1,EUCLID_6:20;
A4: - |( C - A, B - C )| = |(C-B,C-A)| by Th14;
    |(C-A,C-B)| = - |(C-A,B-C)| by Th14;
    then the_foot_of_the_altitude(A,C,B) = C by A2,A3,Th43;
    hence thesis by A2,A3,A4,Th43,Th34;
  end;
