
theorem Th9:
  for a being Real, i being Integer holds cos a = cos (2*PI *i+a)
proof
  let r be Real, i being Integer;
A1: cos.r = cos r by SIN_COS:def 19;
A2: cos.(2*PI*i+r) = cos (2*PI*i+r) by SIN_COS:def 19;
A3: cos.(2*PI*(-i)+(2*PI*i+r)) = cos (2*PI*(-i)+(2*PI*i+r)) by SIN_COS:def 19;
  per cases;
  suppose
    i >= 0;
    then reconsider iN = i as Element of NAT by INT_1:3;
    cos r = cos (2*PI*iN+r) by A1,A2,SIN_COS2:11;
    hence thesis;
  end;
  suppose
    i < 0;
    then reconsider iN = -i as Element of NAT by INT_1:3;
    set aa = 2*PI*i+r;
    cos (aa) = cos (2*PI*iN+aa) by A2,A3,SIN_COS2:11;
    hence thesis;
  end;
end;
