theorem
  A,B,C is_a_triangle & |(B-A,B-C)| <> 0
  implies |.A-B.| * sin angle(A,B,the_foot_of_the_altitude(A,B,C))
    = |.the_foot_of_the_altitude(A,B,C)-A.| or
  |.A-B.| * (- sin angle(A,B,the_foot_of_the_altitude(A,B,C)))
    = |.the_foot_of_the_altitude(A,B,C)-A.|
  proof
    assume that
A1: A,B,C is_a_triangle and
A2: |(B-A,B-C)| <> 0;
    A,B,C are_mutually_distinct by A1,EUCLID_6:20;
    hence thesis by A1,A2,Th67,Th45;
  end;
