
theorem
  for A,B,C be Point of TOP-REAL 2 st
  A,B,C is_a_triangle & angle(B,A,C) = PI / 2 holds
   tan angle(A,C,B) = |.A-B.| / |.A-C.| &
   tan angle(C,B,A) = |.A-C.| / |.A-B.|
   proof
     let A,B,C be Point of TOP-REAL 2 such that
A1: A,B,C is_a_triangle and
A2: angle(B,A,C) = PI/2;
A3: A,B,C are_mutually_distinct by A1,EUCLID_6:20;
     |.C-B.| * sin angle (A,C,B)=|.A-B.| &
     |.C-B.| * cos angle (A,C,B)=|.A-C.| by A1,A2,Thm21;
     then (|.C-B.|/|.C-B.|) * sin angle (A,C,B) / cos angle (A,C,B) =
     |.A-B.|/|.A-C.| by XCMPLX_1:83; then
A4:  1 * sin angle (A,C,B) / cos angle (A,C,B) =
     |.A-B.|/|.A-C.| by A3,EUCLID_6:42,XCMPLX_1:60;
     hence  tan angle(A,C,B) = |.A-B.| / |.A-C.| by SIN_COS4:def 1;
     angle(B,A,C)+angle(A,C,B)+angle(C,B,A)=PI by A2,A3,COMPTRIG:5,EUCLID_3:47;
     then tan angle(C,B,A)
     =(sin (PI/2-angle(A,C,B))) / cos (PI/2-angle(A,C,B)) by A2,SIN_COS4:def 1
     .=cos (angle(A,C,B)) / cos (PI/2-angle(A,C,B)) by SIN_COS:79
     .=cos angle(A,C,B) / sin angle(A,C,B) by SIN_COS:79
     .=1/(sin angle(A,C,B) / cos angle(A,C,B)) by XCMPLX_1:57
     .=|.A-C.|/|.A-B.| by A4,XCMPLX_1:57;
     hence thesis;
  end;
