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

theorem Th29:
  exp((-2*PI*n)*<i>) = 1
proof
  thus exp((-2*PI*n)*<i>) = cos(-2*PI*n)+sin(-2*PI*n)*<i> by SIN_COS:25
    .=cos(2*PI*n)+sin(-(2*PI*n))*<i> by SIN_COS:31
    .=cos(0+2*PI*n)+(-sin(2*PI*n))*<i> by SIN_COS:31
    .=cos.(0+2*PI*n)+-sin(0+2*PI*n)*<i> by SIN_COS:def 19
    .=cos.(0+2*PI*n)+-sin.(0+2*PI*n)*<i> by SIN_COS:def 17
    .=cos.(0+2*PI*n)+-sin.(0)*<i> by SIN_COS2:10
    .= 1 by SIN_COS:30,SIN_COS2:11;
end;
