theorem Th34:
  B <> C implies
  the_foot_of_the_altitude(A,B,C) = the_foot_of_the_altitude(A,C,B)
  proof
    assume
A1: B <> C;
    consider D being Point of TOP-REAL 2 such that
A2: the_foot_of_the_altitude(A,B,C) = D and
A3: the_altitude(A,B,C) /\ Line(B,C) = {D} by A1,Def2;
    consider E being Point of TOP-REAL 2 such that
A4: the_foot_of_the_altitude(A,C,B) = E and
A5: the_altitude(A,C,B) /\ Line(C,B) = {E} by A1,Def2;
    {D} = {E} by A3,A5,A1,Th32;
    hence thesis by A2,A4,ZFMISC_1:3;
  end;
