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This holds in any ordered field; the only property we need to take advantage of is translation invariance and transitivity. That is, the properties that $$a<b\Leftrightarrow a+c<b+c$$ $$a<b \text{ and } b<c\Rightarrow a<c$$

Starting with $$a<b$$ we can, using translation invariance, add a constant to both sides: $$a+c<b+c$$ We can, using translation invariance, we can also establish, from $c<d$ translated by $b$ get that $$b+c<b+d$$ and using transitivity gives $$a+c<b+d$$

This holds in any ordered field (or more generally, partially ordered group); the only property we need to take advantage of is translation invariance and transitivity. That is, the properties that $$a<b\Leftrightarrow a+c<b+c$$ $$a<b \text{ and } b<c\Rightarrow a<c$$

Starting with $$a<b$$ we can, using translation invariance, add a constant to both sides: $$a+c<b+c$$ We can, using translation invariance, we can also establish, from $c<d$ translated by $b$ get that $$b+c<b+d$$ and using transitivity gives $$a+c<b+d$$

This holds in any ordered field; the only property we need to take advantage of is translation invariance and transitivity. That is, the properties that $$a<b\Leftrightarrow a+c<b+c$$ $$a<b \text{ and } b<c\Rightarrow a<c$$

Starting with $$a<b$$ we can, using translation invariance, add a constant to both sides: $$a+c<b+c$$ We can, using translation invariance, we can also establish, from the second inequality translated by $b$ get that $$b+c<b+d$$ and using transitivity gives $$a+c<b+d$$

This holds in any ordered field; the only property we need to take advantage of is translation invariance and transitivity. That is, the properties that $$a<b\Leftrightarrow a+c<b+c$$ $$a<b \text{ and } b<c\Rightarrow a<c$$

Starting with $$a<b$$ we can, using translation invariance, add a constant to both sides: $$a+c<b+c$$ We can, using translation invariance, we can also establish, from $c<d$ translated by $b$ get that $$b+c<b+d$$ and using transitivity gives $$a+c<b+d$$

This holds in any ordered field (or, more generally, partially ordered group); the only property we need to take advantage of is translation invariance and transitivity. That is, the properties that $$a<b\Leftrightarrow a+c<b+c$$ $$a<b \text{ and } b<c\Rightarrow a<c$$

Starting with $$a<b$$ we can, using translation invariance, add a constant to both sides: $$a+c<b+c$$ We can, using translation invariance, we can also establish, from the second inequality translated by $b$ get that $$b+c<b+d$$ and using transitivity gives $$a+c<b+d$$

This holds in any ordered field; the only property we need to take advantage of is translation invariance and transitivity. That is, the properties that $$a<b\Leftrightarrow a+c<b+c$$ $$a<b \text{ and } b<c\Rightarrow a<c$$

Starting with $$a<b$$ we can, using translation invariance, add a constant to both sides: $$a+c<b+c$$ We can, using translation invariance, we can also establish, from the second inequality translated by $b$ get that $$b+c<b+d$$ and using transitivity gives $$a+c<b+d$$