reserve n for Nat;
reserve i for Integer;
reserve r,s,t for Real;
reserve An,Bn,Cn,Dn for Point of TOP-REAL n;
reserve L1,L2 for Element of line_of_REAL n;
reserve A,B,C for Point of TOP-REAL 2;
reserve D for Point of TOP-REAL 2;
reserve a,b,c,d for Real;

theorem Th40:
  B <> C implies |(A - the_foot_of_the_altitude(A,B,C),
                   C - the_foot_of_the_altitude(A,B,C))| = 0
  proof
    assume
A1: B <> C;
    then the_foot_of_the_altitude(A,C,B)
           = the_foot_of_the_altitude(A,B,C) by Th34;
    hence thesis by A1,Th39;
  end;
