if-m-n-z-N-so-m-lt-n-and-z-lt-m-then-z-lt-n-

Question Number 215944 by mokys last updated on 22/Jan/25

$${if}\:{m},{n},{z}\:\in{N}\:{so}\:,\:{m}<{n}\:{and}\:{z}<{m}\:{then}\:{z}<{n}\:?\: \\ $$

Answered by MrGaster last updated on 22/Jan/25

$$\forall{m},{n},{z}\in\mathbb{N} \\ $$$${m}<{n\&x}<{m} \\ $$$$\Rightarrow{z}<{n} \\ $$

Commented by mokys last updated on 22/Jan/25

$${i}\:{want}\:{prove}\:{by}\:{steps} \\ $$

Commented by MrGaster last updated on 22/Jan/25

Let others solve this request later.

Answered by mehdee7396 last updated on 22/Jan/25

$${n}−{m}>\mathrm{0}\:\:\&\:\:{m}−{z}>\mathrm{0} \\ $$$$\Rightarrow\left({n}−{m}\right)+\left({m}−{z}\right)>\mathrm{0} \\ $$$$\Rightarrow{n}−{z}>\mathrm{0}\Rightarrow{n}>{z} \\ $$

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