
theorem Th8:
  for a being Real, i being Integer holds sin a = sin (2*PI* i+a)
proof
  let r being Real, i being Integer;
A1: sin.r = sin r by SIN_COS:def 17;
A2: sin.(2*PI*i+r) = sin (2*PI*i+r) by SIN_COS:def 17;
A3: sin.(2*PI*(-i)+(2*PI*i+r)) = sin (2*PI*(-i)+(2*PI*i+r)) by SIN_COS:def 17;
  per cases;
  suppose
    i >= 0;
    then reconsider iN = i as Element of NAT by INT_1:3;
    sin r = sin (2*PI*iN+r) by A1,A2,SIN_COS2:10;
    hence thesis;
  end;
  suppose
    i < 0;
    then reconsider iN = -i as Element of NAT by INT_1:3;
    set aa = 2*PI*i+r;
    sin (aa) = sin (2*PI*iN+aa) by A2,A3,SIN_COS2:10;
    hence thesis;
  end;
end;
