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<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>set </b><font color="Maroon">F</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">xx</font></span>)</span>,<span class="p3">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Olive">y</font></span>)</span></span>)</span> where <font color="Olive">xx</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K3" target="_self">CSp</a> <font color="Maroon">X</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K3" target="_self">CSp</a> <font color="Maroon">X</font> : verum </span>}</span> ;<br/>
<b>thus </b><a NAME="E1:26"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="coh_sp.html#K4" target="_self">FuncsC</a> <font color="Maroon">X</font> implies  ex <font color="Olive">C1</font>, <font color="Olive">C2</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K3" target="_self">CSp</a> <font color="Maroon">X</font> st <br/>( (  <a href="tarski.html#K3">union</a> <font color="Olive">C2</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">C1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) &amp; <font color="Maroon">x</font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="tarski.html#K3">union</a> <font color="Olive">C1</font>, <a href="tarski.html#K3">union</a> <font color="Olive">C2</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">x</font> <a href="hidden.html#R2">in</a>  <a href="coh_sp.html#K4" target="_self">FuncsC</a> <font color="Maroon">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon">C</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:26_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E4:26_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">C</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">xx</font></span>)</span>,<span class="p3">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Olive">y</font></span>)</span></span>)</span> where <font color="Olive">xx</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K3" target="_self">CSp</a> <font color="Maroon">X</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K3" target="_self">CSp</a> <font color="Maroon">X</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">A</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K3" target="_self">CSp</a> <font color="Maroon">X</font>, <font color="Maroon">B</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K3" target="_self">CSp</a> <font color="Maroon">X</font><b> such that </b><br/><a NAME="E6:26_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">C</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">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">A</font>
;<br/>

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

<b>thus </b><a NAME="E7:26_1"/>
( (  <a href="tarski.html#K3">union</a> <font color="Maroon">B</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">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) &amp; <font color="Maroon">x</font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="tarski.html#K3">union</a> <font color="Maroon">A</font>, <a href="tarski.html#K3">union</a> <font color="Maroon">B</font> )
 <b>by </b><i>, , <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">A</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K3" target="_self">CSp</a> <font color="Maroon">X</font>, <font color="Maroon">B</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K3" target="_self">CSp</a> <font color="Maroon">X</font><b> such that </b><a NAME="E2:26"/><i><font color="Green">E27</font></i>: 
( (  <a href="tarski.html#K3">union</a> <font color="Maroon">B</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">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) &amp; <font color="Maroon">x</font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="tarski.html#K3">union</a> <font color="Maroon">A</font>, <a href="tarski.html#K3">union</a> <font color="Maroon">B</font> )
 ;<br/>

<a NAME="E3:26"/>
( <font color="Maroon">x</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">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">B</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">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">B</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">xx</font></span>)</span>,<span class="p3">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Olive">y</font></span>)</span></span>)</span> where <font color="Olive">xx</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K3" target="_self">CSp</a> <font color="Maroon">X</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K3" target="_self">CSp</a> <font color="Maroon">X</font> : verum </span>}</span>  )
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<b>hence </b><a NAME="E4:26"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="coh_sp.html#K4" target="_self">FuncsC</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


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