theorem Th47:
  B <> C & |(B-A,B-C)| = 0 implies
  |.the_foot_of_the_altitude(A,B,C)-A.| = |.A-B.|
  proof
    assume that
A1: B <> C and
A2: |(B-A,B-C)| = 0;
    |.the_foot_of_the_altitude(A,B,C)-A.| = |.B-A.| by A1,A2,Th43;
    hence thesis by EUCLID_6:43;
  end;
