theorem Th35:
  B <> C implies the_foot_of_the_altitude(A,B,C) in Line(B,C) &
  the_foot_of_the_altitude(A,B,C) in the_altitude(A,B,C)
  proof
    assume B <> C;
    then consider P be Point of TOP-REAL 2 such that
A1: P = the_foot_of_the_altitude(A,B,C) and
A2: the_altitude(A,B,C) /\ Line(B,C) = {P} by Def2;
    P in {P} by TARSKI:def 1;
    hence thesis by A1,A2,XBOOLE_0:def 4;
  end;
