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Split-Isa-AF-MMA

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The imaginary part of the factorial of any natural number is zero.
If $f$ is a function from a set $A$ to the set of real numbers, then the real part of the product of $f$ over $A$ is equal to the product of the real parts of $f$ over $A$.
The complex number $a + bi$ is equal to $a + \iota b$.
Every complex number $a$ can be written as $a = \text{Re}(a) + i \cdot \text{Im}(a)$.
Every complex-valued function $f$ is equal to the function $x \mapsto \mathrm{Re}(f(x)) + i \mathrm{Im}(f(x))$.
$i^2 = -1$.
$\i^2 = -1$.
The inverse of $\i$ is $-\i$.
$\frac{x}{i} = -i \cdot x$
$i^2 = -1$.
The imaginary unit $\i$ is not zero.
$i$ is not equal to $1$.
The complex number $\i$ is not a numeral.
$i$ is not equal to $-1$.
Every complex number can be written as $r(\cos a + i \sin a)$ for some real numbers $r$ and $a$.
$i^{2n} = (-1)^n$.
The real part of $\i z$ is $-\text{Im}(z)$.
The imaginary part of $\iota z$ is equal to the real part of $z$.
$\i * w = z$ if and only if $w = -\i * z$.
For any complex number $z$ and any natural number $n$, we have $z / (n \cdot i) = -i \cdot z / n$.
If $y$ and $x$ are real numbers, then $\i y = x$ if and only if $x=0$ and $y=0$.
If $x$ and $y$ are real numbers, then $x = iy$ if and only if $x = 0$ and $y = 0$.
The norm of $\i$ is $1$.
The complex number $e^{i\theta}$ has absolute value $1$.
The modulus of a complex number in polar form is equal to the absolute value of its radius.
The real part of a complex number is less than or equal to its modulus.
$|-x| \leq |x|$
The absolute value of the sum of two complex numbers is less than or equal to the sum of their absolute values.
The absolute value of the real part of a complex number is less than or equal to the modulus of the complex number.
The absolute value of the imaginary part of a complex number is less than or equal to the complex modulus of the number.
The modulus of a complex number is less than or equal to the sum of the absolute values of its real and imaginary parts.
If $z$ is a real number, then $|z| = |\Re(z)|$.
If $z$ is purely imaginary, then $|z| = |\Im(z)|$.
The square of the absolute value of a complex number is the sum of the squares of its real and imaginary parts.
$\lvert z \rvert + \Re(z) \leq 0$ if and only if $\Re(z) = -\lvert z \rvert$.
If $x$ and $y$ are complex numbers with the same imaginary part, then $|x| \leq |y|$ if and only if $|\Re(x)| \leq |\Re(y)|$.
If two complex numbers have the same real part, then their absolute values are equal if and only if their imaginary parts have the same absolute value.
If the real part of a complex number is equal to its modulus, then the imaginary part is zero.
If $P(x, x^2)$ holds for all nonnegative $x$, then $P(\lvert x \rvert, x^2)$ holds for all $x$.
The absolute value of the real part of a complex number plus the absolute value of the imaginary part of a complex number is less than or equal to the square root of 2 times the norm of the complex number.
If $z \neq 0$, then $(\text{Re}(z) / |z|)^2 + (\text{Im}(z) / |z|)^2 = 1$.
The signum function of a complex number is equal to the complex number divided by its modulus.
The real part of the signum function is the real part of the argument divided by the modulus of the argument.
The imaginary part of the signum function is the imaginary part of the argument divided by the modulus of the argument.
The real part of a complex number is a bounded linear function.
The imaginary part of a complex number is a bounded linear function.
The real part of a complex number is a bounded linear function.
The imaginary part of a complex number is a bounded linear function.
If $f$ is a bounded linear operator on a Banach space $X$, then $\Re(f(x)) \to \Re(f(y))$ as $x \to y$.
If $f$ is a bounded linear operator, then $\lim_{n \to \infty} f(x_n) = f(\lim_{n \to \infty} x_n)$.
The real part of a complex number is continuous.
The imaginary part of a complex number is continuous.
The real part of a complex number is continuous.
The imaginary part of a complex number is continuous.
The real part of a complex number is continuous.
The imaginary part of a complex number is continuous.
The real part of a complex number is a bounded linear function.
The imaginary part of a complex number is a bounded linear function.
The real part of a sum of complex numbers is the sum of the real parts.
The sum of the imaginary parts of a bounded linear operator is equal to the imaginary part of the sum of the operator.
If $f$ and $g$ tend to $a$ and $b$, respectively, then the function $x \mapsto (f(x), g(x))$ tends to $(a, b)$.
A complex-valued function $f$ converges to a complex number $x$ if and only if the real and imaginary parts of $f$ converge to the real and imaginary parts of $x$, respectively.
A complex-valued function is continuous if and only if its real and imaginary parts are continuous.
The function $x \mapsto \mathrm{Re}(x)$ is continuous if and only if the function $x \mapsto \mathrm{Im}(x)$ is continuous.
The function $f(x) = x$ is continuous on any set $S$.
A complex-valued function $f$ has a complex derivative $x$ at $a$ if and only if the real and imaginary parts of $f$ have real derivatives $Re(x)$ and $Im(x)$ at $a$, respectively.
If $f$ is differentiable at $x$, then $\Re(f)$ is differentiable at $x$ and $\Re(f)'(x) = \Re(f'(x))$.
If $f$ is differentiable at $x$, then $\operatorname{Im}(f)$ is differentiable at $x$ and $\operatorname{Im}(f)'(x) = \operatorname{Im}(f'(x))$.
The complex conjugate of a complex number is unique.
The complex conjugate of the complex conjugate of $z$ is $z$.
The complex conjugate of $0$ is $0$.
The complex conjugate of $z$ is zero if and only if $z$ is zero.
The complex conjugate of $z$ is $1$ if and only if $z$ is $1$.
The complex conjugate of the sum of two complex numbers is the sum of the complex conjugates of the two complex numbers.
The complex conjugate of a sum is the sum of the complex conjugates.
The complex conjugate of the difference of two complex numbers is the difference of the complex conjugates of the two complex numbers.
The complex conjugate of a negative number is the negative of the complex conjugate of the number.
The complex conjugate of $1$ is $1$.
The complex conjugate of a product is the product of the complex conjugates.
The complex conjugate of a product is the product of the complex conjugates.
The complex conjugate of the inverse of a complex number is the inverse of the complex conjugate of that number.
The complex conjugate of a quotient is the quotient of the complex conjugates.
The complex conjugate of a complex number to the $n$th power is the complex conjugate of the complex number to the $n$th power.
The complex conjugate of a natural number is the same natural number.
The complex conjugate of an integer is the integer itself.
The complex conjugate of a numeral is the numeral itself.
The complex conjugate of a negative numeral is the negative numeral.
The complex conjugate of a real number times a complex number is the complex conjugate of the complex number times the real number.
The modulus of the complex conjugate of a complex number is equal to the modulus of the complex number.
The complex conjugate of a real number is the real number itself.
The complex conjugate of $\i$ is $-\i$.
$z + \overline{z} = 2 \cdot \text{Re}(z)$.
$z - \overline{z} = 2i \cdot \text{Im}(z)$.
If $x$ is an algebraic integer, then $\overline{x}$ is an algebraic integer.
The complex conjugate of an algebraic integer is an algebraic integer if and only if the original number is an algebraic integer.
The product of a complex number $z$ with its conjugate is equal to the square of the real part of $z$ plus the square of the imaginary part of $z$.
$z \overline{w} + \overline{z} w = 2 \operatorname{Re}(z \overline{w})$.
The modulus of the product of a complex number with its conjugate is equal to the square of the modulus of the complex number.
The modulus of a complex number $z$ is equal to the square root of the real part of $z$ times the complex conjugate of $z$.
The imaginary part of the product of a complex number with its conjugate is zero.