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

<b>let </b><font color="Maroon" title="c1">X</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>;<br/><b>let </b><font color="Maroon" title="c2">X1</font>, <font color="Maroon" title="c3">X2</font>, <font color="Maroon" title="c4">Y1</font>, <font color="Maroon" title="c5">Y2</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">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="c6">A1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X1</font>, <font color="Maroon" title="c7">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" title="TSEP_1:th.1">TSEP_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">C1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">Y1</font>, <font color="Maroon" title="c9">C2</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">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" title="TSEP_1:th.1">TSEP_1:1</a></i>;<br/>
<b>assume </b><a NAME="E3:43"/>
( <font color="Maroon" title="c4">Y1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c2">X1</font> &amp; <font color="Maroon" title="c5">Y2</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X2</font> )
 ;<br/>

<a NAME="E4:43"/><b>then </b><i><font color="Green" title="E29">A1</font></i>: 
( <font color="Maroon" title="c8">C1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">A1</font> &amp; <font color="Maroon" title="c9">C2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A2</font> )
 
<b>by </b><i><a class="ref" href="tsep_1.html#T4" title="TSEP_1:th.4">TSEP_1:4</a></i>;<br/>
<b>assume </b><a NAME="E5:43"/><i><font color="Green" title="E30">A2</font></i>: 
<font color="Maroon" title="c4">Y1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c5">Y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c3">X2</font>
 ;<br/>

<b>assume </b><a NAME="E6:43"/>
<font color="Maroon" title="c4">Y1</font>,<font color="Maroon" title="c5">Y2</font> <a href="tsep_1.html#R4" title="TSEP_1:pred.4">are_weakly_separated</a> 
 ;<br/>

<a NAME="E7:43"/><b>then </b><i><font color="Green" title="E31">A3</font></i>: 
<font color="Maroon" title="c8">C1</font>,<font color="Maroon" title="c9">C2</font> <a href="tsep_1.html#R2" title="TSEP_1:pred.2">are_weakly_separated</a> 
 
<b>by </b><i><a class="ref" href="tsep_1.html#D9" title="TSEP_1:def.9">TSEP_1:def 9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">A1</font>, <font color="Maroon" title="c11">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="E1:43_1"/><i><font color="Green" title="E32">A4</font></i>: 
( <font color="Maroon" title="c10">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="c11">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="E2:43_1"/><b>then </b>
<font color="Maroon" title="c10">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c11">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 <span class="p1">(<span class="default"><font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c3">X2</font></span>)</span>
 <b>by </b><i><a class="ref" href="tsep_1.html#D2" title="TSEP_1:def.2">TSEP_1:def 2</a></i>;<br/><a NAME="E3:43_1"/><b>then </b>
<font color="Maroon" title="c10">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c11">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">C1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">C2</font>
 <b>by </b><i><a class="txt" href="tsep_2.html#E5:43"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="tsep_1.html#D2" title="TSEP_1:def.2">TSEP_1:def 2</a></i>;<br/><b>hence </b><a NAME="E4:43_1"/>
<font color="Maroon" title="c10">A1</font>,<font color="Maroon" title="c11">A2</font> <a href="tsep_1.html#R2" title="TSEP_1:pred.2">are_weakly_separated</a> 
 <b>by </b><i><a class="txt" href="tsep_2.html#E4:43"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="tsep_2.html#E7:43"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="tsep_2.html#E1:43_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="tsep_2.html#T23" target="_self" title="TSEP_2:th.23">Th23</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:43"/>
<font color="Maroon" title="c2">X1</font>,<font color="Maroon" title="c3">X2</font> <a href="tsep_1.html#R4" title="TSEP_1:pred.4">are_weakly_separated</a> 
 <b>by </b><i><a class="ref" href="tsep_1.html#D9" title="TSEP_1:def.9">TSEP_1:def 9</a></i>;<br/>


</div>
