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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="pre_topc.html#L1">TopStruct</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/>



<b>assume </b><a NAME="E1:14"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>3</sub></font> #)
 ;<br/>

<b>thus </b><a NAME="E2:14"/>
( <font color="Maroon">c<sub>2</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>3</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:14_1"/><i><font color="Green">E13</font></i>: 
(  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E3:14_1"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:14"><i><font color="Green">E11</font></i></a>, <a class="txt" href="tmap_1.html#E1:14_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a><br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>

<b>thus </b><a NAME="E4:14_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>3</sub></font> implies ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:14"><i><font color="Green">E11</font></i></a>, <a class="txt" href="tmap_1.html#E1:14_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="tmap_1.html#E2:14_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>

<b>given </b><font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E5:14_1"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 ;<br/>

<b>thus </b><a NAME="E6:14_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:14"><i><font color="Green">E11</font></i></a>, <a class="txt" href="tmap_1.html#E1:14_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="tmap_1.html#E2:14_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="tmap_1.html#E5:14_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>


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

<b>thus </b><a NAME="E3:14"/>
( <font color="Maroon">c<sub>3</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>2</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:14"><i><font color="Green">E11</font></i></a>, <a class="ref" href="tmap_1.html#T11" target="_self">Th11</a></i>;<br/>


</div>
