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<b>let </b><font color="Maroon">it1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">it2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:38_1_2"/><i><font color="Green">E53</font></i>: 
(  <a href="graphsp.html#K8" target="_self">OuterVx</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K6" target="_self">repeat</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">it1</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st  <a href="graphsp.html#K8" target="_self">OuterVx</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K6" target="_self">repeat</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/><font color="Maroon">it1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> ) &amp;  <a href="graphsp.html#K8" target="_self">OuterVx</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K6" target="_self">repeat</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">it2</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st  <a href="graphsp.html#K8" target="_self">OuterVx</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K6" target="_self">repeat</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/><font color="Maroon">it2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> ) &amp; not <font color="Maroon">it1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">it2</font> )
 ;<br/>

<a NAME="E2:38_1_2"/><b>then </b>
( <font color="Maroon">it1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">it2</font> &amp; <font color="Maroon">it2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">it1</font> )
 
;<br/>
<b>hence </b><a NAME="E3:38_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


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