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 Th32:
  B <> C implies the_altitude(A,B,C) = the_altitude(A,C,B)
  proof
    assume
A1: B <> C;
    then consider L11,L12 being Element of line_of_REAL 2 such that
A2: the_altitude(A,B,C) = L11 and
A3: L12 = Line(B,C) and
A4: A in L11 and
A5: L11 _|_ L12 by Def1;
    consider L21,L22 being Element of line_of_REAL 2 such that
A6: the_altitude(A,C,B) = L21 and
A7: L22 = Line(C,B) and
A8: A in L21 and
A9: L21 _|_ L22 by A1,Def1;
    L11 // L21 by A5,A9,A3,A7,EUCLID12:16,EUCLIDLP:111;
    hence thesis by A2,A6,A4,A8,XBOOLE_0:3,EUCLIDLP:71;
  end;
