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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="coh_sp.html#NM1">Coherence_Space</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:116"/><i><font color="Green">E66</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font> being <a href="cohsp_1.html#V10" target="_self">U-linear</a> <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="cohsp_1.html#K11" target="_self">LinTrace</a> <font color="Olive">b<sub>3</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<a NAME="E2:116"/><i><font color="Green">E67</font></i>: 
 <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></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="E1:116_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:116_1"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:116_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being  <a href="cohsp_1.html#V10" target="_self">U-linear</a> <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E6:116_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="cohsp_1.html#K11" target="_self">LinTrace</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E1:116"><i><font color="Green">E66</font></i></a>, <a class="txt" href="cohsp_1.html#E3:116_1"><i><font color="Green">E68</font></i></a></i>;<br/>
<b>thus </b><a NAME="E7:116_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E3:116_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="cohsp_1.html#E6:116_1"><i><font color="Green">E69</font></i></a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E68</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="116_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:116_2"/><i><font color="Green">E69</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E2:116_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:116_2"/><i><font color="Green">E70</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>assume </b><a NAME="E5:116_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:116_2"/><i><font color="Green">E71</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>11</sub></font> being  <a href="cohsp_1.html#V10" target="_self">U-linear</a> <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E9:116_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="cohsp_1.html#K11" target="_self">LinTrace</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E1:116"><i><font color="Green">E66</font></i></a>, <a class="txt" href="cohsp_1.html#E4:116_2"><i><font color="Green">E70</font></i></a>, <a class="txt" href="cohsp_1.html#E7:116_2"><i><font color="Green">E71</font></i></a></i>;<br/><a NAME="E10:116_2"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E11:116_2"/>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E1:116_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="cohsp_1.html#E4:116_2"><i><font color="Green">E70</font></i></a>, <a class="txt" href="cohsp_1.html#E7:116_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="cohsp_1.html#E9:116_2"><i><font color="Green">E72</font></i></a>, <a class="ref" href="cohsp_1.html#T54" target="_self">Th54</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:116_3"/><i><font color="Green">E69</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E2:116_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:116_3"/><i><font color="Green">E70</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>assume </b><a NAME="E5:116_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:116_3"/><i><font color="Green">E71</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>10</sub></font> being  <a href="cohsp_1.html#V10" target="_self">U-linear</a> <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E9:116_3"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="cohsp_1.html#K11" target="_self">LinTrace</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E1:116"><i><font color="Green">E66</font></i></a>, <a class="txt" href="cohsp_1.html#E4:116_3"><i><font color="Green">E70</font></i></a>, <a class="txt" href="cohsp_1.html#E7:116_3"><i><font color="Green">E71</font></i></a></i>;<br/><a NAME="E10:116_3"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E11:116_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E1:116_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="cohsp_1.html#E4:116_3"><i><font color="Green">E70</font></i></a>, <a class="txt" href="cohsp_1.html#E7:116_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="cohsp_1.html#E9:116_3"><i><font color="Green">E72</font></i></a>, <a class="ref" href="cohsp_1.html#T55" target="_self">Th55</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:116"/>
ex <font color="Olive">b<sub>1</sub></font> being <a href="cohsp_1.html#V10" target="_self">U-linear</a> <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> st  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="cohsp_1.html#K11" target="_self">LinTrace</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E2:116"><i><font color="Green">E67</font></i></a>, <a class="txt" href="cohsp_1.html#E3:116"><i><font color="Green">E68</font></i></a>, <a class="ref" href="cohsp_1.html#T57" target="_self">Th57</a></i>;<br/>


</div>
