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<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<a NAME="E1:3_1_2"/>
( <font color="Maroon">A</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> ) &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> ) implies <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:3_1_2_1"/>
( <font color="Maroon">A</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> ) &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> ) )
 ;<br/>

<a NAME="E2:3_1_2_1"/><b>then </b>
( <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> )
 
;<br/>
<b>hence </b><a NAME="E3:3_1_2_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:3_1_2"/>
( ( <font color="Maroon">A</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> ) &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> ) implies <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> ) &amp; ( <font color="Maroon">A</font> is  not  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> ) )
 ;<br/>


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