theorem Th53:
  A,B,C is_a_triangle & 0 < angle(C,B,A) < PI implies
  the_diameter_of_the_circumcircle(A,B,C) > 0
  proof
    assume that
A1: A,B,C is_a_triangle and
A2: 0 < angle(C,B,A) < PI;
    A,B,C are_mutually_distinct by A1,EUCLID_6:20; then
A3: |.C-A.| >=0 & |.C-A.| <> 0 by EUCLID_6:42;
    2 * PI * 0 < angle(C,B,A) & angle(C,B,A) < PI + 2 * PI * 0 by A2;
    then sin(angle(C,B,A)) > 0 & |.C-A.| > 0 by A3,SIN_COS6:11;
    then |.C-A.| / sin(angle(C,B,A)) > 0 by XREAL_1:139;
    hence thesis by A1,EUCLID10:44;
end;
