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

<b>let </b><font color="Maroon">T1</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">T2</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">D</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">T1</font>,<font color="Maroon">T2</font></span><a href="borsuk_1.html#K5">:]</a></span>;<br/>





<b>assume </b><a NAME="E1:6"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">D</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>let </b><font color="Maroon">x1</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T1</font>;<br/><b>let </b><font color="Maroon">x2</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T2</font>;<br/>



<b>set </b><font color="Maroon">cT1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>;<br/>
<b>set </b><font color="Maroon">cT2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font>;<br/>
<b>let </b><font color="Maroon">X1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T1</font>;<br/><b>let </b><font color="Maroon">X2</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T2</font>;<br/>



<b>consider </b><font color="Maroon">FA</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">T1</font>,<font color="Maroon">T2</font></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E3:6"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">D</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">FA</font> &amp; ( for <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FA</font> holds <br/> ex <font color="Olive">B1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T1</font> ex <font color="Olive">B2</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T2</font> st <br/>( <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">B1</font>,<font color="Olive">B2</font></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <font color="Olive">B1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">B2</font> is <a href="pre_topc.html#V3">open</a> ) ) )
 <b>by </b><i>, <a class="ref" href="borsuk_1.html#T45">BORSUK_1:45</a></i>;<br/>
<b>thus </b><a NAME="E4:6"/>
( <font color="Maroon">X1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K9">pr1</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="subset_1.html#K8">/\</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x2</font></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> implies <font color="Maroon">X1</font> is <a href="pre_topc.html#V3">open</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">X1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K9">pr1</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="subset_1.html#K8">/\</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x2</font></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 ;<br/>

<a NAME="E2:6_1"/>
for <font color="Olive">t</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> iff  ex <font color="Olive">U</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T1</font> st <br/>( <font color="Olive">U</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">U</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X1</font> &amp; <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Olive">U</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">t</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<a NAME="E1:6_1_1"/>
( <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> implies  ex <font color="Olive">U</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T1</font> st <br/>( <font color="Olive">U</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">U</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X1</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Olive">U</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_1_1_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">tx</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:6_1_1_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">tx</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="funct_3.html#K9">pr1</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font></span>)</span> &amp; <font color="Maroon">tx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> <a href="subset_1.html#K8">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x2</font></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K9">pr1</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">tx</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon">tx1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">tx2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:6_1_1_1"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">tx1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font> &amp; <font color="Maroon">tx2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font> &amp; <font color="Maroon">tx</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">tx1</font>,<font color="Maroon">tx2</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E6:6_1_1_1"/>
<font color="Maroon">tx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">tX</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:6_1_1_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">tx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">tX</font> &amp; <font color="Maroon">tX</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FA</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">tX1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T1</font>, <font color="Maroon">tX2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T2</font><b> such that </b><br/><a NAME="E10:6_1_1_1"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">tX</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">tX1</font>,<font color="Maroon">tX2</font></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <font color="Maroon">tX1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">tX2</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">tX1</font>
;<br/>

<b>thus </b><a NAME="E11:6_1_1_1"/>
<font color="Maroon">tX1</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i/>;<br/>

<a NAME="E12:6_1_1_1"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">tx1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">tX1</font> &amp; <span class="p1">(<span class="default"><a href="funct_3.html#K9">pr1</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">tx1</font>,<font color="Maroon">tx2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">tx1</font> )
 
<b>by </b><i>, , , <a class="ref" href="funct_3.html#D5">FUNCT_3:def 5</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>thus </b><a NAME="E13:6_1_1_1"/>
<font color="Maroon">tX1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:6_1_1_1_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">tX1</font>
 ;<br/>

<a NAME="E2:6_1_1_1_1"/>
<font color="Maroon">tx</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x2</font></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:6_1_1_1_1"/><b>then </b>
( <font color="Maroon">tx2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">tx2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">tX2</font> )
 
<b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a>, <a class="ref" href="zfmisc_1.html#T129">ZFMISC_1:129</a></i>;<br/>
<a NAME="E4:6_1_1_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">tX1</font>,<font color="Maroon">tX2</font></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E5:6_1_1_1_1"/><b>then </b><i><font color="Green">E69</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">FA</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x2</font></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="funct_3.html#K9">pr1</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font></span>)</span> )
 
<b>by </b><i>, , , <a class="ref" href="funct_3.html#D5">FUNCT_3:def 5</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a>, <a class="ref" href="zfmisc_1.html#T129">ZFMISC_1:129</a></i>;<br/>
<a NAME="E6:6_1_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> <a href="subset_1.html#K8">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x2</font></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E7:6_1_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="funct_3.html#K9">pr1</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_3.html#K9">pr1</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="subset_1.html#K8">/\</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x2</font></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="funct_3.html#K9">pr1</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a>, <a class="ref" href="funct_3.html#D5">FUNCT_3:def 5</a></i>;<br/>
<b>hence </b><a NAME="E8:6_1_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 <b>by </b><i/>;<br/>


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

<a NAME="E14:6_1_1_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K9">pr1</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">tx1</font>,<font color="Maroon">tx2</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E15:6_1_1_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">tX1</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:6_1_1"/>
( <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> iff  ex <font color="Olive">U</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T1</font> st <br/>( <font color="Olive">U</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">U</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X1</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Olive">U</font> ) )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:6_1"/>
<font color="Maroon">X1</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T57">TOPS_1:57</a></i>;<br/>


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

<b>thus </b><a NAME="E5:6"/>
( <font color="Maroon">X2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K10">pr2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="subset_1.html#K8">/\</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x1</font></span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> implies <font color="Maroon">X2</font> is <a href="pre_topc.html#V3">open</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">X2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K10">pr2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="subset_1.html#K8">/\</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x1</font></span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 ;<br/>

<a NAME="E2:6_2"/>
for <font color="Olive">t</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> iff  ex <font color="Olive">U</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T2</font> st <br/>( <font color="Olive">U</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">U</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X2</font> &amp; <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Olive">U</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">t</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<a NAME="E1:6_2_1"/>
( <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> implies  ex <font color="Olive">U</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T2</font> st <br/>( <font color="Olive">U</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">U</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X2</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Olive">U</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_2_1_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font>
 ;<br/>

<b>then consider </b><font color="Maroon">tx</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:6_2_1_1"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">tx</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="funct_3.html#K10">pr2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font></span>)</span> &amp; <font color="Maroon">tx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> <a href="subset_1.html#K8">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x1</font></span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K10">pr2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">tx</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon">tx1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">tx2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:6_2_1_1"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">tx1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font> &amp; <font color="Maroon">tx2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font> &amp; <font color="Maroon">tx</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">tx1</font>,<font color="Maroon">tx2</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E6:6_2_1_1"/>
<font color="Maroon">tx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">tX</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:6_2_1_1"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">tx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">tX</font> &amp; <font color="Maroon">tX</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FA</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">tX1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T1</font>, <font color="Maroon">tX2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T2</font><b> such that </b><br/><a NAME="E10:6_2_1_1"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">tX</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">tX1</font>,<font color="Maroon">tX2</font></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <font color="Maroon">tX1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">tX2</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E11:6_2_1_1"/><i><font color="Green">E111</font></i>: 
( <font color="Maroon">tx2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">tX2</font> &amp; <span class="p1">(<span class="default"><a href="funct_3.html#K10">pr2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">tx1</font>,<font color="Maroon">tx2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">tx2</font> )
 
<b>by </b><i>, , , <a class="ref" href="funct_3.html#D6">FUNCT_3:def 6</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E12:6_2_1_1"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K10">pr2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">tx1</font>,<font color="Maroon">tx2</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E13:6_2_1_1"/>
<font color="Maroon">tX2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:6_2_1_1_1"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">tX2</font>
 ;<br/>

<a NAME="E2:6_2_1_1_1"/>
<font color="Maroon">tx</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x1</font></span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:6_2_1_1_1"/><b>then </b>
( <font color="Maroon">tx1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">tx1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">tX1</font> )
 
<b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a>, <a class="ref" href="zfmisc_1.html#T128">ZFMISC_1:128</a></i>;<br/>
<a NAME="E4:6_2_1_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">tX1</font>,<font color="Maroon">tX2</font></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E5:6_2_1_1_1"/><b>then </b><i><font color="Green">E114</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">FA</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x1</font></span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="funct_3.html#K10">pr2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font></span>)</span> )
 
<b>by </b><i>, , , <a class="ref" href="funct_3.html#D6">FUNCT_3:def 6</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a>, <a class="ref" href="zfmisc_1.html#T128">ZFMISC_1:128</a></i>;<br/>
<a NAME="E6:6_2_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> <a href="subset_1.html#K8">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x1</font></span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E7:6_2_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="funct_3.html#K10">pr2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_3.html#K10">pr2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="subset_1.html#K8">/\</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x1</font></span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="funct_3.html#K10">pr2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a>, <a class="ref" href="funct_3.html#D6">FUNCT_3:def 6</a></i>;<br/>
<b>hence </b><a NAME="E8:6_2_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E14:6_2_1_1"/>
 ex <font color="Olive">U</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T2</font> st <br/>( <font color="Olive">U</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">U</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X2</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Olive">U</font> )
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:6_2_1"/>
( <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> iff  ex <font color="Olive">U</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T2</font> st <br/>( <font color="Olive">U</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">U</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X2</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Olive">U</font> ) )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:6_2"/>
<font color="Maroon">X2</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T57">TOPS_1:57</a></i>;<br/>


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


</div>
