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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/>







<b>thus </b><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:()"><i><font color="Red">::  thesis: </font></i></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"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:69_1"/><i><font color="Green" title="E47">A1</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>consider </b><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><b> such that </b><br/><a NAME="E3:69_1"/><i><font color="Green" title="E48">A2</font></i>: 
<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>
 <b>by </b><i><a class="ref" href="rmod_2.html#D6" target="_self" title="RMOD_2:def.6">Def6</a></i>;<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> 
 
<b>by </b><i><a class="txt" href="rmod_2.html#E3:69_1"><i><font color="Green" title="E48">A2</font></i></a>, <a class="ref" href="rmod_2.html#T59" target="_self" title="RMOD_2:th.59">Th59</a></i>;<br/>
<a NAME="E5:69_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> <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>
 
<b>by </b><i><a class="txt" href="rmod_2.html#E1:69_1"><i><font color="Green" title="E47">A1</font></i></a>, <a class="txt" href="rmod_2.html#E3:69_1"><i><font color="Green" title="E48">A2</font></i></a>, <a class="ref" href="rmod_2.html#T4" target="_self" title="RMOD_2:th.4">Th4</a></i>;<br/>
<b>hence </b><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>
 <b>by </b><i><a class="txt" href="rmod_2.html#E3:69_1"><i><font color="Green" title="E48">A2</font></i></a>, <a class="ref" href="rmod_2.html#T63" target="_self" title="RMOD_2:th.63">Th63</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><b>end;</b></div>

<b>thus </b><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> )
 <b>by </b><i><a class="txt" href="rmod_2.html#E27"><i><font color="Green" title="E12">Lm1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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