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<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:11"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E2:11"/><i><font color="Green">E30</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<a NAME="E3:11"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E4:11"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">n</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="card_1.html#T56">CARD_1:56</a></i>;<br/>
<a NAME="E5:11"/><b>then </b><i><font color="Green">E42</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<a NAME="E6:11"/><i><font color="Green">E43</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<a NAME="E7:11"/><i><font color="Green">E44</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<a NAME="E8:11"/><i><font color="Green">E45</font></i>: 
 <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a>  <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:11_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E2:11_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">m</font></span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="wellord2.html#T8">WELLORD2:8</a></i>;<br/>
<b>hence </b><a NAME="E3:11_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:11"/>
<span class="p1">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">m</font></span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="wellord2.html#T8">WELLORD2:8</a></i>;<br/>
<a NAME="E10:11"/><b>then </b>
the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">m</font></span>)</span> <a href="wellord1.html#K2">|_2</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<b>hence </b><a NAME="E11:11"/>
 <a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">n</font> is  <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of  <a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">m</font>
 <b>by </b><i>, , , , <a class="ref" href="yellow_0.html#D13">YELLOW_0:def 13</a>, <a class="ref" href="yellow_0.html#D14">YELLOW_0:def 14</a></i>;<br/>


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