reserve x,y for Real;
reserve z,z1,z2 for Complex;
reserve n for Element of NAT;

theorem
  sin_C/.(z + 2*n*PI) = sin_C/.z
proof
  sin_C/.(z + 2*n*PI) = (exp(<i>*z + <i>*(2*n*PI))-(exp(-<i> * (z + 2*n*PI
  ))))/(2 * <i>) by Def1
    .= (exp(<i>*z) * exp(2*PI*n*<i>)-(exp(-<i> * (z + 2*n*PI))))/(2 * <i>)
  by SIN_COS:23
    .= (exp(<i>*z) * 1-(exp(-<i> * (z + 2*n*PI))))/(2 * <i>) by Th28
    .= (exp(<i>*z)-(exp((-<i>)*z + (-<i>)*(2*n*PI))))/(2 * <i>)
    .= (exp(<i>*z)-(exp((-<i>)*z) * exp((-2*PI*n)*<i>)))/(2 * <i>) by
SIN_COS:23
    .= (exp(<i>*z)-(exp((-<i>)*z) * 1))/(2 * <i>) by Th29
    .= (exp(<i>*z) - exp(-<i>*z))/(2*<i>);
  hence thesis by Def1;
end;
