theorem Th65:
  A,C,B is_a_triangle & angle(C,A,B) = PI/2 implies
  the_length_of_the_altitude(C,A,B) = |.A-B.| * tan angle(A,B,C)
  proof
    assume that
A1: A,C,B is_a_triangle and
A2: angle(C,A,B)=PI/2;
A3: A,C,B are_mutually_distinct by A1,EUCLID_6:20;
    then |(B-A,C-A)| = 0 by A2,EUCLID_3:45;
    then
A4: |(A-B,A-C)| = 0 by Th10;
    tan angle(A,B,C) * |.A-B.| = |.A-C.|/|.A-B.| * |.A-B.|
                                         by A1,A2,EUCLID10:35
                              .= |.A-C.| by A3,EUCLID_6:42,XCMPLX_1:87
                              .= |.C-A.| by EUCLID_6:43;
    then tan angle(A,B,C)*|.A-B.|
             = |.the_foot_of_the_altitude(C,A,B) - C.| by A4,A3,Th47
            .= |.C - the_foot_of_the_altitude(C,A,B).| by EUCLID_6:43;
    hence thesis by A3,Def3;
  end;
