<?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>, <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>assume </b><a NAME="E1:38"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">T1</font> <a href="urysohn1.html#NR1" target="_self">is_T1</a>  &amp; <font color="Maroon">T2</font> <a href="urysohn1.html#NR1" target="_self">is_T1</a>  )
 ;<br/>

<b>set </b><font color="Maroon">T</font> = <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/>
<a NAME="E2:38"/>
<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> <a href="urysohn1.html#NR1" target="_self">is_T1</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</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>, <font color="Maroon">q</font> be   <a href="pre_topc.html#NM2">Point</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>; <i><font color="Red">:: according to </font></i><a class="ref" href="urysohn1.html#D9">URYSOHN1:def 9</a><br/>

<b>assume </b><a NAME="E1:38_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font>
 ;<br/>

<a NAME="E2:38_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</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> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</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/>
<a NAME="E3:38_1"/><b>then </b><i><font color="Green">E36</font></i>: 
( <font color="Maroon">p</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>,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">q</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>,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="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:38_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">x</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">y</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">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</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/>
<b>consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">v</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:38_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">z</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">v</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">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">v</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/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font>, <font color="Maroon">z</font> = <font color="Maroon">z</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T1</font> <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font>, <font color="Maroon">v</font> = <font color="Maroon">v</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T2</font> <b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1"><b>suppose </b></a><a NAME="E1:38_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">Y</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T1</font>, <font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T1</font><b> such that </b><br/><a NAME="E3:38_1_1_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">Y</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> )
 <b>by </b><i>, <a class="ref" href="urysohn1.html#D9">URYSOHN1:def 9</a></i>;<br/><a NAME="E4:38_1_1_1"/><i><font color="Green">E51</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T2</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T53">TOPS_1:53</a></i>;<br/><b>set </b><font color="Maroon">X</font> = <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">Y</font>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T2</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>;<br/><b>set </b><font color="Maroon">Z</font> = <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">V</font>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T2</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>;<br/><a NAME="E5:38_1_1_1"/><i><font color="Green">E52</font></i>: 
( <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">Y</font>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T2</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span> is <a href="pre_topc.html#V3">open</a> &amp; <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">V</font>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T2</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i>, , <a class="ref" href="borsuk_1.html#T46">BORSUK_1:46</a></i>;<br/><a NAME="E6:38_1_1_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">Y</font>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T2</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">V</font>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T2</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span> )
 <b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E7:38_1_1_1"/>
( not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">Y</font>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T2</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span> &amp; not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">V</font>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T2</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span> )
 <b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E8:38_1_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</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> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_2"><b>suppose </b></a><a NAME="E1:38_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">Y</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T2</font>, <font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T2</font><b> such that </b><br/><a NAME="E3:38_1_1_2"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">Y</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> )
 <b>by </b><i>, , , , <a class="ref" href="urysohn1.html#D9">URYSOHN1:def 9</a></i>;<br/><b>reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">Y</font>, <font color="Maroon">V</font> = <font color="Maroon">V</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T2</font> ;<br/><a NAME="E5:38_1_1_2"/><i><font color="Green">E55</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T53">TOPS_1:53</a></i>;<br/><b>set </b><font color="Maroon">X</font> = <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font></span>)</span>,<font color="Maroon">Y</font></span><a href="borsuk_1.html#K6">:]</a></span>;<br/><b>set </b><font color="Maroon">Z</font> = <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font></span>)</span>,<font color="Maroon">V</font></span><a href="borsuk_1.html#K6">:]</a></span>;<br/><a NAME="E6:38_1_1_2"/><i><font color="Green">E56</font></i>: 
( <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font></span>)</span>,<font color="Maroon">Y</font></span><a href="borsuk_1.html#K6">:]</a></span> is <a href="pre_topc.html#V3">open</a> &amp; <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font></span>)</span>,<font color="Maroon">V</font></span><a href="borsuk_1.html#K6">:]</a></span> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i>, , <a class="ref" href="borsuk_1.html#T46">BORSUK_1:46</a></i>;<br/><a NAME="E7:38_1_1_2"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font></span>)</span>,<font color="Maroon">Y</font></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font></span>)</span>,<font color="Maroon">V</font></span><a href="borsuk_1.html#K6">:]</a></span> )
 <b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E8:38_1_1_2"/>
( not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font></span>)</span>,<font color="Maroon">V</font></span><a href="borsuk_1.html#K6">:]</a></span> &amp; not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font></span>)</span>,<font color="Maroon">Y</font></span><a href="borsuk_1.html#K6">:]</a></span> )
 <b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E9:38_1_1_2"/>
 ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</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> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E3:38"/>
<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> <a href="urysohn1.html#NR1" target="_self">is_T1</a> 
 ;<br/>


</div>
