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

<b>set </b><font color="Maroon" title="c5">VV</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>;<br/>
<b>set </b><font color="Maroon" title="c6">VW1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">W1</font>;<br/>
<b>set </b><font color="Maroon" title="c7">VW2</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">W2</font>;<br/>
<a NAME="E1:2_1_1"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">W1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">W2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font> )
 
<b>by </b><i><a class="ref" href="rmod_2.html#D2" title="RMOD_2:def.2">RMOD_2:def 2</a></i>;<br/>
<a NAME="E2:2_1_1"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">W1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">W2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">V1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">W1</font>, <font color="Maroon" title="c9">V2</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">W2</font>, <font color="Maroon" title="c10">V3</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">W1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">W2</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="ref" href="rmod_2.html#D2" title="RMOD_2:def.2">RMOD_2:def 2</a></i>;<br/>
<a NAME="E4:2_1_1"/>
(  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W2</font> )
 
<b>by </b><i><a class="ref" href="rmod_2.html#T25" title="RMOD_2:th.25">RMOD_2:25</a></i>;<br/>
<a NAME="E5:2_1_1"/><b>then </b>
(  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">W1</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">W2</font> )
 
<b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E6:2_1_1"/><b>then </b><i><font color="Green" title="E2">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">W1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">W2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E7:2_1_1"/>
( <font color="Maroon" title="c8">V1</font> is <a href="rmod_2.html#V1" title="RMOD_2:attr.1">linearly-closed</a> &amp; <font color="Maroon" title="c9">V2</font> is <a href="rmod_2.html#V1" title="RMOD_2:attr.1">linearly-closed</a> )
 
<b>by </b><i><a class="ref" href="rmod_2.html#T41" title="RMOD_2:th.41">RMOD_2:41</a></i>;<br/>
<a NAME="E8:2_1_1"/><b>then </b>
<font color="Maroon" title="c10">V3</font> is <a href="rmod_2.html#V1" title="RMOD_2:attr.1">linearly-closed</a>
 
<b>by </b><i><a class="ref" href="rmod_2.html#T10" title="RMOD_2:th.10">RMOD_2:10</a></i>;<br/>
<b>hence </b><a NAME="E9:2_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being <a href="vectsp_2.html#V2" title="VECTSP_2:attr.2">strict</a>  <a href="rmod_2.html#M1" title="RMOD_2:mode.1">Submodule</a> of <font color="Maroon" title="c2">V</font> st the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive">b<sub>1</sub></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">W1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">W2</font>
 <b>by </b><i><a class="txt" href="#E6:2_1_1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="rmod_2.html#T42" title="RMOD_2:th.42">RMOD_2:42</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
