theorem
  B <> C implies
  the_length_of_the_altitude(A,B,C) = the_length_of_the_altitude(A,C,B)
  proof
    assume
A1: B <> C;
    then the_length_of_the_altitude(A,B,C)
             = |.A - the_foot_of_the_altitude(A,B,C).| by Def3
            .= |.A - the_foot_of_the_altitude(A,C,B).| by A1,Th34
            .= the_length_of_the_altitude(A,C,B) by A1,Def3;
    hence thesis;
  end;
