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<b>consider </b><font color="Maroon">f</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">C1</font>,<font color="Maroon">C2</font>;<br/>
<b>set </b><font color="Maroon">C</font> =  <a href="cohsp_1.html#K12" target="_self">LinCoh</a> <font color="Maroon">C1</font>,<font color="Maroon">C2</font>;<br/>
<a NAME="E2:117_1_1"/>
 <a href="cohsp_1.html#K11" target="_self">LinTrace</a> <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="cohsp_1.html#K12" target="_self">LinCoh</a> <font color="Maroon">C1</font>,<font color="Maroon">C2</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:117_1_1"/>
not  <a href="cohsp_1.html#K12" target="_self">LinCoh</a> <font color="Maroon">C1</font>,<font color="Maroon">C2</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>

<b>thus </b><a NAME="E4:117_1_1"/>
 <a href="cohsp_1.html#K12" target="_self">LinCoh</a> <font color="Maroon">C1</font>,<font color="Maroon">C2</font> is <a href="classes1.html#V1">subset-closed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1_1_1"><b>proof </b></a><div class="add">

<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="classes1.html#D1">CLASSES1:def 1</a><br/>

<b>assume </b><a NAME="E1:117_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="cohsp_1.html#K12" target="_self">LinCoh</a> <font color="Maroon">C1</font>,<font color="Maroon">C2</font>
 ;<br/>

<b>then consider </b><font color="Maroon">f</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">C1</font>,<font color="Maroon">C2</font><b> such that </b><br/><a NAME="E3:117_1_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="cohsp_1.html#K11" target="_self">LinTrace</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E4:117_1_1_1"/>
<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font>
 ;<br/>

<a NAME="E5:117_1_1_1"/><b>then </b>
 ex <font color="Olive">g</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">C1</font>,<font color="Maroon">C2</font> st  <a href="cohsp_1.html#K11" target="_self">LinTrace</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="cohsp_1.html#T17" target="_self">Th17</a>, <a class="ref" href="cohsp_1.html#D7" target="_self">Def7</a></i>;<br/>
<b>hence </b><a NAME="E6:117_1_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="cohsp_1.html#K12" target="_self">LinCoh</a> <font color="Maroon">C1</font>,<font color="Maroon">C2</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="cohsp_1.html#D1" target="_self">COHSP_1:def 1</a><br/>

<b>assume </b><a NAME="E5:117_1_1"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Olive">a</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="cohsp_1.html#K12" target="_self">LinCoh</a> <font color="Maroon">C1</font>,<font color="Maroon">C2</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:117_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/><a NAME="E2:117_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="cohsp_1.html#K12" target="_self">LinCoh</a> <font color="Maroon">C1</font>,<font color="Maroon">C2</font>
 <b>by </b><i><a class="ref" href="cohsp_1.html#T17" target="_self">Th17</a></i>;<br/><b>hence </b><a NAME="E3:117_1_1_2"/>
 ex <font color="Olive">f</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">C1</font>,<font color="Maroon">C2</font> st <font color="Maroon">x</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="cohsp_1.html#K11" target="_self">LinTrace</a> <font color="Olive">f</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E7:117_1_1"/><b>then </b>
 ex <font color="Olive">f</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">C1</font>,<font color="Maroon">C2</font> st  <a href="tarski.html#K3">union</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="cohsp_1.html#K11" target="_self">LinTrace</a> <font color="Olive">f</font>
 
<b>by </b><i><a class="ref" href="cohsp_1.html#T15" target="_self">Th15</a></i>;<br/>
<b>hence </b><a NAME="E8:117_1_1"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="cohsp_1.html#K12" target="_self">LinCoh</a> <font color="Maroon">C1</font>,<font color="Maroon">C2</font>
 <b>by </b><i/>;<br/>


</div>
