theorem Th40:
  B <> C implies |(A - the_foot_of_the_altitude(A,B,C),
                   C - the_foot_of_the_altitude(A,B,C))| = 0
  proof
    assume
A1: B <> C;
    then the_foot_of_the_altitude(A,C,B)
           = the_foot_of_the_altitude(A,B,C) by Th34;
    hence thesis by A1,Th39;
  end;
