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

<b>set </b><font color="Maroon" title="c4">A1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Olive" title="b1">l</font></span>)</span> where <font color="Olive" title="b1">l</font> is    <a href="rmod_4.html#M2" title="RMOD_4:mode.2">Linear_Combination</a> of <font color="Maroon" title="c3">A</font> : verum </span>}</span> ;<br/>
<a NAME="E1:7_1_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Olive" title="b1">l</font></span>)</span> where <font color="Olive" title="b1">l</font> is    <a href="rmod_4.html#M2" title="RMOD_4:mode.2">Linear_Combination</a> of <font color="Maroon" title="c3">A</font> : verum </span>}</span>  <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:7_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Olive" title="b1">l</font></span>)</span> where <font color="Olive" title="b1">l</font> is    <a href="rmod_4.html#M2" title="RMOD_4:mode.2">Linear_Combination</a> of <font color="Maroon" title="c3">A</font> : verum </span>}</span> 
 ;<br/>

<a NAME="E2:7_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">l</font> being   <a href="rmod_4.html#M2" title="RMOD_4:mode.2">Linear_Combination</a> of <font color="Maroon" title="c3">A</font> st <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Olive" title="b1">l</font>
 
;<br/>
<b>hence </b><a NAME="E3:7_1_1_1"/>
<font color="Maroon" title="c5">x</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="c2">V</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c5">A1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Olive" title="b1">l</font></span>)</span> where <font color="Olive" title="b1">l</font> is    <a href="rmod_4.html#M2" title="RMOD_4:mode.2">Linear_Combination</a> of <font color="Maroon" title="c3">A</font> : verum </span>}</span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c6">l</font> =  <a href="rmod_4.html#K3" title="RMOD_4:func.3">ZeroLC</a> <font color="Maroon" title="c2">V</font> as    <a href="rmod_4.html#M2" title="RMOD_4:mode.2">Linear_Combination</a> of <font color="Maroon" title="c3">A</font> <b>by </b><i><a class="ref" href="rmod_4.html#T26" title="RMOD_4:th.26">RMOD_4:26</a></i>;<br/>
<a NAME="E4:7_1_1"/>
 <a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Maroon" title="c6">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font>
 
<b>by </b><i><a class="ref" href="rmod_4.html#T41" title="RMOD_4:th.41">RMOD_4:41</a></i>;<br/>
<a NAME="E5:7_1_1"/><b>then </b><i><font color="Green" title="E4">A1</font></i>: 
 <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">A1</font>
 
;<br/>
<a NAME="E6:7_1_1"/>
<font color="Maroon" title="c5">A1</font> is <a href="rmod_2.html#V1" title="RMOD_2:attr.1">linearly-closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:7_1_1_2"/>
for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">u</font> being  <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A1</font> &amp; <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A1</font> holds <br/><font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A1</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rmod_2.html#D1" title="RMOD_2:def.1">RMOD_2:def 1</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">v</font>, <font color="Maroon" title="c8">u</font> be   <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:7_1_1_2_1"/><i><font color="Green" title="E5">A2</font></i>: 
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A1</font>
 <b>and </b><br/><a NAME="E2:7_1_1_2_1"/><i><font color="Green" title="E6">A3</font></i>: 
<font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A1</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c9">l1</font> being    <a href="rmod_4.html#M2" title="RMOD_4:mode.2">Linear_Combination</a> of <font color="Maroon" title="c3">A</font><b> such that </b><br/><a NAME="E4:7_1_1_2_1"/><i><font color="Green" title="E7">A4</font></i>: 
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Maroon" title="c9">l1</font>
 <b>by </b><i><a class="txt" href="rmod_5.html#E1:7_1_1_2_1"><i><font color="Green" title="E5">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">l2</font> being    <a href="rmod_4.html#M2" title="RMOD_4:mode.2">Linear_Combination</a> of <font color="Maroon" title="c3">A</font><b> such that </b><br/><a NAME="E6:7_1_1_2_1"/><i><font color="Green" title="E8">A5</font></i>: 
<font color="Maroon" title="c8">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Maroon" title="c10">l2</font>
 <b>by </b><i><a class="txt" href="rmod_5.html#E2:7_1_1_2_1"><i><font color="Green" title="E6">A3</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">f</font> = <font color="Maroon" title="c9">l1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c10">l2</font> as    <a href="rmod_4.html#M2" title="RMOD_4:mode.2">Linear_Combination</a> of <font color="Maroon" title="c3">A</font> <b>by </b><i><a class="ref" href="rmod_4.html#T52" title="RMOD_4:th.52">RMOD_4:52</a></i>;<br/>
<a NAME="E8:7_1_1_2_1"/>
<font color="Maroon" title="c7">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Maroon" title="c11">f</font>
 
<b>by </b><i><a class="txt" href="rmod_5.html#E4:7_1_1_2_1"><i><font color="Green" title="E7">A4</font></i></a>, <a class="txt" href="rmod_5.html#E6:7_1_1_2_1"><i><font color="Green" title="E8">A5</font></i></a>, <a class="ref" href="rmod_4.html#T77" title="RMOD_4:th.77">RMOD_4:77</a></i>;<br/>
<b>hence </b><a NAME="E9:7_1_1_2_1"/>
<font color="Maroon" title="c7">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A1</font>
 ;<br/>


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

<b>let </b><font color="Maroon" title="c7">a</font> be   <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">R</font>;<br/><b>let </b><font color="Maroon" title="c8">v</font> be   <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font>;<br/>



<b>assume </b><a NAME="E2:7_1_1_2"/>
<font color="Maroon" title="c8">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A1</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">l</font> being    <a href="rmod_4.html#M2" title="RMOD_4:mode.2">Linear_Combination</a> of <font color="Maroon" title="c3">A</font><b> such that </b><br/><a NAME="E4:7_1_1_2"/><i><font color="Green" title="E5">A6</font></i>: 
<font color="Maroon" title="c8">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Maroon" title="c9">l</font>
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c10">f</font> = <font color="Maroon" title="c9">l</font> <a href="rmod_4.html#K7" title="RMOD_4:func.7">*</a> <font color="Maroon" title="c7">a</font> as    <a href="rmod_4.html#M2" title="RMOD_4:mode.2">Linear_Combination</a> of <font color="Maroon" title="c3">A</font> <b>by </b><i><a class="ref" href="rmod_4.html#T61" title="RMOD_4:th.61">RMOD_4:61</a></i>;<br/>
<a NAME="E6:7_1_1_2"/>
<font color="Maroon" title="c8">v</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Maroon" title="c10">f</font>
 
<b>by </b><i><a class="txt" href="rmod_5.html#E4:7_1_1_2"><i><font color="Green" title="E5">A6</font></i></a>, <a class="ref" href="rmod_4.html#T78" title="RMOD_4:th.78">RMOD_4:78</a></i>;<br/>
<b>hence </b><a NAME="E7:7_1_1_2"/>
<font color="Maroon" title="c8">v</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A1</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:7_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> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Olive" title="b2">l</font></span>)</span> where <font color="Olive" title="b2">l</font> is    <a href="rmod_4.html#M2" title="RMOD_4:mode.2">Linear_Combination</a> of <font color="Maroon" title="c3">A</font> : verum </span>}</span> 
 <b>by </b><i><a class="txt" href="rmod_5.html#E5:7_1_1"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="rmod_2.html#T42" title="RMOD_2:th.42">RMOD_2:42</a></i>;<br/>


</div>
