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(PI*2*n+PI)+sin(PI*(2*n)+1*PI)*<i> by SIN_COS:25
    .= 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;
