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

<span class="kw">let </span><font color="Maroon" title="c1">R</font> be   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">V</font> being   <a href="vectsp_2.html#NM7" title="VECTSP_2:NM.7">RightMod</a> of <font color="Maroon" title="c1">R</font><br/>  for <font color="Olive" title="b2">W</font> being    <a href="rmod_2.html#M1" title="RMOD_2:mode.1">Submodule</a> of <font color="Olive" title="b1">V</font><br/>  for <font color="Olive" title="b3">C</font> being    <a href="rmod_2.html#M2" title="RMOD_2:mode.2">Coset</a> of <font color="Olive" title="b2">W</font> holds <br/> ( <font color="Olive" title="b3">C</font> is  <a href="rmod_2.html#V1" title="RMOD_2:attr.1">linearly-closed</a>  iff <font color="Olive" title="b3">C</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="Olive" title="b2">W</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">V</font> be   <a href="vectsp_2.html#NM7" title="VECTSP_2:NM.7">RightMod</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">W</font> being    <a href="rmod_2.html#M1" title="RMOD_2:mode.1">Submodule</a> of <font color="Maroon" title="c2">V</font><br/>  for <font color="Olive" title="b2">C</font> being    <a href="rmod_2.html#M2" title="RMOD_2:mode.2">Coset</a> of <font color="Olive" title="b1">W</font> holds <br/> ( <font color="Olive" title="b2">C</font> is  <a href="rmod_2.html#V1" title="RMOD_2:attr.1">linearly-closed</a>  iff <font color="Olive" title="b2">C</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="Olive" title="b1">W</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">W</font> be    <a href="rmod_2.html#M1" title="RMOD_2:mode.1">Submodule</a> of <font color="Maroon" title="c2">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">C</font> being    <a href="rmod_2.html#M2" title="RMOD_2:mode.2">Coset</a> of <font color="Maroon" title="c3">W</font> holds <br/> ( <font color="Olive" title="b1">C</font> is  <a href="rmod_2.html#V1" title="RMOD_2:attr.1">linearly-closed</a>  iff <font color="Olive" title="b1">C</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">W</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">C</font> be    <a href="rmod_2.html#M2" title="RMOD_2:mode.2">Coset</a> of <font color="Maroon" title="c3">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">C</font> is  <a href="rmod_2.html#V1" title="RMOD_2:attr.1">linearly-closed</a>  iff <font color="Maroon" title="c4">C</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">W</font> )</span><br/>







<span class="kw">thus </span><a NAME="E1:69"/>
( <font color="Maroon" title="c4">C</font> is  <a href="rmod_2.html#V1" title="RMOD_2:attr.1">linearly-closed</a>  implies <font color="Maroon" title="c4">C</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">W</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">C</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">W</font> implies <font color="Maroon" title="c4">C</font> is  <a href="rmod_2.html#V1" title="RMOD_2:attr.1">linearly-closed</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:69_1"/><span class="lab"><font color="Green" title="E47">A1</font></span>: 
<font color="Maroon" title="c4">C</font> is  <a href="rmod_2.html#V1" title="RMOD_2:attr.1">linearly-closed</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">C</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">W</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">v</font> being   <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E3:69_1"/><span class="lab"><font color="Green" title="E48">A2</font></span>: 
<font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rmod_2.html#D6" target="_self" title="RMOD_2:def.6">Def6</a></span>;<br/>
<a NAME="E4:69_1"/>
<font color="Maroon" title="c4">C</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_1"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="ref" href="rmod_2.html#T44" target="_self" title="RMOD_2:th.44">Th59</a></span>;<br/>
<a NAME="E5:69_1"/><span class="kw">then </span>
 <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> <font color="Maroon" title="c5">v</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c3">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="txt" href="#E3:69_1"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="ref" href="rmod_2.html#T1" target="_self" title="RMOD_2:th.1">Th4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:69_1"/>
<font color="Maroon" title="c4">C</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">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_1"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="ref" href="rmod_2.html#T48" target="_self" title="RMOD_2:th.48">Th63</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E2:69"/>
( <font color="Maroon" title="c4">C</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">W</font> implies <font color="Maroon" title="c4">C</font> is  <a href="rmod_2.html#V1" title="RMOD_2:attr.1">linearly-closed</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="rmod_2.html#E19"><span class="lab"><font color="Green" title="E12">Lm1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
