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<b>let </b><font color="Maroon" title="c1">C1</font>, <font color="Maroon" title="c2">C2</font> be   <a href="coh_sp.html#NM1" title="COH_SP:NM.1">Coherence_Space</a>;<br/><b>let </b><font color="Maroon" title="c3">f</font> be  <a href="cohsp_1.html#V9" title="COHSP_1:attr.9">U-stable</a> <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>;<br/><b>let </b><font color="Maroon" title="c4">X</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="cohsp_1.html#K8" title="COHSP_1:func.8">Trace</a> <font color="Maroon" title="c3">f</font></span>)</span>;<br/>





<a NAME="E1:91"/><i><font color="Green" title="E48">A1</font></i>: 
<font color="Maroon" title="c4">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">C1</font>,<span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">C2</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<div><i><font color="Green" title="E49">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:91_1"/><i><font color="Green" title="E50">A3</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:91_1"/><i><font color="Green" title="E51">A4</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> &amp; ( for <font color="Olive" title="b1">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Olive" title="b1">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> ) )
 <b>by </b><i><a class="ref" href="cohsp_1.html#D18" target="_self" title="COHSP_1:def.18">Def18</a></i>;<br/><a NAME="E4:91_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">C1</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E5:91_1"/><b>then </b>
<font color="Maroon" title="c6">a</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E1:91_1"><i><font color="Green" title="E50">A3</font></i></a>, <a class="txt" href="cohsp_1.html#E3:91_1"><i><font color="Green" title="E51">A4</font></i></a>, <a class="ref" href="cohsp_1.html#T34" target="_self" title="COHSP_1:th.34">Th34</a></i>;<br/><b>hence </b><a NAME="E6:91_1"/>
<font color="Maroon" title="c5">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E3:91_1"><i><font color="Green" title="E51">A4</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:91"/><i><font color="Green" title="E50">A5</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font>  st <font color="Olive" title="b1">a</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font> holds <br/>for <font color="Olive" title="b3">y1</font>, <font color="Olive" title="b4">y2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">b</font>,<font color="Olive" title="b4">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> holds <br/><span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b3">y1</font>,<font color="Olive" title="b4">y2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C2</font>
 
<b>by </b><i><a class="ref" href="cohsp_1.html#T35" target="_self" title="COHSP_1:th.35">Th35</a></i>;<br/>
<a NAME="E4:91"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font>  st <font color="Olive" title="b1">a</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font> holds <br/>for <font color="Olive" title="b3">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">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> <font color="Maroon" title="c4">X</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">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> <font color="Maroon" title="c4">X</font> holds <br/><font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font>
 
<b>by </b><i><a class="ref" href="cohsp_1.html#T36" target="_self" title="COHSP_1:th.36">Th36</a></i>;<br/>
<b>hence </b><a NAME="E5:91"/>
 ex <font color="Olive" title="b1">g</font> being <a href="cohsp_1.html#V9" title="COHSP_1:attr.9">U-stable</a> <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> st  <a href="cohsp_1.html#K8" title="COHSP_1:func.8">Trace</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E1:91"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="cohsp_1.html#E2:91"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="cohsp_1.html#E3:91"><i><font color="Green" title="E50">A5</font></i></a>, <a class="ref" href="cohsp_1.html#T39" target="_self" title="COHSP_1:th.39">Th39</a></i>;<br/>


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