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<b>let </b><font color="Maroon">Z</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1"/>
( <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1"><b>suppose </b></a><a NAME="E1:29_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:29_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
(  <a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a>  <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> &amp;  <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <a href="xboole_0.html#K1">{}</a> </span>)</span>,<span class="p1">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span>)</span> #) &amp; <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>,<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> #) )
 <b>by </b><i><a class="ref" href="yellow_1.html#T4">YELLOW_1:4</a>, <a class="ref" href="yellow_1.html#T27">YELLOW_1:27</a>, <a class="ref" href="yellow_1.html#D1">YELLOW_1:def 1</a></i>;<br/><a NAME="E3:29_1_1"/><i><font color="Green">E39</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="relat_1.html#T71">RELAT_1:71</a></i>;<br/><b>reconsider </b><font color="Maroon">f</font> =  <a href="partfun1.html#K6">id</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">Z</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">Z</font></span>)</span> <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T1">ZFMISC_1:1</a></i>;<br/><b>take </b>
<font color="Maroon">f</font>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">Z</font></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">Z</font></span>)</span>;<br/><a NAME="E1:29_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="zfmisc_1.html#T1">ZFMISC_1:1</a></i>;<br/><b>hence </b><a NAME="E2:29_1_1_1"/>
( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> iff <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> )
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:29_1_1"/>
<font color="Maroon">f</font> is <a href="waybel_0.html#V23">isomorphic</a>
 <b>by </b><i>, , <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/><b>let </b><font color="Maroon">Y</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Z</font>;<br/><a NAME="E7:29_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><a NAME="E8:29_1_1"/><b>then </b>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E9:29_1_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <font color="Maroon">Y</font>,<font color="Maroon">Z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a>, <a class="ref" href="funct_3.html#D3">FUNCT_3:def 3</a></i>;<br/><b>hence </b><a NAME="E10:29_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="funct_3.html#K5">chi</a> <font color="Maroon">Y</font>,<font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2"><b>suppose </b></a><a NAME="E1:29_1_2"/>
<font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">Z'</font> = <font color="Maroon">Z</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><a NAME="E3:29_1_2"/>
<span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a>  <a href="yellow_1.html#K5">product</a> <span class="p1">(<span class="default"><font color="Maroon">Z'</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_1.html#D5">YELLOW_1:def 5</a></i>;<br/><b>hence </b><a NAME="E4:29_1_2"/>
 ex <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">Z</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">Z</font></span>)</span> st <br/>( <font color="Olive">f</font> is <a href="waybel_0.html#V23">isomorphic</a> &amp; ( for <font color="Olive">Y</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Z</font> holds  <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">Y</font> <a href="hidden.html#R1">=</a>  <a href="funct_3.html#K5">chi</a> <font color="Olive">Y</font>,<font color="Maroon">Z</font> ) )
 <b>by </b><i><a class="ref" href="waybel18.html#T15">WAYBEL18:15</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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