Let we have following two if statements:
\begin{eqnarray*} a &\leqslant &b\Rightarrow x\leqslant y \\ c &\leqslant &d\Rightarrow z\leqslant t \end{eqnarray*} Do the following inequality is correct ? \begin{equation*} a+c\leqslant b+d\Rightarrow x+z\leqslant y+t \end{equation*} Any help would be highly appreciated.
I looks very false. Let us take real numbers. It is true that $a\le b \Rightarrow a\le b$ and that $c\le d \Rightarrow c/2 \le d/2$ but I don't think that \begin{equation} a + c \le b + d \Rightarrow a + c/2 \le b + d/2 \end{equation} for example $3 + 2\le 2 + 3$ but $4 = 3 + 2/2 > 2 + 3/2 = 3.5$