
theorem Th10:
  for a being Real st sin a = 0 holds cos a <> 0
proof
  let a be Real;
  assume that
A1: sin a = 0 and
A2: cos a = 0;
  consider r being Real such that
A3: r = 2*PI*-[\ a/(2*PI) /]+a and
A4: 0 <= r & r < 2*PI by Th1,COMPTRIG:5;
A5: cos a = cos r by A3,Th9;
  sin a = sin r by A3,Th8;
  then r = 0 or r = PI by A4,A1,COMPTRIG:17;
  hence thesis by A5,A2,COMPTRIG:5,18;
end;
