Pregunta: 13. If x

13. If $x<y$, then $x+z<y+z$. 14. If $x<y$ and $z<w$, then $x+z<y+w$. 15. If $x=y$ and $0<z$, then $y<x+z$. 16. If $x=y$ and $z<0$, then $x+z<y$. 17. If $x<y$ and $y<z$, then $x<z$. 18. If $x<y$ and $0<z$, then $xz<yz$. 19. If $x<y$ and $z<0$, then $yz<xz$. 1. If $0<x<y$ and $0<z<w$, then $xz<yw$. 22. If $0<z$ and $xz<yz$, then $x<y$. 23. $z-x<z-y$ if and only if $y<x$. 24. If $x<y$, then $-y<-x$. 25. Prove that if $x$ and $y$ are integers and $xy=0$, then either $x=0$ or $y=0$. (Hint: If $x\ne 0$, then either $x\in {\mathbf{Z}}^{+}$or $-x\in {\mathbf{Z}}^{\ast }$, and similarly for $y$. Consider $xy$ for the various cases.) 26. Prove that the cancellation law for multiplication holds in $\mathbf{Z}$. That is, if $xy=xz$ and $x\ne 0$, then $y=z$ 27. Let $x$ and $y$ be in $\mathbf{Z}$, not both zero, then $x^2+y^2\in Z^{\text{: }}$. For an integer $x$, the absolute value of $x$ is denoted by $|x|$ and is defined by