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<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">m</font> implies  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c1">n</font> is  <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c2">m</font> )</span><br/>

<b>assume </b><a NAME="E1:11"/><i><font color="Green" title="E4">A1</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c1">n</font> is  <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c2">m</font></span><br/>

<a NAME="E2:11"/><i><font color="Green" title="E5">A2</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/>
<a NAME="E3:11"/><i><font color="Green" title="E6">A3</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">m</font>
 
<b>by </b><i><a class="txt" href="orders_4.html#E1:11"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="nat_1.html#T40" title="NAT_1:th.40">NAT_1:40</a></i>;<br/>
<a NAME="E4:11"/><b>then </b><i><font color="Green" title="E7">A4</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c2">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="orders_4.html#E2:11"><i><font color="Green" title="E5">A2</font></i></a>, <a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/>
<a NAME="E5:11"/><i><font color="Green" title="E8">A5</font></i>: 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/>
<a NAME="E6:11"/><i><font color="Green" title="E9">A6</font></i>: 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c2">m</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/>
<a NAME="E7:11"/><i><font color="Green" title="E10">A7</font></i>: 
 <a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c2">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" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c2">m</font> )</span><br/>

<b>assume </b><a NAME="E1:11_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c2">m</font></span><br/>

<a NAME="E2:11_1"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="orders_4.html#E3:11"><i><font color="Green" title="E6">A3</font></i></a>, <a class="ref" href="wellord2.html#T8" title="WELLORD2:th.8">WELLORD2:8</a></i>;<br/>
<b>hence </b><a NAME="E3:11_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c2">m</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E8:11"/>
<span class="p1">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="orders_4.html#E3:11"><i><font color="Green" title="E6">A3</font></i></a>, <a class="txt" href="orders_4.html#E5:11"><i><font color="Green" title="E8">A5</font></i></a>, <a class="ref" href="wellord2.html#T8" title="WELLORD2:th.8">WELLORD2:8</a></i>;<br/>
<a NAME="E9:11"/><b>then </b>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="orders_4.html#E6:11"><i><font color="Green" title="E9">A6</font></i></a>, <a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/>
<b>hence </b><a NAME="E10:11"/>
 <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c1">n</font> is  <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c2">m</font>
 <b>by </b><i><a class="txt" href="orders_4.html#E4:11"><i><font color="Green" title="E7">A4</font></i></a>, <a class="txt" href="orders_4.html#E5:11"><i><font color="Green" title="E8">A5</font></i></a>, <a class="txt" href="orders_4.html#E6:11"><i><font color="Green" title="E9">A6</font></i></a>, <a class="txt" href="orders_4.html#E7:11"><i><font color="Green" title="E10">A7</font></i></a>, <a class="ref" href="yellow_0.html#D13" title="YELLOW_0:def.13">YELLOW_0:def 13</a>, <a class="ref" href="yellow_0.html#D14" title="YELLOW_0:def.14">YELLOW_0:def 14</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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