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<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">l</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="coh_sp.html#K5" title="COH_SP:func.5">MapsC</a> <font color="Maroon" title="c1">X</font> holds <br/> ( <font color="Olive" title="b1">l</font> <a href="coh_sp.html#K9" title="COH_SP:func.9">*</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K8" title="COH_SP:func.8">id$</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K6" title="COH_SP:func.6">dom</a> <font color="Olive" title="b1">l</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">l</font> &amp; <span class="p1">(<span class="default"><a href="coh_sp.html#K8" title="COH_SP:func.8">id$</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K7" title="COH_SP:func.7">cod</a> <font color="Olive" title="b1">l</font></span>)</span></span>)</span> <a href="coh_sp.html#K9" title="COH_SP:func.9">*</a> <font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">l</font> )</span><br/><b>let </b><font color="Maroon" title="c2">l</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="coh_sp.html#K5" title="COH_SP:func.5">MapsC</a> <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">l</font> <a href="coh_sp.html#K9" title="COH_SP:func.9">*</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K8" title="COH_SP:func.8">id$</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K6" title="COH_SP:func.6">dom</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">l</font> &amp; <span class="p1">(<span class="default"><a href="coh_sp.html#K8" title="COH_SP:func.8">id$</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K7" title="COH_SP:func.7">cod</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span> <a href="coh_sp.html#K9" title="COH_SP:func.9">*</a> <font color="Maroon" title="c2">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">l</font> )</span><br/>



<b>set </b><font color="Maroon" title="c3">i1</font> =  <a href="coh_sp.html#K8" title="COH_SP:func.8">id$</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K6" title="COH_SP:func.6">dom</a> <font color="Maroon" title="c2">l</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c4">i2</font> =  <a href="coh_sp.html#K8" title="COH_SP:func.8">id$</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K7" title="COH_SP:func.7">cod</a> <font color="Maroon" title="c2">l</font></span>)</span>;<br/>
<a NAME="E1:42"/><i><font color="Green" title="E23">A1</font></i>: 
( (  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K7" title="COH_SP:func.7">cod</a> <font color="Maroon" title="c2">l</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K6" title="COH_SP:func.6">dom</a> <font color="Maroon" title="c2">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; <font color="Maroon" title="c2">l</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K6" title="COH_SP:func.6">dom</a> <font color="Maroon" title="c2">l</font></span>)</span>, <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K7" title="COH_SP:func.7">cod</a> <font color="Maroon" title="c2">l</font></span>)</span> &amp; ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="coh_sp.html#K6" title="COH_SP:func.6">dom</a> <font color="Maroon" title="c2">l</font> holds <br/><span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="coh_sp.html#K7" title="COH_SP:func.7">cod</a> <font color="Maroon" title="c2">l</font> ) )
 
<b>by </b><i><a class="ref" href="coh_sp.html#T21" target="_self" title="COH_SP:th.21">Th21</a></i>;<br/>
<a NAME="E2:42"/>
 <a href="coh_sp.html#K7" title="COH_SP:func.7">cod</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K8" title="COH_SP:func.8">id$</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K6" title="COH_SP:func.6">dom</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="coh_sp.html#K6" title="COH_SP:func.6">dom</a> <font color="Maroon" title="c2">l</font>
 
<b>by </b><i><a class="ref" href="coh_sp.html#T24" target="_self" title="COH_SP:th.24">Th24</a></i>;<br/>
<a NAME="E3:42"/><b>then </b>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K6" title="COH_SP:func.6">dom</a> <font color="Maroon" title="c2">l</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="coh_sp.html#K8" title="COH_SP:func.8">id$</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K6" title="COH_SP:func.6">dom</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K6" title="COH_SP:func.6">dom</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span> &amp;  <a href="coh_sp.html#K6" title="COH_SP:func.6">dom</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K8" title="COH_SP:func.8">id$</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K6" title="COH_SP:func.6">dom</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="coh_sp.html#K6" title="COH_SP:func.6">dom</a> <font color="Maroon" title="c2">l</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="coh_sp.html#K9" title="COH_SP:func.9">*</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K8" title="COH_SP:func.8">id$</a> <span class="p3">(<span class="default"><a href="coh_sp.html#K6" title="COH_SP:func.6">dom</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="coh_sp.html#K8" title="COH_SP:func.8">id$</a> <span class="p3">(<span class="default"><a href="coh_sp.html#K6" title="COH_SP:func.6">dom</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> &amp;  <a href="coh_sp.html#K6" title="COH_SP:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="coh_sp.html#K9" title="COH_SP:func.9">*</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K8" title="COH_SP:func.8">id$</a> <span class="p3">(<span class="default"><a href="coh_sp.html#K6" title="COH_SP:func.6">dom</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="coh_sp.html#K6" title="COH_SP:func.6">dom</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K8" title="COH_SP:func.8">id$</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K6" title="COH_SP:func.6">dom</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span> &amp;  <a href="coh_sp.html#K7" title="COH_SP:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="coh_sp.html#K9" title="COH_SP:func.9">*</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K8" title="COH_SP:func.8">id$</a> <span class="p3">(<span class="default"><a href="coh_sp.html#K6" title="COH_SP:func.6">dom</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="coh_sp.html#K7" title="COH_SP:func.7">cod</a> <font color="Maroon" title="c2">l</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">l</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  )
 
<b>by </b><i><a class="txt" href="coh_sp.html#E1:42"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="coh_sp.html#T22" target="_self" title="COH_SP:th.22">Th22</a>, <a class="ref" href="coh_sp.html#T24" target="_self" title="COH_SP:th.24">Th24</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relat_1.html#T78" title="RELAT_1:th.78">RELAT_1:78</a></i>;<br/>
<b>hence </b><a NAME="E4:42"/>
<font color="Maroon" title="c2">l</font> <a href="coh_sp.html#K9" title="COH_SP:func.9">*</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K8" title="COH_SP:func.8">id$</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K6" title="COH_SP:func.6">dom</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">l</font>
 <b>by </b><i><a class="txt" href="coh_sp.html#E32"><i><font color="Green" title="E15">Lm1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="coh_sp.html#K8" title="COH_SP:func.8">id$</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K7" title="COH_SP:func.7">cod</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span> <a href="coh_sp.html#K9" title="COH_SP:func.9">*</a> <font color="Maroon" title="c2">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">l</font></span><br/>

<a NAME="E5:42"/>
(  <a href="coh_sp.html#K6" title="COH_SP:func.6">dom</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K8" title="COH_SP:func.8">id$</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K7" title="COH_SP:func.7">cod</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="coh_sp.html#K7" title="COH_SP:func.7">cod</a> <font color="Maroon" title="c2">l</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K7" title="COH_SP:func.7">cod</a> <font color="Maroon" title="c2">l</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="coh_sp.html#E1:42"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="coh_sp.html#T24" target="_self" title="COH_SP:th.24">Th24</a>, <a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<a NAME="E6:42"/><b>then </b>
( <span class="p1">(<span class="default"><a href="coh_sp.html#K8" title="COH_SP:func.8">id$</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K7" title="COH_SP:func.7">cod</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K7" title="COH_SP:func.7">cod</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span> &amp;  <a href="coh_sp.html#K7" title="COH_SP:func.7">cod</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K8" title="COH_SP:func.8">id$</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K7" title="COH_SP:func.7">cod</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="coh_sp.html#K7" title="COH_SP:func.7">cod</a> <font color="Maroon" title="c2">l</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="coh_sp.html#K8" title="COH_SP:func.8">id$</a> <span class="p3">(<span class="default"><a href="coh_sp.html#K7" title="COH_SP:func.7">cod</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span> <a href="coh_sp.html#K9" title="COH_SP:func.9">*</a> <font color="Maroon" title="c2">l</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="coh_sp.html#K8" title="COH_SP:func.8">id$</a> <span class="p3">(<span class="default"><a href="coh_sp.html#K7" title="COH_SP:func.7">cod</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> &amp;  <a href="coh_sp.html#K6" title="COH_SP:func.6">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="coh_sp.html#K8" title="COH_SP:func.8">id$</a> <span class="p3">(<span class="default"><a href="coh_sp.html#K7" title="COH_SP:func.7">cod</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span> <a href="coh_sp.html#K9" title="COH_SP:func.9">*</a> <font color="Maroon" title="c2">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="coh_sp.html#K6" title="COH_SP:func.6">dom</a> <font color="Maroon" title="c2">l</font> &amp;  <a href="coh_sp.html#K7" title="COH_SP:func.7">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="coh_sp.html#K8" title="COH_SP:func.8">id$</a> <span class="p3">(<span class="default"><a href="coh_sp.html#K7" title="COH_SP:func.7">cod</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span> <a href="coh_sp.html#K9" title="COH_SP:func.9">*</a> <font color="Maroon" title="c2">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="coh_sp.html#K7" title="COH_SP:func.7">cod</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K8" title="COH_SP:func.8">id$</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K7" title="COH_SP:func.7">cod</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="coh_sp.html#K7" title="COH_SP:func.7">cod</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">l</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  )
 
<b>by </b><i><a class="ref" href="coh_sp.html#T22" target="_self" title="COH_SP:th.22">Th22</a>, <a class="ref" href="coh_sp.html#T24" target="_self" title="COH_SP:th.24">Th24</a>, <a class="ref" href="relat_1.html#T79" title="RELAT_1:th.79">RELAT_1:79</a></i>;<br/>
<b>hence </b><a NAME="E7:42"/>
<span class="p1">(<span class="default"><a href="coh_sp.html#K8" title="COH_SP:func.8">id$</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K7" title="COH_SP:func.7">cod</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span> <a href="coh_sp.html#K9" title="COH_SP:func.9">*</a> <font color="Maroon" title="c2">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">l</font>
 <b>by </b><i><a class="txt" href="coh_sp.html#E32"><i><font color="Green" title="E15">Lm1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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