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

<span class="kw">let </span><font color="Maroon" title="c1">T1</font>, <font color="Maroon" title="c2">T2</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">T1</font> is  <a href="pre_topc.html#V8" title="PRE_TOPC:attr.8">T_2</a>  &amp; <font color="Maroon" title="c2">T2</font> is  <a href="pre_topc.html#V8" title="PRE_TOPC:attr.8">T_2</a>  implies <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> is  <a href="pre_topc.html#V8" title="PRE_TOPC:attr.8">T_2</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:42"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
<font color="Maroon" title="c1">T1</font> is  <a href="pre_topc.html#V8" title="PRE_TOPC:attr.8">T_2</a> 
 <span class="kw">and </span><br/><a NAME="E2:42"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
<font color="Maroon" title="c2">T2</font> is  <a href="pre_topc.html#V8" title="PRE_TOPC:attr.8">T_2</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> is  <a href="pre_topc.html#V8" title="PRE_TOPC:attr.8">T_2</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">T</font> = <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>;<br/>
<a NAME="E3:42"/>
<span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> is  <a href="pre_topc.html#V8" title="PRE_TOPC:attr.8">T_2</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">p</font>, <font color="Maroon" title="c5">q</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="pre_topc.html#D16" title="PRE_TOPC:def.16">PRE_TOPC:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> or  ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> st <br/>( <font color="Olive">b<sub>1</sub></font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive">b<sub>2</sub></font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive">b<sub>2</sub></font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:42_1"/><span class="lab"><font color="Green" title="E24">A3</font></span>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> st <br/>( <font color="Olive">b<sub>1</sub></font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive">b<sub>2</sub></font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive">b<sub>2</sub></font> )</span><br/>

<a NAME="E2:42_1"/>
<font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>
 
;<br/>
<a NAME="E3:42_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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">T1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">z</font>, <font color="Maroon" title="c7">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:42_1"/><span class="lab"><font color="Green" title="E25">A4</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T1</font>
 <span class="kw">and </span><br/><a NAME="E6:42_1"/><span class="lab"><font color="Green" title="E26">A5</font></span>: 
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T2</font>
 <span class="kw">and </span><br/><a NAME="E7:42_1"/><span class="lab"><font color="Green" title="E27">A6</font></span>: 
<font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c7">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<a NAME="E8:42_1"/>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>
 
;<br/>
<a NAME="E9:42_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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">T1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c9">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E11:42_1"/><span class="lab"><font color="Green" title="E28">A7</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T1</font>
 <span class="kw">and </span><br/><a NAME="E12:42_1"/><span class="lab"><font color="Green" title="E29">A8</font></span>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T2</font>
 <span class="kw">and </span><br/><a NAME="E13:42_1"/><span class="lab"><font color="Green" title="E30">A9</font></span>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">y</font> = <font color="Maroon" title="c9">y</font>, <font color="Maroon" title="c11">v</font> = <font color="Maroon" title="c7">v</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">T2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:42_1"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E6:42_1"><span class="lab"><font color="Green" title="E26">A5</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">x</font> = <font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c13">z</font> = <font color="Maroon" title="c6">z</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:42_1"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E5:42_1"><span class="lab"><font color="Green" title="E25">A4</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:42_1_1"/>
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">z</font> or <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">z</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:42_1_1_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> st <br/>( <font color="Olive">b<sub>1</sub></font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive">b<sub>2</sub></font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive">b<sub>2</sub></font> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">Y</font>, <font color="Maroon" title="c15">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T1</font><span class="kw"> such that </span><br/><a NAME="E3:42_1_1_1"/><span class="lab"><font color="Green" title="E31">A10</font></span>: 
( <font color="Maroon" title="c14">Y</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c15">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )
 <span class="kw">and </span><br/><a NAME="E4:42_1_1_1"/><span class="lab"><font color="Green" title="E32">A11</font></span>: 
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Y</font> &amp; <font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">V</font> )
 <span class="kw">and </span><br/><a NAME="E5:42_1_1_1"/><span class="lab"><font color="Green" title="E33">A12</font></span>: 
<font color="Maroon" title="c14">Y</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c15">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="pre_topc.html#D16" title="PRE_TOPC:def.16">PRE_TOPC:def 16</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c16">Y</font> = <font color="Maroon" title="c14">Y</font>, <font color="Maroon" title="c17">V</font> = <font color="Maroon" title="c15">V</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T1</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c18">X</font> = <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c16">Y</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">T2</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>, <font color="Maroon" title="c19">Z</font> = <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c17">V</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">T2</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> ;<br/><a NAME="E8:42_1_1_1"/><span class="lab"><font color="Green" title="E34">A13</font></span>: 
<font color="Maroon" title="c18">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c19">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:42_1_1_1"><span class="lab"><font color="Green" title="E33">A12</font></span></a>, <a class="ref" href="zfmisc_1.html#T127" title="ZFMISC_1:th.127">ZFMISC_1:127</a></span>;<br/><a NAME="E9:42_1_1_1"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">T2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T53" title="TOPS_1:th.53">TOPS_1:53</a></span>;<br/><a NAME="E10:42_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A14</font></span>: 
( <font color="Maroon" title="c18">X</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c19">Z</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_1_1_1"><span class="lab"><font color="Green" title="E31">A10</font></span></a>, <a class="ref" href="borsuk_1.html#T46" title="BORSUK_1:th.46">BORSUK_1:46</a></span>;<br/><a NAME="E11:42_1_1_1"/>
( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">X</font> &amp; <font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">Z</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:42_1"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E13:42_1"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="txt" href="#E6:42_1"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E7:42_1"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="txt" href="#E4:42_1_1_1"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><span class="kw">hence </span><a NAME="E12:42_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" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> st <br/>( <font color="Olive">b<sub>1</sub></font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive">b<sub>2</sub></font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive">b<sub>2</sub></font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:42_1_1_1"><span class="lab"><font color="Green" title="E35">A14</font></span></a>, <a class="txt" href="#E8:42_1_1_1"><span class="lab"><font color="Green" title="E34">A13</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:42_1_1_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> st <br/>( <font color="Olive">b<sub>1</sub></font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive">b<sub>2</sub></font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive">b<sub>2</sub></font> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">Y</font>, <font color="Maroon" title="c15">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T2</font><span class="kw"> such that </span><br/><a NAME="E3:42_1_1_2"/><span class="lab"><font color="Green" title="E31">A15</font></span>: 
( <font color="Maroon" title="c14">Y</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c15">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )
 <span class="kw">and </span><br/><a NAME="E4:42_1_1_2"/><span class="lab"><font color="Green" title="E32">A16</font></span>: 
( <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Y</font> &amp; <font color="Maroon" title="c11">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">V</font> )
 <span class="kw">and </span><br/><a NAME="E5:42_1_1_2"/><span class="lab"><font color="Green" title="E33">A17</font></span>: 
<font color="Maroon" title="c14">Y</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c15">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E1:42_1"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E13:42_1"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="txt" href="#E7:42_1"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="ref" href="pre_topc.html#D16" title="PRE_TOPC:def.16">PRE_TOPC:def 16</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c16">Y</font> = <font color="Maroon" title="c14">Y</font>, <font color="Maroon" title="c17">V</font> = <font color="Maroon" title="c15">V</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T2</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c18">X</font> = <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T1</font></span>)</span>,<font color="Maroon" title="c16">Y</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>, <font color="Maroon" title="c19">Z</font> = <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T1</font></span>)</span>,<font color="Maroon" title="c17">V</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> ;<br/><a NAME="E8:42_1_1_2"/><span class="lab"><font color="Green" title="E34">A18</font></span>: 
<font color="Maroon" title="c18">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c19">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:42_1_1_2"><span class="lab"><font color="Green" title="E33">A17</font></span></a>, <a class="ref" href="zfmisc_1.html#T127" title="ZFMISC_1:th.127">ZFMISC_1:127</a></span>;<br/><a NAME="E9:42_1_1_2"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T53" title="TOPS_1:th.53">TOPS_1:53</a></span>;<br/><a NAME="E10:42_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A19</font></span>: 
( <font color="Maroon" title="c18">X</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c19">Z</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_1_1_2"><span class="lab"><font color="Green" title="E31">A15</font></span></a>, <a class="ref" href="borsuk_1.html#T46" title="BORSUK_1:th.46">BORSUK_1:46</a></span>;<br/><a NAME="E11:42_1_1_2"/>
( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">X</font> &amp; <font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">Z</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:42_1"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E13:42_1"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="txt" href="#E5:42_1"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E7:42_1"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="txt" href="#E4:42_1_1_2"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><span class="kw">hence </span><a NAME="E12:42_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" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> st <br/>( <font color="Olive">b<sub>1</sub></font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive">b<sub>2</sub></font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive">b<sub>2</sub></font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:42_1_1_2"><span class="lab"><font color="Green" title="E35">A19</font></span></a>, <a class="txt" href="#E8:42_1_1_2"><span class="lab"><font color="Green" title="E34">A18</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:42"/>
<span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> is  <a href="pre_topc.html#V8" title="PRE_TOPC:attr.8">T_2</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
