<s>[INST] Prove that if $a < b$ and $c < d$ then $a + c < b + d$ I'm having a tough time with this questions. Could someone perhaps give me a hint? I'm only allowed to use the axioms of real numbers:
A1. $a + b = b + a$
A2. $a + (b + c) = (a+b) + c$
A3. $a + 0 = a$
A4. $a + (-a) = 0$
M1. $a \cdot b = b \cdot a$
M2. $a \cdot (b \cdot c) = (a \cdot b) \cdot c$
M3. $a \cdot 1 = a$
M4. if $a \neq 0$ then $a \cdot \frac{1}{a} = 1$
D. $a \cdot (b+c) = a \cdot b + a \cdot c$
 [/INST] Use $x < y\iff x + d < y+d$, to prove that $a < b \iff a+c < b+ c$ and to prove that $c< d \iff b + c < b + d$.
and finish it off with $x < y; y< z \implies x < z$.
to prove $a + c < b+c$ and $b+c  < b+d\implies a+c < b+d$.
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