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
  A,C,B is_a_triangle & angle(A,C,B) < PI & D,A,C is_a_triangle &
  angle(A,D,C)=PI/2 & A in LSeg(B,D) & A <> D implies
  |.D-C.| = |.A-B.| * sin angle(C,B,A) / sin (angle(C,A,D) - angle(C,B,A))
                    * sin angle(C,A,D)
  proof
    assume that
A1: A,C,B is_a_triangle and
A2: angle(A,C,B) < PI and
A3: D,A,C is_a_triangle and
A4: angle(A,D,C)=PI/2 and
A5: A in LSeg(B,D) and
A6: A <> D;
    A,C,B are_mutually_distinct by A1,EUCLID_6:20;
    then angle(B,A,C) + angle(C,A,D) = PI or
    angle(B,A,C) + angle(C,A,D) = 3*PI by A5,A6,EUCLID_6:13;
    then sin (angle(B,A,C) + angle(C,B,A))
             = sin (PI - (angle(C,A,D) - angle(C,B,A))) or
         sin (angle(B,A,C) + angle(C,B,A))
             = sin (2*PI*1 + (PI - (angle(C,A,D) - angle(C,B,A))));
    then sin (angle(B,A,C) + angle(C,B,A))
             = sin (PI - (angle(C,A,D) - angle(C,B,A))) or
         sin (angle(B,A,C) + angle(C,B,A))
             = sin (PI - (angle(C,A,D) - angle(C,B,A))) by COMPLEX2:8;
    then sin (angle(B,A,C) + angle(C,B,A))
             = sin (angle(C,A,D) - angle(C,B,A)) by EUCLID10:1;
    hence thesis by A1,A2,A3,A4,Th70;
  end;
