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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">C1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K3" target="_self">CSp</a> <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">C2</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K3" target="_self">CSp</a> <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="tarski.html#K3">union</a> <font color="Maroon">C1</font>, <a href="tarski.html#K3">union</a> <font color="Maroon">C2</font>;<br/>







<b>assume </b><a NAME="E1:30"/><i><font color="Green">E17</font></i>: 
( (  <a href="tarski.html#K3">union</a> <font color="Maroon">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="Maroon">C1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) &amp; ( for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> holds <br/><span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C2</font> ) )
 ;<br/>

<b>then reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K4" target="_self">FuncsC</a> <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<a NAME="E3:30"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> holds <br/><span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:30"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">C1</font>,<font color="Maroon">C2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">T</font>,<font color="Olive">TT</font></span><a href="tarski.html#K4">]</a></span>,<font color="Olive">g</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">T</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">TT</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">g</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K4" target="_self">FuncsC</a> <font color="Maroon">X</font> : ( (  <a href="tarski.html#K3">union</a> <font color="Olive">TT</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">T</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) &amp; <font color="Olive">g</font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="tarski.html#K3">union</a> <font color="Olive">T</font>, <a href="tarski.html#K3">union</a> <font color="Olive">TT</font> &amp; ( for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">T</font> holds <br/><span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">TT</font> ) ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:30"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">C1</font>,<font color="Maroon">C2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="coh_sp.html#K5" target="_self">MapsC</a> <font color="Maroon">X</font>
 ;<br/>


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