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

theorem
  cos_C/.(z + 2*n*PI) = cos_C/.z
proof
  cos_C/.(z + 2*n*PI) = (exp(<i>*z + <i>*(2*n*PI)) + exp(-<i>*(z + 2*n*PI)
  ))/2 by Def2
    .= (exp(<i>*z) * exp(2*PI*n*<i>) + exp(-<i>*(z + 2*n*PI)))/2 by SIN_COS:23
    .= (exp(<i>*z) * 1 + exp(-<i>*(z + 2*n*PI)))/2 by Th28
    .= (exp(<i>*z) + exp((-<i>)*z + (-<i>)*(2*n*PI)))/2
    .= (exp(<i>*z) + exp((-<i>)*z) * exp((-2*PI*n)*<i>))/2 by SIN_COS:23
    .= (exp(<i>*z)*1 + exp(-<i>*z)*1)/2 by Th29
    .= (exp(<i>*z) + exp(-<i>*z))/2;
  hence thesis by Def2;
end;
