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<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">X0</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">X1</font> be    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>, <font color="Maroon">X2</font> be    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">X01</font> be    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X0</font>, <font color="Maroon">X02</font> be    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X0</font>;<br/>







<b>assume </b><a NAME="E1:47"/><i><font color="Green">E12</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X01</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X02</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">A1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">B1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X01</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X0</font> <b>by </b><i><a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">B2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X02</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X0</font> <b>by </b><i><a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/>
<b>thus </b><a NAME="E6:47"/>
( <font color="Maroon">X1</font>,<font color="Maroon">X2</font> <a href="tsep_1.html#R4">are_weakly_separated</a>  implies <font color="Maroon">X01</font>,<font color="Maroon">X02</font> <a href="tsep_1.html#R4">are_weakly_separated</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:47_1"/>
<font color="Maroon">X1</font>,<font color="Maroon">X2</font> <a href="tsep_1.html#R4">are_weakly_separated</a> 
 ;<br/>

<a NAME="E2:47_1"/><b>then </b>
<font color="Maroon">A1</font>,<font color="Maroon">A2</font> <a href="tsep_1.html#R2">are_weakly_separated</a> 
 
<b>by </b><i><a class="ref" href="tsep_1.html#D9">TSEP_1:def 9</a></i>;<br/>
<a NAME="E3:47_1"/><b>then </b>
for <font color="Olive">B1</font>, <font color="Olive">B2</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X0</font>  st <font color="Olive">B1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X01</font> &amp; <font color="Olive">B2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X02</font> holds <br/><font color="Olive">B1</font>,<font color="Olive">B2</font> <a href="tsep_1.html#R2">are_weakly_separated</a> 
 
<b>by </b><i>, <a class="ref" href="tsep_2.html#T6" target="_self">Th6</a></i>;<br/>
<b>hence </b><a NAME="E4:47_1"/>
<font color="Maroon">X01</font>,<font color="Maroon">X02</font> <a href="tsep_1.html#R4">are_weakly_separated</a> 
 <b>by </b><i><a class="ref" href="tsep_1.html#D9">TSEP_1:def 9</a></i>;<br/>


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

<b>thus </b><a NAME="E7:47"/>
( <font color="Maroon">X01</font>,<font color="Maroon">X02</font> <a href="tsep_1.html#R4">are_weakly_separated</a>  implies <font color="Maroon">X1</font>,<font color="Maroon">X2</font> <a href="tsep_1.html#R4">are_weakly_separated</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:47_2"/>
<font color="Maroon">X01</font>,<font color="Maroon">X02</font> <a href="tsep_1.html#R4">are_weakly_separated</a> 
 ;<br/>

<a NAME="E2:47_2"/><b>then </b>
<font color="Maroon">B1</font>,<font color="Maroon">B2</font> <a href="tsep_1.html#R2">are_weakly_separated</a> 
 
<b>by </b><i><a class="ref" href="tsep_1.html#D9">TSEP_1:def 9</a></i>;<br/>
<a NAME="E3:47_2"/><b>then </b>
for <font color="Olive">A1</font>, <font color="Olive">A2</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">A1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> &amp; <font color="Olive">A2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font> holds <br/><font color="Olive">A1</font>,<font color="Olive">A2</font> <a href="tsep_1.html#R2">are_weakly_separated</a> 
 
<b>by </b><i>, <a class="ref" href="tsep_2.html#T6" target="_self">Th6</a></i>;<br/>
<b>hence </b><a NAME="E4:47_2"/>
<font color="Maroon">X1</font>,<font color="Maroon">X2</font> <a href="tsep_1.html#R4">are_weakly_separated</a> 
 <b>by </b><i><a class="ref" href="tsep_1.html#D9">TSEP_1:def 9</a></i>;<br/>


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


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