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 Th42:
  B <> C & D in Line(B,C) & D <> C implies
  the_foot_of_the_altitude(A,B,C) = the_foot_of_the_altitude(A,D,C)
  proof
    assume that
A1: B <> C and
A2: D in Line(B,C) and
A3: D <> C;
    consider P1 being Point of TOP-REAL 2 such that
A4: the_foot_of_the_altitude(A,B,C) = P1 and
A5: the_altitude(A,B,C) /\ Line(B,C) = {P1} by A1,Def2;
    consider P2 being Point of TOP-REAL 2 such that
A6: the_foot_of_the_altitude(A,D,C) = P2 and
A7: the_altitude(A,D,C) /\ Line(D,C) = {P2} by A3,Def2;
    Line(D,C) = Line(B,C) by A2,A3,Th7;
    then {P1} = {P2} by A7,A5,A1,A2,A3,Th33;
    hence thesis by A4,A6,ZFMISC_1:3;
  end;
