theorem
  B <> C & B <> A & angle(A,B,C) = PI/2 implies
  the_foot_of_the_altitude(A,B,C) = B
  proof
    assume that
A1: B <> C and
A2: B <> A and
A3: angle(A,B,C) = PI/2;
    |(A-B,C-B)| = 0 by A1,A2,A3,EUCLID_3:45;
    then |(B-A,B-C)| = 0 by Th10;
    hence thesis by A1,Th43;
  end;
