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<b>let </b><font color="Maroon">C1</font> be   <a href="coh_sp.html#NM1">Coherence_Space</a>, <font color="Maroon">C2</font> be   <a href="coh_sp.html#NM1">Coherence_Space</a>;<br/><b>let </b><font color="Maroon">f</font> be  <a href="cohsp_1.html#V8" target="_self">U-continuous</a> <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">C1</font>,<font color="Maroon">C2</font>, <font color="Maroon">g</font> be  <a href="cohsp_1.html#V8" target="_self">U-continuous</a> <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">C1</font>,<font color="Maroon">C2</font>;<br/>



<b>assume </b><a NAME="E1:70"/><i><font color="Green">E26</font></i>: 
 <a href="cohsp_1.html#K6" target="_self">graph</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="cohsp_1.html#K6" target="_self">graph</a> <font color="Maroon">g</font>
 ;<br/>

<a NAME="E2:70"/><i><font color="Green">E27</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C1</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C1</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><i><font color="Green">E36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be  <a href="finset_1.html#V1">finite</a>  <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font>;<br/><b>let </b><font color="Maroon">f</font> be  <a href="cohsp_1.html#V8" target="_self">U-continuous</a> <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">C1</font>,<font color="Maroon">C2</font>, <font color="Maroon">g</font> be  <a href="cohsp_1.html#V8" target="_self">U-continuous</a> <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">C1</font>,<font color="Maroon">C2</font>;<br/><b>assume </b><a NAME="E1:70_1"/><i><font color="Green">E41</font></i>: 
 <a href="cohsp_1.html#K6" target="_self">graph</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="cohsp_1.html#K6" target="_self">graph</a> <font color="Maroon">g</font>
 ;<br/><a NAME="E2:70_1"/><i><font color="Green">E42</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C1</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>thus </b><a NAME="E3:70_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:70_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:70_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="cohsp_1.html#K6" target="_self">graph</a> <font color="Maroon">f</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:70_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<div><i><font color="Green">E43</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="70_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font>;<br/><b>let </b><font color="Maroon">f</font> be  <a href="cohsp_1.html#V8" target="_self">U-continuous</a> <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">C1</font>,<font color="Maroon">C2</font>, <font color="Maroon">g</font> be  <a href="cohsp_1.html#V8" target="_self">U-continuous</a> <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">C1</font>,<font color="Maroon">C2</font>;<br/><b>assume </b><a NAME="E1:70_2"/><i><font color="Green">E44</font></i>: 
 <a href="cohsp_1.html#K6" target="_self">graph</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="cohsp_1.html#K6" target="_self">graph</a> <font color="Maroon">g</font>
 ;<br/><a NAME="E2:70_2"/><i><font color="Green">E45</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C1</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C1</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>thus </b><a NAME="E3:70_2"/>
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="cohsp_1.html#K3" target="_self">Fin</a> <font color="Maroon">a</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">g</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="cohsp_1.html#K3" target="_self">Fin</a> <font color="Maroon">a</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:70_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="cohsp_1.html#K3" target="_self">Fin</a> <font color="Maroon">a</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:70_2_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">x</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">x</font> <a href="hidden.html#R2">in</a>  <a href="cohsp_1.html#K3" target="_self">Fin</a> <font color="Maroon">a</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E4:70_2_1"/>
<font color="Maroon">x</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i>, <a class="ref" href="finsub_1.html#D5">FINSUB_1:def 5</a></i>;<br/>
<a NAME="E5:70_2_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 
<b>by </b><i><a class="ref" href="cohsp_1.html#T2" target="_self">Th2</a>, , </i>;<br/>
<a NAME="E6:70_2_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E7:70_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="cohsp_1.html#K3" target="_self">Fin</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font>;<br/><a NAME="E1:70_3"/>
( <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="cohsp_1.html#K3" target="_self">Fin</a> <font color="Maroon">a</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">g</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="cohsp_1.html#K3" target="_self">Fin</a> <font color="Maroon">a</font></span>)</span> &amp; <font color="Maroon">g</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="cohsp_1.html#K3" target="_self">Fin</a> <font color="Maroon">a</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="cohsp_1.html#K3" target="_self">Fin</a> <font color="Maroon">a</font></span>)</span> )
 <b>by </b><i>, </i>;<br/><a NAME="E2:70_3"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="cohsp_1.html#K3" target="_self">Fin</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="cohsp_1.html#K3" target="_self">Fin</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><b>thus </b><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> = 
 <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> <span class="p2">(<span class="default"><a href="cohsp_1.html#K3" target="_self">Fin</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="cohsp_1.html#E3:70"><i><font color="Green">E36</font></i></a>, <a class="txt" href="cohsp_1.html#E1:70"><i><font color="Green">E26</font></i></a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
<b>by </b><i><a class="txt" href="cohsp_1.html#E3:70"><i><font color="Green">E36</font></i></a>, , <a class="txt" href="cohsp_1.html#E1:70"><i><font color="Green">E26</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:70"/>
<font color="Maroon">f</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


</div>
