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

<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">X1</font>, <font color="Maroon" title="c3">X2</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>;<br/>





<b>reconsider </b><font color="Maroon" title="c4">A1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="tsep_1.html#T1" target="_self" title="TSEP_1:th.1">Th1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">A2</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X2</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="tsep_1.html#T1" target="_self" title="TSEP_1:th.1">Th1</a></i>;<br/>
<b>thus </b><a NAME="E3:91"/>
( <font color="Maroon" title="c2">X1</font>,<font color="Maroon" title="c3">X2</font> <a href="tsep_1.html#R3" title="TSEP_1:pred.3">are_separated</a>  implies  ex <font color="Olive" title="b1">Y1</font>, <font color="Olive" title="b2">Y2</font> being non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="borsuk_1.html#V1" title="BORSUK_1:attr.1">closed</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c2">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Olive" title="b1">Y1</font> &amp; <font color="Maroon" title="c3">X2</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Olive" title="b2">Y2</font> &amp; <font color="Olive" title="b1">Y1</font> <a href="tsep_1.html#R1" title="TSEP_1:pred.1">misses</a> <font color="Maroon" title="c3">X2</font> &amp; <font color="Olive" title="b2">Y2</font> <a href="tsep_1.html#R1" title="TSEP_1:pred.1">misses</a> <font color="Maroon" title="c2">X1</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:91_1"/>
<font color="Maroon" title="c2">X1</font>,<font color="Maroon" title="c3">X2</font> <a href="tsep_1.html#R3" title="TSEP_1:pred.3">are_separated</a> 
 ;<br/>

<a NAME="E2:91_1"/><b>then </b>
<font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">A2</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> 
 
<b>by </b><i><a class="ref" href="tsep_1.html#D8" target="_self" title="TSEP_1:def.8">Def8</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">C1</font>, <font color="Maroon" title="c7">C2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E4:91_1"/><i><font color="Green" title="E57">A1</font></i>: 
( <font color="Maroon" title="c4">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">C1</font> &amp; <font color="Maroon" title="c5">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">C2</font> )
 <b>and </b><br/><a NAME="E5:91_1"/><i><font color="Green" title="E58">A2</font></i>: 
( <font color="Maroon" title="c6">C1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">A2</font> &amp; <font color="Maroon" title="c7">C2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">A1</font> )
 <b>and </b><br/><a NAME="E6:91_1"/><i><font color="Green" title="E59">A3</font></i>: 
( <font color="Maroon" title="c6">C1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c7">C2</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )
 <b>by </b><i><a class="ref" href="tsep_1.html#T46" target="_self" title="TSEP_1:th.46">Th46</a></i>;<br/>
<a NAME="E7:91_1"/><i><font color="Green" title="E60">A4</font></i>: 
( not <font color="Maroon" title="c6">C1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not <font color="Maroon" title="c7">C2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 
<b>by </b><i><a class="txt" href="tsep_1.html#E4:91_1"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">Y1</font> being  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a> <a href="borsuk_1.html#V1" title="BORSUK_1:attr.1">closed</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E9:91_1"/><i><font color="Green" title="E61">A5</font></i>: 
<font color="Maroon" title="c6">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y1</font>
 <b>by </b><i><a class="txt" href="tsep_1.html#E6:91_1"><i><font color="Green" title="E59">A3</font></i></a>, <a class="ref" href="tsep_1.html#T15" target="_self" title="TSEP_1:th.15">Th15</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">Y2</font> being  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a> <a href="borsuk_1.html#V1" title="BORSUK_1:attr.1">closed</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E11:91_1"/><i><font color="Green" title="E62">A6</font></i>: 
<font color="Maroon" title="c7">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">Y2</font>
 <b>by </b><i><a class="txt" href="tsep_1.html#E6:91_1"><i><font color="Green" title="E59">A3</font></i></a>, <a class="txt" href="tsep_1.html#E7:91_1"><i><font color="Green" title="E60">A4</font></i></a>, <a class="ref" href="tsep_1.html#T15" target="_self" title="TSEP_1:th.15">Th15</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">Y1</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c9">Y2</font>
;<br/>

<b>thus </b><a NAME="E12:91_1"/>
( <font color="Maroon" title="c2">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c8">Y1</font> &amp; <font color="Maroon" title="c3">X2</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c9">Y2</font> &amp; <font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#R1" title="TSEP_1:pred.1">misses</a> <font color="Maroon" title="c3">X2</font> &amp; <font color="Maroon" title="c9">Y2</font> <a href="tsep_1.html#R1" title="TSEP_1:pred.1">misses</a> <font color="Maroon" title="c2">X1</font> )
 <b>by </b><i><a class="txt" href="tsep_1.html#E4:91_1"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="tsep_1.html#E5:91_1"><i><font color="Green" title="E58">A2</font></i></a>, <a class="txt" href="tsep_1.html#E9:91_1"><i><font color="Green" title="E61">A5</font></i></a>, <a class="txt" href="tsep_1.html#E11:91_1"><i><font color="Green" title="E62">A6</font></i></a>, <a class="ref" href="tsep_1.html#D3" target="_self" title="TSEP_1:def.3">Def3</a>, <a class="ref" href="tsep_1.html#T4" target="_self" title="TSEP_1:th.4">Th4</a></i>;<br/>


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

<b>given </b><font color="Maroon" title="c6">Y1</font>, <font color="Maroon" title="c7">Y2</font> being  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="borsuk_1.html#V1" title="BORSUK_1:attr.1">closed</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font><b> such that </b><a NAME="E4:91"/><i><font color="Green" title="E57">A7</font></i>: 
( <font color="Maroon" title="c2">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c6">Y1</font> &amp; <font color="Maroon" title="c3">X2</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c7">Y2</font> )
 <b>and </b><br/><a NAME="E5:91"/><i><font color="Green" title="E58">A8</font></i>: 
( <font color="Maroon" title="c6">Y1</font> <a href="tsep_1.html#R1" title="TSEP_1:pred.1">misses</a> <font color="Maroon" title="c3">X2</font> &amp; <font color="Maroon" title="c7">Y2</font> <a href="tsep_1.html#R1" title="TSEP_1:pred.1">misses</a> <font color="Maroon" title="c2">X1</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">A1</font>, <font color="Maroon" title="c9">A2</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/><b>assume </b><a NAME="E1:91_2"/><i><font color="Green" title="E59">A9</font></i>: 
( <font color="Maroon" title="c8">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X1</font> &amp; <font color="Maroon" title="c9">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X2</font> )
 ;<br/><a NAME="E2:91_2"/>
 ex <font color="Olive" title="b1">C1</font>, <font color="Olive" title="b2">C2</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c8">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">C1</font> &amp; <font color="Maroon" title="c9">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">C2</font> &amp; <font color="Olive" title="b1">C1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c9">A2</font> &amp; <font color="Olive" title="b2">C2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c8">A1</font> &amp; <font color="Olive" title="b1">C1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Olive" title="b2">C2</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_2_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c10">C1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">Y1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="tsep_1.html#T1" target="_self" title="TSEP_1:th.1">Th1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">C2</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">Y2</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="tsep_1.html#T1" target="_self" title="TSEP_1:th.1">Th1</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c10">C1</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c11">C2</font>
;<br/>

<b>thus </b><a NAME="E3:91_2_1"/>
( <font color="Maroon" title="c8">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">C1</font> &amp; <font color="Maroon" title="c9">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">C2</font> &amp; <font color="Maroon" title="c10">C1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c9">A2</font> &amp; <font color="Maroon" title="c11">C2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c8">A1</font> &amp; <font color="Maroon" title="c10">C1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c11">C2</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )
 <b>by </b><i><a class="txt" href="tsep_1.html#E4:91"><i><font color="Green" title="E57">A7</font></i></a>, <a class="txt" href="tsep_1.html#E5:91"><i><font color="Green" title="E58">A8</font></i></a>, <a class="txt" href="tsep_1.html#E1:91_2"><i><font color="Green" title="E59">A9</font></i></a>, <a class="ref" href="tsep_1.html#D3" target="_self" title="TSEP_1:def.3">Def3</a>, <a class="ref" href="tsep_1.html#T4" target="_self" title="TSEP_1:th.4">Th4</a>, <a class="ref" href="tsep_1.html#T11" target="_self" title="TSEP_1:th.11">Th11</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:91_2"/>
<font color="Maroon" title="c8">A1</font>,<font color="Maroon" title="c9">A2</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> 
 <b>by </b><i><a class="ref" href="tsep_1.html#T46" target="_self" title="TSEP_1:th.46">Th46</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:91"/>
<font color="Maroon" title="c2">X1</font>,<font color="Maroon" title="c3">X2</font> <a href="tsep_1.html#R3" title="TSEP_1:pred.3">are_separated</a> 
 <b>by </b><i><a class="ref" href="tsep_1.html#D8" target="_self" title="TSEP_1:def.8">Def8</a></i>;<br/>


</div>
