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<a NAME="E1:38_1_1"/><span class="lab"><font color="Green" title="E19">A2</font></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="c3">l2</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="c3">l2</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>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="coh_sp.html#T21" target="_self" title="COH_SP:th.21">Th21</a></span>;<br/>
<a NAME="E2:38_1_1"/><span class="lab"><font color="Green" title="E20">A3</font></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">l1</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">l1</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>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="coh_sp.html#T21" target="_self" title="COH_SP:th.21">Th21</a></span>;<br/>
<a NAME="E3:38_1_1"/><span class="lab"><font color="Green" title="E21">A4</font></span>: 
<font color="Maroon" title="c2">l1</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 <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">l1</font></span>)</span></span>)</span>,<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">l1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="coh_sp.html#T21" target="_self" title="COH_SP:th.21">Th21</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E22">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">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">l1</font> implies <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"><span class="p4">(<span class="default"><font color="Maroon" title="c3">l2</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="p4">(<span class="default"><font color="Maroon" title="c2">l1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">l2</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="p4">(<span class="default"><font color="Maroon" title="c2">l1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">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="c3">l2</font> )</span><br/><span class="kw">assume </span><a NAME="E1:38_1_1_1"/><span class="lab"><font color="Green" title="E23">A6</font></span>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">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">l1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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"><span class="p4">(<span class="default"><font color="Maroon" title="c3">l2</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="p4">(<span class="default"><font color="Maroon" title="c2">l1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">l2</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="p4">(<span class="default"><font color="Maroon" title="c2">l1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">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="c3">l2</font></span><br/><a NAME="E2:38_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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">l1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="coh_sp.html#T16" target="_self" title="COH_SP:th.16">Th16</a></span>;<br/><a NAME="E3:38_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A7</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">l1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38_1_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E3:38_1_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E4:38_1_1_1"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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">l1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_1_1_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="ref" href="coh_sp.html#T16" target="_self" title="COH_SP:th.16">Th16</a></span>;<br/><a NAME="E5:38_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A8</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">l1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38_1_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E3:38_1_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E6:38_1_1_1"/>
<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">l1</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="Maroon" title="c4">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">l1</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="Maroon" title="c5">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">l1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_1_1_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="ref" href="coh_sp.html#T21" target="_self" title="COH_SP:th.21">Th21</a></span>;<br/><a NAME="E7:38_1_1_1"/><span class="kw">then </span>
<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="c3">l2</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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">l1</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="Maroon" title="c4">x</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">l2</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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">l1</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="Maroon" title="c5">y</font></span>)</span></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="c3">l2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="coh_sp.html#T21" target="_self" title="COH_SP:th.21">Th21</a></span>;<br/><a NAME="E8:38_1_1_1"/><span class="kw">then </span>
<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"><span class="p4">(<span class="default"><font color="Maroon" title="c3">l2</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="p4">(<span class="default"><font color="Maroon" title="c2">l1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">l2</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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">l1</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="Maroon" title="c5">y</font></span>)</span></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="c3">l2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_1_1_1"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E9:38_1_1_1"/>
<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"><span class="p4">(<span class="default"><font color="Maroon" title="c3">l2</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="p4">(<span class="default"><font color="Maroon" title="c2">l1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">l2</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="p4">(<span class="default"><font color="Maroon" title="c2">l1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">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="c3">l2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:38_1_1_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E5:38_1_1"/>
<font color="Maroon" title="c3">l2</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 <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="c3">l2</font></span>)</span></span>)</span>,<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="c3">l2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="coh_sp.html#T21" target="_self" title="COH_SP:th.21">Th21</a></span>;<br/>
<a NAME="E6:38_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">l2</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"><font color="Maroon" title="c2">l1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a></span>)</span> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <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">l1</font></span>)</span></span>)</span>,<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="c3">l2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E2:38_1_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E3:38_1_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="funct_2.html#T13" title="FUNCT_2:th.13">FUNCT_2:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:38_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="coh_sp.html#K6" title="COH_SP:func.6">dom</a> <font color="Maroon" title="c2">l1</font></span>)</span>,<span class="p3">(<span class="default"><a href="coh_sp.html#K7" title="COH_SP:func.7">cod</a> <font color="Maroon" title="c3">l2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">l2</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="p3">(<span class="default"><font color="Maroon" title="c2">l1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is    <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E2:38_1_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E1:38_1_1"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E4:38_1_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="ref" href="coh_sp.html#T19" target="_self" title="COH_SP:th.19">Th19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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