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

theorem
  exp(z+2*PI*<i>) = exp(z)
proof
  z+2*PI*<i> = (Re z+(Im z)*<i>)+(2*PI+0*<i>)*<i> by COMPLEX1:13
    .= Re z + 0 + (Im z + 2*PI)*<i>;
  then
  exp(z+2*PI*<i>) =exp_R.(Re z)*cos.(Im z + 2*PI*1)+exp_R.(Re z)*sin.(Im z
  + 2*PI)*<i> by Th19
    .=exp_R.(Re z)*cos.(Im z)+exp_R.(Re z)*sin.(Im z + 2*PI*1)*<i> by
SIN_COS2:11
    .=exp_R.(Re z)*cos.(Im z)+exp_R.(Re z)*sin.(Im z)*<i> by SIN_COS2:10
    .=exp(Re z+Im z*<i>) by Th19;
  hence thesis by COMPLEX1:13;
end;
