theorem Th9:
  sin(r) = sin(s) implies ex i st r = s + 2*PI*i or r = PI-s + 2*PI*i
  proof
    assume
A1: sin(r) = sin(s);
A2: sin(r) - sin(s) = 2*(cos((r+s)/2)*sin((r-s)/2)) by SIN_COS4:16;
    per cases by A1,A2;
    suppose sin((r-s)/2) = 0;
      hence thesis by Lm5;
    end;
    suppose cos((r+s)/2) = 0;
      then consider i such that
A3:   (r+s)/2 = PI/2+PI*i by Th8;
      r = PI-s+2*PI*i by A3;
      hence thesis;
    end;
  end;
