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<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">X0</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>let </b><font color="Maroon" title="c3">X1</font>, <font color="Maroon" title="c4">X2</font> be    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c5">X01</font>, <font color="Maroon" title="c6">X02</font> be    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c2">X0</font>;<br/>







<b>assume </b><a NAME="E1:45"/><i><font color="Green" title="E31">A1</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X1</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="c5">X01</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">X2</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="c6">X02</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c7">A1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X1</font>, <font color="Maroon" title="c8">A2</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">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="c9">B1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">X01</font>, <font color="Maroon" title="c10">B2</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">X02</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">X0</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>thus </b><a NAME="E4:45"/>
( <font color="Maroon" title="c3">X1</font>,<font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#R3" title="TSEP_1:pred.3">are_separated</a>  implies <font color="Maroon" title="c5">X01</font>,<font color="Maroon" title="c6">X02</font> <a href="tsep_1.html#R3" title="TSEP_1:pred.3">are_separated</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:45_1"/>
<font color="Maroon" title="c3">X1</font>,<font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#R3" title="TSEP_1:pred.3">are_separated</a> 
 ;<br/>

<a NAME="E2:45_1"/><b>then </b>
<font color="Maroon" title="c7">A1</font>,<font color="Maroon" title="c8">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" title="TSEP_1:def.8">TSEP_1:def 8</a></i>;<br/>
<a NAME="E3:45_1"/><b>then </b>
for <font color="Olive" title="b1">B1</font>, <font color="Olive" title="b2">B2</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">X0</font>  st <font color="Olive" title="b1">B1</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="c5">X01</font> &amp; <font color="Olive" title="b2">B2</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="c6">X02</font> holds <br/><font color="Olive" title="b1">B1</font>,<font color="Olive" title="b2">B2</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> 
 
<b>by </b><i><a class="txt" href="tsep_2.html#E1:45"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="tsep_2.html#T26" target="_self" title="TSEP_2:th.26">Th26</a></i>;<br/>
<b>hence </b><a NAME="E4:45_1"/>
<font color="Maroon" title="c5">X01</font>,<font color="Maroon" title="c6">X02</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" title="TSEP_1:def.8">TSEP_1:def 8</a></i>;<br/>


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

<b>thus </b><a NAME="E5:45"/>
( <font color="Maroon" title="c5">X01</font>,<font color="Maroon" title="c6">X02</font> <a href="tsep_1.html#R3" title="TSEP_1:pred.3">are_separated</a>  implies <font color="Maroon" title="c3">X1</font>,<font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#R3" title="TSEP_1:pred.3">are_separated</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:45_2"/>
<font color="Maroon" title="c5">X01</font>,<font color="Maroon" title="c6">X02</font> <a href="tsep_1.html#R3" title="TSEP_1:pred.3">are_separated</a> 
 ;<br/>

<a NAME="E2:45_2"/><b>then </b>
<font color="Maroon" title="c9">B1</font>,<font color="Maroon" title="c10">B2</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" title="TSEP_1:def.8">TSEP_1:def 8</a></i>;<br/>
<a NAME="E3:45_2"/><b>then </b>
for <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">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="c3">X1</font> &amp; <font color="Olive" title="b2">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="c4">X2</font> holds <br/><font color="Olive" title="b1">A1</font>,<font color="Olive" title="b2">A2</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> 
 
<b>by </b><i><a class="txt" href="tsep_2.html#E1:45"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="tsep_2.html#T26" target="_self" title="TSEP_2:th.26">Th26</a></i>;<br/>
<b>hence </b><a NAME="E4:45_2"/>
<font color="Maroon" title="c3">X1</font>,<font color="Maroon" title="c4">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" title="TSEP_1:def.8">TSEP_1:def 8</a></i>;<br/>


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


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