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 Th37:
  B <> C & A in Line(B,C) implies the_foot_of_the_altitude(A,B,C) = A
  proof
    assume that
A1: B <> C and
A2: A in Line(B,C);
    consider P being Point of TOP-REAL 2 such that
A3: the_foot_of_the_altitude(A,B,C) = P and
A4: the_altitude(A,B,C) /\ Line(B,C) = {P} by A1,Def2;
    consider L1,L2 being Element of line_of_REAL 2 such that
A5: the_altitude(A,B,C) = L1 and
A6: L2 = Line(B,C) and
A7: A in L1 and
    L1 _|_ L2 by A1,Def1;
    A in L1 /\ L2 by A6,A7,A2,XBOOLE_0:def 4;
    hence thesis by A3,A4,A5,A6,TARSKI:def 1;
  end;
