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<a NAME="E1:38_1_1"/><i><font color="Green">E19</font></i>: 
( (  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K7" target="_self">cod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K6" target="_self">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) &amp; <font color="Maroon">c<sub>2</sub></font> <a href="mcart_1.html#K2">`2</a>  is   <a href="funct_2.html#NM1">Function</a> of  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K6" target="_self">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>, <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K7" target="_self">cod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; ( 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 <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a>  <a href="coh_sp.html#K6" target="_self">dom</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a>  <a href="coh_sp.html#K7" target="_self">cod</a> <font color="Maroon">c<sub>2</sub></font> ) &amp; (  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K7" target="_self">cod</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K6" target="_self">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) &amp; <font color="Maroon">c<sub>3</sub></font> <a href="mcart_1.html#K2">`2</a>  is   <a href="funct_2.html#NM1">Function</a> of  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K6" target="_self">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>, <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K7" target="_self">cod</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; ( 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 <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a>  <a href="coh_sp.html#K6" target="_self">dom</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a>  <a href="coh_sp.html#K7" target="_self">cod</a> <font color="Maroon">c<sub>3</sub></font> ) )
 
<b>by </b><i><a class="ref" href="coh_sp.html#T21" target="_self">Th21</a></i>;<br/>
<a NAME="E2:38_1_1"/><b>then </b><i><font color="Green">E20</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> is   <a href="funct_2.html#NM1">Function</a> of  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K6" target="_self">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>, <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K7" target="_self">cod</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="coh_sp.html#E1:38"><i><font color="Green">E18</font></i></a>, <a class="ref" href="funct_2.html#T19">FUNCT_2:19</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:38_1_1_1"/><i><font color="Green">E21</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a>  <a href="coh_sp.html#K6" target="_self">dom</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:38_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a>  <a href="coh_sp.html#K7" target="_self">cod</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="coh_sp.html#T21" target="_self">Th21</a></i>;<br/><a NAME="E3:38_1_1_1"/><b>then </b><i><font color="Green">E22</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a>  <a href="coh_sp.html#K7" target="_self">cod</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="coh_sp.html#E1:38"><i><font color="Green">E18</font></i></a>, <a class="ref" href="coh_sp.html#T21" target="_self">Th21</a></i>;<br/><a NAME="E4:38_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K6" target="_self">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K6" target="_self">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="coh_sp.html#E1:38_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="coh_sp.html#T16" target="_self">Th16</a></i>;<br/><a NAME="E5:38_1_1_1"/><b>then </b><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="coh_sp.html#E1:38_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E6:38_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a>  <a href="coh_sp.html#K7" target="_self">cod</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="coh_sp.html#E3:38_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><b>hence </b><a NAME="E7:38_1_1_1"/>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a>  <a href="coh_sp.html#K7" target="_self">cod</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="coh_sp.html#E5:38_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:38_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="coh_sp.html#K6" target="_self">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="coh_sp.html#K7" target="_self">cod</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> is    <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K5" target="_self">MapsC</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="coh_sp.html#E1:38"><i><font color="Green">E18</font></i></a>, <a class="txt" href="coh_sp.html#E1:38_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="coh_sp.html#E2:38_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="coh_sp.html#T19" target="_self">Th19</a></i>;<br/>


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