<?xml version="1.0"?>
<div class="add">

<b>set </b><font color="Maroon" title="c3">t</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> : <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b2">X1</font>,<font color="Olive" title="b3">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> where <font color="Olive" title="b2">X1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b3">Y1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> : ( <font color="Olive" title="b2">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b3">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> ) </span>}</span>  </span>}</span> ;<br/>
<a NAME="E1:38_1_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> : <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b2">X1</font>,<font color="Olive" title="b3">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> where <font color="Olive" title="b2">X1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b3">Y1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> : ( <font color="Olive" title="b2">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b3">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> ) </span>}</span>  </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> : <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b2">X1</font>,<font color="Olive" title="b3">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> where <font color="Olive" title="b2">X1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b3">Y1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> : ( <font color="Olive" title="b2">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b3">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> ) </span>}</span>  </span>}</span>  or <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )</span><br/>

<b>assume </b><a NAME="E1:38_1_1_1"/>
<font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> : <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b2">X1</font>,<font color="Olive" title="b3">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> where <font color="Olive" title="b2">X1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b3">Y1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> : ( <font color="Olive" title="b2">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b3">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> ) </span>}</span>  </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><br/>

<a NAME="E2:38_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">A</font> being  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> st <br/>( <font color="Maroon" title="c4">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b2">X1</font>,<font color="Olive" title="b3">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> where <font color="Olive" title="b2">X1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b3">Y1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> : ( <font color="Olive" title="b2">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b3">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> ) </span>}</span>  )
 
;<br/>
<b>hence </b><a NAME="E3:38_1_1_1"/>
<font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c4">t</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> : <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b2">X1</font>,<font color="Olive" title="b3">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> where <font color="Olive" title="b2">X1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b3">Y1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> : ( <font color="Olive" title="b2">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b3">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> ) </span>}</span>  </span>}</span>  as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ;<br/>
<b>set </b><font color="Maroon" title="c5">T</font> =  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">t</font> #);<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_2"><b>now </b></a><div class="add"><b>reconsider </b><font color="Maroon" title="c6">A</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <b>by </b><i><a class="ref" href="zfmisc_1.html#T80" title="ZFMISC_1:th.80">ZFMISC_1:80</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">A</font> = <font color="Maroon" title="c6">A</font> as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ;<br/><a NAME="E3:38_1_1_2"/><i><font color="Green" title="E29">A1</font></i>: 
<font color="Maroon" title="c7">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> where <font color="Olive" title="b1">X1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Y1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> : ( <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> ) </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font> or <font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> where <font color="Olive" title="b1">X1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Y1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> : ( <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> ) </span>}</span>  )</span><br/>

<b>assume </b><a NAME="E1:38_1_1_2_1"/>
<font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> where <font color="Olive" title="b1">X1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Y1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> : ( <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> ) </span>}</span> </span><br/>

<a NAME="E2:38_1_1_2_1"/><b>then </b><i><font color="Green" title="E30">A2</font></i>: 
<font color="Maroon" title="c8">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:38_1_1_2_1"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#D1" title="PRE_TOPC:def.1">PRE_TOPC:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:38_1_1_2_1"/>
<font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> where <font color="Olive" title="b1">X1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Y1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> : ( <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="#E2:38_1_1_2_1"><i><font color="Green" title="E30">A2</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/>


</div><b>end;</b></div><a NAME="E4:38_1_1_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">t</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c7">A</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></i>;<br/><b>hence </b><a NAME="E5:38_1_1_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">t</font> #) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">t</font> #)
 <b>by </b><i><a class="txt" href="#E3:38_1_1_2"><i><font color="Green" title="E29">A1</font></i></a></i>; <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">a</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">t</font> #)  st <font color="Olive" title="b1">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">t</font> #) holds <br/> <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">t</font> #) ) &amp; ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">t</font> #)  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">t</font> #) &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">t</font> #) holds <br/><font color="Olive" title="b1">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">t</font> #) ) )</span><br/><b>thus </b><a NAME="E6:38_1_1_2"/>
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">t</font> #)  st <font color="Olive" title="b1">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">t</font> #) holds <br/> <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">t</font> #)
  <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">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">t</font> #)  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">t</font> #) &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">t</font> #) holds <br/><font color="Olive" title="b1">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">t</font> #)</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">a</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">t</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">t</font> #) implies  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">t</font> #) )</span><br/>

<b>assume </b><a NAME="E1:38_1_1_2_2"/><i><font color="Green" title="E30">A3</font></i>: 
<font color="Maroon" title="c8">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">t</font> #)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">t</font> #)</span><br/>

<b>set </b><font color="Maroon" title="c9">A</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> where <font color="Olive" title="b1">X1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Y1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> : ( <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> &amp;  ex <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">a</font> &amp; <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">x</font> ) ) </span>}</span> ;<br/>
<a NAME="E2:38_1_1_2_2"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> where <font color="Olive" title="b1">X1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Y1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> : ( <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> &amp;  ex <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">a</font> &amp; <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">x</font> ) ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> where <font color="Olive" title="b1">X1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Y1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> : ( <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> &amp;  ex <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">a</font> &amp; <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">x</font> ) ) </span>}</span>  or <font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )</span><br/>

<b>assume </b><a NAME="E1:38_1_1_2_2_1"/>
<font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> where <font color="Olive" title="b1">X1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Y1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> : ( <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> &amp;  ex <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">a</font> &amp; <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">x</font> ) ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><br/>

<a NAME="E2:38_1_1_2_2_1"/><b>then </b>
 ex <font color="Olive" title="b1">X1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ex <font color="Olive" title="b2">Y1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> st <br/>( <font color="Maroon" title="c10">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> &amp; <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> &amp;  ex <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">a</font> &amp; <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">x</font> ) )
 
;<br/>
<b>hence </b><a NAME="E3:38_1_1_2_2_1"/>
<font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c10">A</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> where <font color="Olive" title="b1">X1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Y1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> : ( <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> &amp;  ex <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">a</font> &amp; <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">x</font> ) ) </span>}</span>  as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ;<br/>
<a NAME="E4:38_1_1_2_2"/><i><font color="Green" title="E31">A4</font></i>: 
<font color="Maroon" title="c10">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X2</font>,<font color="Olive" title="b2">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> where <font color="Olive" title="b1">X2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Y2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> : ( <font color="Olive" title="b1">X2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">Y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_2_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font> or <font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X2</font>,<font color="Olive" title="b2">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> where <font color="Olive" title="b1">X2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Y2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> : ( <font color="Olive" title="b1">X2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">Y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> ) </span>}</span>  )</span><br/>

<b>assume </b><a NAME="E1:38_1_1_2_2_2"/>
<font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X2</font>,<font color="Olive" title="b2">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> where <font color="Olive" title="b1">X2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Y2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> : ( <font color="Olive" title="b1">X2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">Y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> ) </span>}</span> </span><br/>

<a NAME="E2:38_1_1_2_2_2"/><b>then </b>
 ex <font color="Olive" title="b1">X1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ex <font color="Olive" title="b2">Y1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> st <br/>( <font color="Maroon" title="c11">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> &amp; <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> &amp;  ex <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">a</font> &amp; <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">x</font> ) )
 
;<br/>
<b>hence </b><a NAME="E3:38_1_1_2_2_2"/>
<font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X2</font>,<font color="Olive" title="b2">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> where <font color="Olive" title="b1">X2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Y2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> : ( <font color="Olive" title="b1">X2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">Y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E5:38_1_1_2_2"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c10">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c8">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_2_2_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:38_1_1_2_2_3"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c10">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c8">a</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c8">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c10">A</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_2_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">e1</font>, <font color="Maroon" title="c12">e2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_1.html#D3" title="RELAT_1:def.3">RELAT_1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">e1</font>,<font color="Maroon" title="c12">e2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c10">A</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">e1</font>,<font color="Maroon" title="c12">e2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c8">a</font> )</span><br/>

<b>assume </b><a NAME="E1:38_1_1_2_2_3_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">e1</font>,<font color="Maroon" title="c12">e2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c10">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">e1</font>,<font color="Maroon" title="c12">e2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c8">a</font></span><br/>

<b>then consider </b><font color="Maroon" title="c13">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:38_1_1_2_2_3_1"/><i><font color="Green" title="E32">A5</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">e1</font>,<font color="Maroon" title="c12">e2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">u</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">X1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c15">Y1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font><b> such that </b><br/><a NAME="E5:38_1_1_2_2_3_1"/><i><font color="Green" title="E33">A6</font></i>: 
<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c14">X1</font>,<font color="Maroon" title="c15">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 <b>and </b><br/><a NAME="E6:38_1_1_2_2_3_1"/>
( <font color="Maroon" title="c14">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c15">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> )
 <b>and </b><br/><a NAME="E7:38_1_1_2_2_3_1"/><i><font color="Green" title="E34">A7</font></i>: 
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">a</font> &amp; <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c14">X1</font>,<font color="Maroon" title="c15">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">x</font> )
 <b>by </b><i><a class="txt" href="#E3:38_1_1_2_2_3_1"><i><font color="Green" title="E32">A5</font></i></a></i>;<br/>
<b>thus </b><a NAME="E8:38_1_1_2_2_3_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">e1</font>,<font color="Maroon" title="c12">e2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c8">a</font>
 <b>by </b><i><a class="txt" href="#E3:38_1_1_2_2_3_1"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="#E5:38_1_1_2_2_3_1"><i><font color="Green" title="E33">A6</font></i></a>, <a class="txt" href="#E7:38_1_1_2_2_3_1"><i><font color="Green" title="E34">A7</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c11">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c8">a</font> or <font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c10">A</font> )</span><br/>

<b>assume </b><a NAME="E2:38_1_1_2_2_3"/>
<font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c8">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c10">A</font></span><br/>

<b>then consider </b><font color="Maroon" title="c12">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:38_1_1_2_2_3"/><i><font color="Green" title="E32">A8</font></i>: 
( <font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">u</font> &amp; <font color="Maroon" title="c12">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">a</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E5:38_1_1_2_2_3"/>
<font color="Maroon" title="c12">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">t</font> #)
 
<b>by </b><i><a class="txt" href="#E1:38_1_1_2_2"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="#E4:38_1_1_2_2_3"><i><font color="Green" title="E32">A8</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">B</font> being   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span><b> such that </b><br/><a NAME="E7:38_1_1_2_2_3"/><i><font color="Green" title="E33">A9</font></i>: 
<font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c13">B</font>
 <b>and </b><br/><a NAME="E8:38_1_1_2_2_3"/><i><font color="Green" title="E34">A10</font></i>: 
<font color="Maroon" title="c13">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> where <font color="Olive" title="b1">X1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Y1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> : ( <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> ) </span>}</span> 
 ;<br/>
<b>consider </b><font color="Maroon" title="c14">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:38_1_1_2_2_3"/><i><font color="Green" title="E35">A11</font></i>: 
( <font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">x</font> &amp; <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">B</font> )
 <b>by </b><i><a class="txt" href="#E4:38_1_1_2_2_3"><i><font color="Green" title="E32">A8</font></i></a>, <a class="txt" href="#E7:38_1_1_2_2_3"><i><font color="Green" title="E33">A9</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E11:38_1_1_2_2_3"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> where <font color="Olive" title="b1">X1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Y1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> : ( <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="#E8:38_1_1_2_2_3"><i><font color="Green" title="E34">A10</font></i></a>, <a class="txt" href="#E10:38_1_1_2_2_3"><i><font color="Green" title="E35">A11</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">X1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c16">Y1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font><b> such that </b><br/><a NAME="E13:38_1_1_2_2_3"/><i><font color="Green" title="E36">A12</font></i>: 
( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c15">X1</font>,<font color="Maroon" title="c16">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> &amp; <font color="Maroon" title="c15">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c16">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> )
 ;<br/>
<a NAME="E14:38_1_1_2_2_3"/>
<span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c15">X1</font>,<font color="Maroon" title="c16">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">u</font>
 
<b>by </b><i><a class="txt" href="#E7:38_1_1_2_2_3"><i><font color="Green" title="E33">A9</font></i></a>, <a class="txt" href="#E10:38_1_1_2_2_3"><i><font color="Green" title="E35">A11</font></i></a>, <a class="txt" href="#E13:38_1_1_2_2_3"><i><font color="Green" title="E36">A12</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E15:38_1_1_2_2_3"/><b>then </b>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font>
 
<b>by </b><i><a class="txt" href="#E4:38_1_1_2_2_3"><i><font color="Green" title="E32">A8</font></i></a>, <a class="txt" href="#E13:38_1_1_2_2_3"><i><font color="Green" title="E36">A12</font></i></a></i>;<br/>
<b>hence </b><a NAME="E16:38_1_1_2_2_3"/>
<font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c10">A</font>
 <b>by </b><i><a class="txt" href="#E10:38_1_1_2_2_3"><i><font color="Green" title="E35">A11</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:38_1_1_2_2"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">t</font> #)
 <b>by </b><i><a class="txt" href="#E4:38_1_1_2_2"><i><font color="Green" title="E31">A4</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/>


</div><b>end;</b></div><b>let </b><font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c9">b</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">t</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">t</font> #) &amp; <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">t</font> #) implies <font color="Maroon" title="c8">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">t</font> #) )</span><br/><b>assume </b><b>that </b><br/><a NAME="E7:38_1_1_2"/><i><font color="Green" title="E30">A13</font></i>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">t</font> #)
 <b>and </b><br/><a NAME="E8:38_1_1_2"/><i><font color="Green" title="E31">A14</font></i>: 
<font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">t</font> #)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">t</font> #)</span><br/><b>consider </b><font color="Maroon" title="c10">A</font> being   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span><b> such that </b><br/><a NAME="E10:38_1_1_2"/><i><font color="Green" title="E32">A15</font></i>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c10">A</font>
 <b>and </b><br/><a NAME="E11:38_1_1_2"/><i><font color="Green" title="E33">A16</font></i>: 
<font color="Maroon" title="c10">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> where <font color="Olive" title="b1">X1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Y1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> : ( <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="#E7:38_1_1_2"><i><font color="Green" title="E30">A13</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c11">B</font> being   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span><b> such that </b><br/><a NAME="E13:38_1_1_2"/><i><font color="Green" title="E34">A17</font></i>: 
<font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c11">B</font>
 <b>and </b><br/><a NAME="E14:38_1_1_2"/><i><font color="Green" title="E35">A18</font></i>: 
<font color="Maroon" title="c11">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> where <font color="Olive" title="b1">X1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Y1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> : ( <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="#E8:38_1_1_2"><i><font color="Green" title="E31">A14</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c12">C</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> where <font color="Olive" title="b1">X1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Y1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> : ( <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> &amp; <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">b</font> ) </span>}</span> ;<br/><a NAME="E15:38_1_1_2"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> where <font color="Olive" title="b1">X1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Y1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> : ( <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> &amp; <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">b</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c13">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> where <font color="Olive" title="b1">X1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Y1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> : ( <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> &amp; <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">b</font> ) </span>}</span>  or <font color="Maroon" title="c13">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )</span><br/>

<b>assume </b><a NAME="E1:38_1_1_2_3"/>
<font color="Maroon" title="c13">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> where <font color="Olive" title="b1">X1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Y1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> : ( <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> &amp; <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">b</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><br/>

<a NAME="E2:38_1_1_2_3"/><b>then </b>
 ex <font color="Olive" title="b1">X1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ex <font color="Olive" title="b2">Y1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> st <br/>( <font color="Maroon" title="c13">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> &amp; <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> &amp; <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">b</font> )
 
;<br/>
<b>hence </b><a NAME="E3:38_1_1_2_3"/>
<font color="Maroon" title="c13">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c13">C</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> where <font color="Olive" title="b1">X1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Y1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> : ( <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> &amp; <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">b</font> ) </span>}</span>  as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ;<br/><a NAME="E17:38_1_1_2"/><i><font color="Green" title="E36">A19</font></i>: 
<font color="Maroon" title="c8">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c13">C</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_2_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:38_1_1_2_4"/>
<font color="Maroon" title="c8">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c13">C</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c13">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">b</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_2_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">b</font> or <font color="Maroon" title="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c13">C</font> )</span><br/>

<b>assume </b><a NAME="E1:38_1_1_2_4_1"/>
<font color="Maroon" title="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c13">C</font></span><br/>

<a NAME="E2:38_1_1_2_4_1"/><b>then </b><i><font color="Green" title="E37">A20</font></i>: 
( <font color="Maroon" title="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">a</font> &amp; <font color="Maroon" title="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">b</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">u1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:38_1_1_2_4_1"/><i><font color="Green" title="E38">A21</font></i>: 
( <font color="Maroon" title="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">u1</font> &amp; <font color="Maroon" title="c15">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font> )
 <b>by </b><i><a class="txt" href="#E10:38_1_1_2"><i><font color="Green" title="E32">A15</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">u2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:38_1_1_2_4_1"/><i><font color="Green" title="E39">A22</font></i>: 
( <font color="Maroon" title="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">u2</font> &amp; <font color="Maroon" title="c16">u2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">B</font> )
 <b>by </b><i><a class="txt" href="#E13:38_1_1_2"><i><font color="Green" title="E34">A17</font></i></a>, <a class="txt" href="#E2:38_1_1_2_4_1"><i><font color="Green" title="E37">A20</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E7:38_1_1_2_4_1"/>
<font color="Maroon" title="c15">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> where <font color="Olive" title="b1">X1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Y1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> : ( <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="#E11:38_1_1_2"><i><font color="Green" title="E33">A16</font></i></a>, <a class="txt" href="#E4:38_1_1_2_4_1"><i><font color="Green" title="E38">A21</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c17">X1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c18">Y1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font><b> such that </b><br/><a NAME="E9:38_1_1_2_4_1"/><i><font color="Green" title="E40">A23</font></i>: 
( <font color="Maroon" title="c15">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c17">X1</font>,<font color="Maroon" title="c18">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> &amp; <font color="Maroon" title="c17">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c18">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> )
 ;<br/>
<a NAME="E10:38_1_1_2_4_1"/>
<font color="Maroon" title="c16">u2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X2</font>,<font color="Olive" title="b2">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> where <font color="Olive" title="b1">X2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Y2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> : ( <font color="Olive" title="b1">X2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">Y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="#E14:38_1_1_2"><i><font color="Green" title="E35">A18</font></i></a>, <a class="txt" href="#E6:38_1_1_2_4_1"><i><font color="Green" title="E39">A22</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c19">X2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c20">Y2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font><b> such that </b><br/><a NAME="E12:38_1_1_2_4_1"/><i><font color="Green" title="E41">A24</font></i>: 
( <font color="Maroon" title="c16">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c19">X2</font>,<font color="Maroon" title="c20">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> &amp; <font color="Maroon" title="c19">X2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c20">Y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> )
 ;<br/>
<a NAME="E13:38_1_1_2_4_1"/><i><font color="Green" title="E42">A25</font></i>: 
( <font color="Maroon" title="c17">X1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c19">X2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c18">Y1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c20">Y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> )
 
<b>by </b><i><a class="txt" href="#E9:38_1_1_2_4_1"><i><font color="Green" title="E40">A23</font></i></a>, <a class="txt" href="#E12:38_1_1_2_4_1"><i><font color="Green" title="E41">A24</font></i></a>, <a class="ref" href="pre_topc.html#D1" title="PRE_TOPC:def.1">PRE_TOPC:def 1</a></i>;<br/>
<a NAME="E14:38_1_1_2_4_1"/>
( <font color="Maroon" title="c15">u1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">a</font> &amp; <font color="Maroon" title="c16">u2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">b</font> )
 
<b>by </b><i><a class="txt" href="#E10:38_1_1_2"><i><font color="Green" title="E32">A15</font></i></a>, <a class="txt" href="#E13:38_1_1_2"><i><font color="Green" title="E34">A17</font></i></a>, <a class="txt" href="#E4:38_1_1_2_4_1"><i><font color="Green" title="E38">A21</font></i></a>, <a class="txt" href="#E6:38_1_1_2_4_1"><i><font color="Green" title="E39">A22</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E15:38_1_1_2_4_1"/><b>then </b>
<span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c17">X1</font>,<font color="Maroon" title="c18">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c19">X2</font>,<font color="Maroon" title="c20">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">b</font>
 
<b>by </b><i><a class="txt" href="#E9:38_1_1_2_4_1"><i><font color="Green" title="E40">A23</font></i></a>, <a class="txt" href="#E12:38_1_1_2_4_1"><i><font color="Green" title="E41">A24</font></i></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></i>;<br/>
<a NAME="E16:38_1_1_2_4_1"/><b>then </b>
<span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">X1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c19">X2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">Y1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c20">Y2</font></span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">b</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T123" title="ZFMISC_1:th.123">ZFMISC_1:123</a></i>;<br/>
<a NAME="E17:38_1_1_2_4_1"/><b>then </b><i><font color="Green" title="E43">A26</font></i>: 
<span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">X1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c19">X2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">Y1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c20">Y2</font></span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">C</font>
 
<b>by </b><i><a class="txt" href="#E13:38_1_1_2_4_1"><i><font color="Green" title="E42">A25</font></i></a></i>;<br/>
<a NAME="E18:38_1_1_2_4_1"/>
<font color="Maroon" title="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">X1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c19">X2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">Y1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c20">Y2</font></span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E4:38_1_1_2_4_1"><i><font color="Green" title="E38">A21</font></i></a>, <a class="txt" href="#E6:38_1_1_2_4_1"><i><font color="Green" title="E39">A22</font></i></a>, <a class="txt" href="#E9:38_1_1_2_4_1"><i><font color="Green" title="E40">A23</font></i></a>, <a class="txt" href="#E12:38_1_1_2_4_1"><i><font color="Green" title="E41">A24</font></i></a>, <a class="ref" href="zfmisc_1.html#T137" title="ZFMISC_1:th.137">ZFMISC_1:137</a></i>;<br/>
<b>hence </b><a NAME="E19:38_1_1_2_4_1"/>
<font color="Maroon" title="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c13">C</font>
 <b>by </b><i><a class="txt" href="#E17:38_1_1_2_4_1"><i><font color="Green" title="E43">A26</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c14">e1</font>, <font color="Maroon" title="c15">e2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_1.html#D3" title="RELAT_1:def.3">RELAT_1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">e1</font>,<font color="Maroon" title="c15">e2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c13">C</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">e1</font>,<font color="Maroon" title="c15">e2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">b</font> )</span><br/>

<b>assume </b><a NAME="E2:38_1_1_2_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">e1</font>,<font color="Maroon" title="c15">e2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c13">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">e1</font>,<font color="Maroon" title="c15">e2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">b</font></span><br/>

<b>then consider </b><font color="Maroon" title="c16">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:38_1_1_2_4"/><i><font color="Green" title="E37">A27</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">e1</font>,<font color="Maroon" title="c15">e2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">u</font> &amp; <font color="Maroon" title="c16">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">C</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E5:38_1_1_2_4"/>
 ex <font color="Olive" title="b1">X1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ex <font color="Olive" title="b2">Y1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> st <br/>( <font color="Maroon" title="c16">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> &amp; <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> &amp; <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">b</font> )
 
<b>by </b><i><a class="txt" href="#E4:38_1_1_2_4"><i><font color="Green" title="E37">A27</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:38_1_1_2_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">e1</font>,<font color="Maroon" title="c15">e2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">b</font>
 <b>by </b><i><a class="txt" href="#E4:38_1_1_2_4"><i><font color="Green" title="E37">A27</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/>


</div><b>end;</b></div><a NAME="E18:38_1_1_2"/>
<font color="Maroon" title="c13">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> where <font color="Olive" title="b1">X1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Y1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> : ( <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> ) </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">C</font> or <font color="Maroon" title="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> where <font color="Olive" title="b1">X1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Y1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> : ( <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> ) </span>}</span>  )</span><br/>

<b>assume </b><a NAME="E1:38_1_1_2_5"/>
<font color="Maroon" title="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> where <font color="Olive" title="b1">X1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Y1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> : ( <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> ) </span>}</span> </span><br/>

<a NAME="E2:38_1_1_2_5"/><b>then </b>
 ex <font color="Olive" title="b1">X1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ex <font color="Olive" title="b2">Y1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> st <br/>( <font color="Maroon" title="c14">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> &amp; <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> &amp; <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">b</font> )
 
;<br/>
<b>hence </b><a NAME="E3:38_1_1_2_5"/>
<font color="Maroon" title="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> where <font color="Olive" title="b1">X1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Y1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> : ( <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E19:38_1_1_2"/>
<font color="Maroon" title="c8">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">t</font> #)
 <b>by </b><i><a class="txt" href="#E17:38_1_1_2"><i><font color="Green" title="E36">A19</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/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c6">T</font> =  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">t</font> #) as  <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a> <b>by </b><i><a class="ref" href="pre_topc.html#D1" title="PRE_TOPC:def.1">PRE_TOPC:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">T</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c6">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c6">T</font> : <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b2">X1</font>,<font color="Olive" title="b3">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> where <font color="Olive" title="b2">X1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b3">Y1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> : ( <font color="Olive" title="b2">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b3">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> ) </span>}</span>  </span>}</span>  )</span><br/>

<b>thus </b><a NAME="E5:38_1_1"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c6">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c6">T</font> : <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b2">X1</font>,<font color="Olive" title="b3">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> where <font color="Olive" title="b2">X1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b3">Y1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> : ( <font color="Olive" title="b2">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b3">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> ) </span>}</span>  </span>}</span>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
