<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">Y</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">a</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">Y</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> ;<br/><b>let </b><font color="Maroon" title="c3">G</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="bvfunc_2.html#K1" title="BVFUNC_2:func.1">PARTITIONS</a> <font color="Maroon" title="c1">Y</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c4">P</font>, <font color="Maroon" title="c5">Q</font> be    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font>;<br/>







<b>assume </b><a NAME="E1:19"/><i><font color="Green" title="E13">A1</font></i>: 
<font color="Maroon" title="c3">G</font> is <a href="bvfunc_2.html#V2" title="BVFUNC_2:attr.2">independent</a>
 ;<br/>

<b>set </b><font color="Maroon" title="c6">A</font> = <font color="Maroon" title="c3">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="bvfunc_2.html#K4" title="BVFUNC_2:func.4">{</a><span class="default"><font color="Maroon" title="c4">P</font></span><a href="bvfunc_2.html#K4" title="BVFUNC_2:func.4">}</a></span>;<br/>
<b>set </b><font color="Maroon" title="c7">B</font> = <font color="Maroon" title="c3">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="bvfunc_2.html#K4" title="BVFUNC_2:func.4">{</a><span class="default"><font color="Maroon" title="c5">Q</font></span><a href="bvfunc_2.html#K4" title="BVFUNC_2:func.4">}</a></span>;<br/>
<a NAME="E2:19"/><i><font color="Green" title="E14">A2</font></i>: 
 <a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="bvfunc_2.html#K4" title="BVFUNC_2:func.4">{</a><span class="default"><font color="Maroon" title="c4">P</font></span><a href="bvfunc_2.html#K4" title="BVFUNC_2:func.4">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="bvfunc_2.html#D7" title="BVFUNC_2:def.7">BVFUNC_2:def 7</a></i>;<br/>
<a NAME="E3:19"/><i><font color="Green" title="E15">A3</font></i>: 
 <a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="bvfunc_2.html#K4" title="BVFUNC_2:func.4">{</a><span class="default"><font color="Maroon" title="c5">Q</font></span><a href="bvfunc_2.html#K4" title="BVFUNC_2:func.4">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="bvfunc_2.html#D7" title="BVFUNC_2:def.7">BVFUNC_2:def 7</a></i>;<br/>
<a NAME="E4:19"/>
( <font color="Maroon" title="c3">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="bvfunc_2.html#K4" title="BVFUNC_2:func.4">{</a><span class="default"><font color="Maroon" title="c4">P</font></span><a href="bvfunc_2.html#K4" title="BVFUNC_2:func.4">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">G</font> &amp; <font color="Maroon" title="c3">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="bvfunc_2.html#K4" title="BVFUNC_2:func.4">{</a><span class="default"><font color="Maroon" title="c5">Q</font></span><a href="bvfunc_2.html#K4" title="BVFUNC_2:func.4">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">G</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E5:19"/><b>then </b><i><font color="Green" title="E16">A4</font></i>: 
<span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="bvfunc_2.html#K4" title="BVFUNC_2:func.4">{</a><span class="default"><font color="Maroon" title="c4">P</font></span><a href="bvfunc_2.html#K4" title="BVFUNC_2:func.4">}</a></span></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">*</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="bvfunc_2.html#K4" title="BVFUNC_2:func.4">{</a><span class="default"><font color="Maroon" title="c5">Q</font></span><a href="bvfunc_2.html#K4" title="BVFUNC_2:func.4">}</a></span></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="bvfunc_2.html#K4" title="BVFUNC_2:func.4">{</a><span class="default"><font color="Maroon" title="c5">Q</font></span><a href="bvfunc_2.html#K4" title="BVFUNC_2:func.4">}</a></span></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">*</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="bvfunc_2.html#K4" title="BVFUNC_2:func.4">{</a><span class="default"><font color="Maroon" title="c4">P</font></span><a href="bvfunc_2.html#K4" title="BVFUNC_2:func.4">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="partit_2.html#E1:19"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="partit_2.html#T16" target="_self" title="PARTIT_2:th.16">Th16</a></i>;<br/>
<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="bvfunc_1.html#D15" title="BVFUNC_1:def.15">BVFUNC_1:def 15</a><br/>

<b>assume </b><a NAME="E6:19"/><i><font color="Green" title="E17">A5</font></i>: 
<span class="p1">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">G</font></span>)</span>,<font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c3">G</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 ;<br/>

<a NAME="E7:19"/><i><font color="Green" title="E18">A6</font></i>: 
 <a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">G</font></span>)</span>,<font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c3">G</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K22" title="BVFUNC_1:func.22">B_SUP</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c3">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="bvfunc_2.html#D10" title="BVFUNC_2:def.10">BVFUNC_2:def 10</a></i>;<br/>
<a NAME="E8:19"/><i><font color="Green" title="E19">A7</font></i>: 
 <a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c3">G</font></span>)</span>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">G</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K21" title="BVFUNC_1:func.21">B_INF</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c3">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="bvfunc_2.html#D9" title="BVFUNC_2:def.9">BVFUNC_2:def 9</a></i>;<br/>
<a NAME="E9:19"/>
for <font color="Olive" title="b1">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>  st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c8">x</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">G</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c3">G</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>;<br/>

<b>assume </b><a NAME="E1:19_1"/>
<font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c8">x</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">G</font></span>)</span>
 ;<br/>

<a NAME="E2:19_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">z</font>,<font color="Maroon" title="c8">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="bvfunc_2.html#K4" title="BVFUNC_2:func.4">{</a><span class="default"><font color="Maroon" title="c4">P</font></span><a href="bvfunc_2.html#K4" title="BVFUNC_2:func.4">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="partit_2.html#E2:19"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="partit_2.html#T5" target="_self" title="PARTIT_2:th.5">Th5</a></i>;<br/>
<a NAME="E3:19_1"/><b>then </b><i><font color="Green" title="E20">A8</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="bvfunc_2.html#K4" title="BVFUNC_2:func.4">{</a><span class="default"><font color="Maroon" title="c4">P</font></span><a href="bvfunc_2.html#K4" title="BVFUNC_2:func.4">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T12" title="EQREL_1:th.12">EQREL_1:12</a></i>;<br/>
<a NAME="E4:19_1"/><i><font color="Green" title="E21">A9</font></i>: 
 <a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c3">G</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K22" title="BVFUNC_1:func.22">B_SUP</a> <font color="Maroon" title="c2">a</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c3">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="bvfunc_2.html#D10" title="BVFUNC_2:def.10">BVFUNC_2:def 10</a></i>;<br/>
<a NAME="E5:19_1"/><i><font color="Green" title="E22">A10</font></i>: 
 <a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">G</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K21" title="BVFUNC_1:func.21">B_INF</a> <font color="Maroon" title="c2">a</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="bvfunc_2.html#D9" title="BVFUNC_2:def.9">BVFUNC_2:def 9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font><b> such that </b><br/><a NAME="E7:19_1"/><i><font color="Green" title="E23">A11</font></i>: 
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c8">x</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c3">G</font></span>)</span>
 <b>and </b><br/><a NAME="E8:19_1"/><i><font color="Green" title="E24">A12</font></i>: 
<span class="p1">(<span class="default"><a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">G</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <b>by </b><i><a class="txt" href="partit_2.html#E6:19"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="partit_2.html#E7:19"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="bvfunc_1.html#D20" title="BVFUNC_1:def.20">BVFUNC_1:def 20</a></i>;<br/>
<a NAME="E9:19_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c8">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="bvfunc_2.html#K4" title="BVFUNC_2:func.4">{</a><span class="default"><font color="Maroon" title="c5">Q</font></span><a href="bvfunc_2.html#K4" title="BVFUNC_2:func.4">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="partit_2.html#E3:19"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="partit_2.html#E7:19_1"><i><font color="Green" title="E23">A11</font></i></a>, <a class="ref" href="partit_2.html#T5" target="_self" title="PARTIT_2:th.5">Th5</a></i>;<br/>
<a NAME="E10:19_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c9">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="bvfunc_2.html#K4" title="BVFUNC_2:func.4">{</a><span class="default"><font color="Maroon" title="c4">P</font></span><a href="bvfunc_2.html#K4" title="BVFUNC_2:func.4">}</a></span></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">*</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="bvfunc_2.html#K4" title="BVFUNC_2:func.4">{</a><span class="default"><font color="Maroon" title="c5">Q</font></span><a href="bvfunc_2.html#K4" title="BVFUNC_2:func.4">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="partit_2.html#E5:19"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="partit_2.html#E3:19_1"><i><font color="Green" title="E20">A8</font></i></a>, <a class="ref" href="relat_1.html#D8" title="RELAT_1:def.8">RELAT_1:def 8</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E12:19_1"/><i><font color="Green" title="E25">A13</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c11">u</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="bvfunc_2.html#K4" title="BVFUNC_2:func.4">{</a><span class="default"><font color="Maroon" title="c4">P</font></span><a href="bvfunc_2.html#K4" title="BVFUNC_2:func.4">}</a></span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E13:19_1"/><i><font color="Green" title="E26">A14</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">u</font>,<font color="Maroon" title="c9">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="bvfunc_2.html#K4" title="BVFUNC_2:func.4">{</a><span class="default"><font color="Maroon" title="c5">Q</font></span><a href="bvfunc_2.html#K4" title="BVFUNC_2:func.4">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#D8" title="RELAT_1:def.8">RELAT_1:def 8</a></i>;<br/>
<a NAME="E14:19_1"/>
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="bvfunc_2.html#K4" title="BVFUNC_2:func.4">{</a><span class="default"><font color="Maroon" title="c5">Q</font></span><a href="bvfunc_2.html#K4" title="BVFUNC_2:func.4">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="partit_2.html#E13:19_1"><i><font color="Green" title="E26">A14</font></i></a>, <a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">u</font> = <font color="Maroon" title="c11">u</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font> <b>by </b><i><a class="ref" href="orders_1.html#T97" title="ORDERS_1:th.97">ORDERS_1:97</a></i>;<br/>
<a NAME="E16:19_1"/><i><font color="Green" title="E27">A15</font></i>: 
<font color="Maroon" title="c12">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c9">z</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c3">G</font></span>)</span>
 
<b>by </b><i><a class="txt" href="partit_2.html#E3:19"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="partit_2.html#E13:19_1"><i><font color="Green" title="E26">A14</font></i></a>, <a class="ref" href="partit_2.html#T5" target="_self" title="PARTIT_2:th.5">Th5</a></i>;<br/>
<a NAME="E17:19_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">u</font>,<font color="Maroon" title="c10">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="bvfunc_2.html#K4" title="BVFUNC_2:func.4">{</a><span class="default"><font color="Maroon" title="c4">P</font></span><a href="bvfunc_2.html#K4" title="BVFUNC_2:func.4">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="partit_2.html#E12:19_1"><i><font color="Green" title="E25">A13</font></i></a>, <a class="ref" href="eqrel_1.html#T12" title="EQREL_1:th.12">EQREL_1:12</a></i>;<br/>
<a NAME="E18:19_1"/><b>then </b>
<font color="Maroon" title="c12">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c10">y</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">G</font></span>)</span>
 
<b>by </b><i><a class="txt" href="partit_2.html#E2:19"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="partit_2.html#T5" target="_self" title="PARTIT_2:th.5">Th5</a></i>;<br/>
<a NAME="E19:19_1"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="partit_2.html#E5:19_1"><i><font color="Green" title="E22">A10</font></i></a>, <a class="txt" href="partit_2.html#E8:19_1"><i><font color="Green" title="E24">A12</font></i></a>, <a class="ref" href="bvfunc_1.html#D19" title="BVFUNC_1:def.19">BVFUNC_1:def 19</a></i>;<br/>
<b>hence </b><a NAME="E20:19_1"/>
<span class="p1">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c3">G</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <b>by </b><i><a class="txt" href="partit_2.html#E4:19_1"><i><font color="Green" title="E21">A9</font></i></a>, <a class="txt" href="partit_2.html#E16:19_1"><i><font color="Green" title="E27">A15</font></i></a>, <a class="ref" href="bvfunc_1.html#D20" title="BVFUNC_1:def.20">BVFUNC_1:def 20</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:19"/>
<span class="p1">(<span class="default"><a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c3">G</font></span>)</span>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">G</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <b>by </b><i><a class="txt" href="partit_2.html#E8:19"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="bvfunc_1.html#D19" title="BVFUNC_1:def.19">BVFUNC_1:def 19</a></i>;<br/>


</div>
