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<b>let </b><font color="Maroon">f</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>

<b>assume </b><a NAME="E1:43_1_1"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">A</font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span>  st <font color="Olive">A</font> is <a href="cohsp_1.html#V1" target="_self">c=directed</a> &amp;  <a href="tarski.html#K3">union</a> <font color="Olive">A</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Olive">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Olive">A</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="cohsp_1.html#D11" target="_self">COHSP_1:def 11</a><br/>

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="cohsp_1.html#D12" target="_self">COHSP_1:def 12</a><br/>

<b>assume </b><a NAME="E2:43_1_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">A</font> = <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K2">}</a></span> as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <b>by </b><i><a class="ref" href="zfmisc_1.html#T38">ZFMISC_1:38</a></i>;<br/>
<i><font color="Green">E36</font></i>:  <a href="tarski.html#K3">union</a> <font color="Maroon">A</font> = 
<font color="Maroon">a</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T93">ZFMISC_1:93</a></i>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="cohsp_1.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
;<br/>
<a NAME="E5:43_1_1"/>
<font color="Maroon">A</font> is <a href="cohsp_1.html#V1" target="_self">c=directed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Y</font> be  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">A</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="cohsp_1.html#D4" target="_self">COHSP_1:def 4</a><br/>

<b>take </b>
<font color="Maroon">b</font>
;<br/>

<a NAME="E1:43_1_1_2"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T95">ZFMISC_1:95</a></i>;<br/>
<a NAME="E2:43_1_1_2"/><b>then </b>
 <a href="tarski.html#K3">union</a> <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T93">ZFMISC_1:93</a></i>;<br/>
<b>hence </b><a NAME="E3:43_1_1_2"/>
(  <a href="tarski.html#K3">union</a> <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="cohsp_1.html#D1" target="_self">Def1</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:43_1_1"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="cohsp_1.html#D1" target="_self">Def1</a>, </i>;<br/>
<a NAME="E7:43_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E8:43_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E9:43_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>


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