<?xml version="1.0"?>
<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="hidden.html#M1">set</a> ;<br/>



<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <span class="p3">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K3" target="_self">CSp</a> <font color="Maroon">c<sub>2</sub></font>, B is    <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K3" target="_self">CSp</a> <font color="Maroon">c<sub>2</sub></font> : verum </span>}</span> ;<br/>
<b>thus </b><a NAME="E1:26"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="coh_sp.html#K4" target="_self">FuncsC</a> <font color="Maroon">c<sub>2</sub></font> implies ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K3" target="_self">CSp</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( (  <a href="tarski.html#K3">union</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="tarski.html#K3">union</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) &amp; <font color="Maroon">c<sub>1</sub></font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="tarski.html#K3">union</a> <font color="Olive">b<sub>1</sub></font>, <a href="tarski.html#K3">union</a> <font color="Olive">b<sub>2</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:26_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="coh_sp.html#K4" target="_self">FuncsC</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:26_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E4:26_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <span class="p3">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K3" target="_self">CSp</a> <font color="Maroon">c<sub>2</sub></font>, B is    <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K3" target="_self">CSp</a> <font color="Maroon">c<sub>2</sub></font> : verum </span>}</span> 
 <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>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K3" target="_self">CSp</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E6:26_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="coh_sp.html#E4:26_1"><i><font color="Green">E13</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>thus </b><a NAME="E7:26_1"/>
( (  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) &amp; <font color="Maroon">c<sub>1</sub></font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>5</sub></font>, <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="coh_sp.html#E3:26_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="coh_sp.html#E6:26_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="funct_2.html#T14">FUNCT_2:14</a>, <a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/>


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

<b>given </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K3" target="_self">CSp</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><a NAME="E2:26"/><i><font color="Green">E12</font></i>: 
( (  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) &amp; <font color="Maroon">c<sub>1</sub></font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>4</sub></font>, <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/>

<a NAME="E3:26"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp;  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <span class="p3">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K3" target="_self">CSp</a> <font color="Maroon">c<sub>2</sub></font>, B is    <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K3" target="_self">CSp</a> <font color="Maroon">c<sub>2</sub></font> : verum </span>}</span>  )
 
<b>by </b><i><a class="txt" href="coh_sp.html#E2:26"><i><font color="Green">E12</font></i></a>, <a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<b>hence </b><a NAME="E4:26"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="coh_sp.html#K4" target="_self">FuncsC</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div>
