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

theorem
  exp((-(2*n+1)*PI)*<i>) = -1
proof
  exp((-(2*n+1)*PI)*<i>) = cos(-(2*n+1)*PI)+sin(-(2*n+1)*PI)*<i> by SIN_COS:25
    .= cos((2*n+1)*PI)+sin(-(2*n+1)*PI)*<i> by SIN_COS:31
    .= cos(PI*2*n+PI)+(-sin(PI*(2*n)+PI))*<i> by SIN_COS:31
    .= cos.(PI*2*n+PI)+-sin(PI*2*n+PI)*<i> by SIN_COS:def 19
    .= cos.(PI*2*n+PI)+-sin.(PI*2*n+PI)*<i> by SIN_COS:def 17
    .= cos.(PI)+-sin.(PI*2*n+PI)*<i> by SIN_COS2:11
    .= -1+-sin.(PI)*<i> by SIN_COS:76,SIN_COS2:10;
  hence thesis by SIN_COS:76;
end;
