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

<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">W</font> be    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">W1</font> being <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">strict</a>  <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W1</font> &amp; <font color="Maroon" title="c2">W</font> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <font color="Olive" title="b1">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlsub_1.html#K1" title="RLSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font> );<br/>
<b>consider </b><font color="Maroon" title="c3">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:47"/><i><font color="Green" title="E39">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlsub_2.html#K3" title="RLSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E3:47"/>
( <font color="Maroon" title="c2">W</font> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <span class="p1">(<span class="default"><a href="rlsub_1.html#K1" title="RLSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlsub_1.html#K1" title="RLSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font> &amp;  <a href="rlsub_1.html#K1" title="RLSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlsub_2.html#K3" title="RLSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font> )
 
<b>by </b><i><a class="ref" href="rlsub_2.html#D3" target="_self" title="RLSUB_2:def.3">Def3</a>, <a class="ref" href="rlsub_2.html#T22" target="_self" title="RLSUB_2:th.22">Th22</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">X</font> = <font color="Maroon" title="c3">X</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="rlsub_2.html#E2:47"><i><font color="Green" title="E39">A1</font></i></a></i>;<br/>
<div><i><font color="Green" title="E40">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font>;<br/><a NAME="E1:47_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlsub_2.html#K3" title="RLSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E2:47"><i><font color="Green" title="E39">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E2:47_1"/>
<font color="Maroon" title="c5">x</font> is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">strict</a>  <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="ref" href="rlsub_2.html#D3" target="_self" title="RLSUB_2:def.3">Def3</a></i>;<br/></div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">W1</font>, <font color="Olive" title="b2">W2</font> being <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">strict</a>  <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W1</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">W2</font> &amp; <font color="Olive" title="b1">W1</font> is    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Olive" title="b2">W2</font> );<br/>
<b>consider </b><font color="Maroon" title="c5">R</font> being   <a href="relset_1.html#NM3" title="RELSET_1:NM.3">Relation</a> of <font color="Maroon" title="c4">X</font><b> such that </b><br/><a NAME="E7:47"/><i><font color="Green" title="E41">A3</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font> iff <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 <b>from </b><i><a class="ref" href="relset_1.html#S2" title="RELSET_1:sch.2">RELSET_1:sch 2</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">$1,$2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font>;<br/>
<a NAME="E8:47"/><i><font color="Green" title="E42">A4</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font> holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b1">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font>;<br/><b>reconsider </b><font color="Maroon" title="c7">W</font> = <font color="Maroon" title="c6">x</font> as  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">strict</a>  <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="rlsub_2.html#E5:47"><i><font color="Green" title="E40">A2</font></i></a></i>;<br/><a NAME="E2:47_2_1"/>
<font color="Maroon" title="c7">W</font> is    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c7">W</font>
 <b>by </b><i><a class="ref" href="rlsub_1.html#T33" title="RLSUB_1:th.33">RLSUB_1:33</a></i>;<br/><b>hence </b><a NAME="E3:47_2_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c6">x</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E7:47"><i><font color="Green" title="E41">A3</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:47_2"/>
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font> holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b1">x</font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E9:47"/><i><font color="Green" title="E43">A5</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>] holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:47_3_1"/><i><font color="Green" title="E44">A6</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font>
 <b>and </b><br/><a NAME="E2:47_3_1"/><i><font color="Green" title="E45">A7</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">x</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font>
 ;<br/><a NAME="E3:47_3_1"/><i><font color="Green" title="E46">A8</font></i>: 
 ex <font color="Olive" title="b1">W1</font>, <font color="Olive" title="b2">W2</font> being <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">strict</a>  <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W1</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">W2</font> &amp; <font color="Olive" title="b1">W1</font> is    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Olive" title="b2">W2</font> )
 <b>by </b><i><a class="txt" href="rlsub_2.html#E7:47"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="rlsub_2.html#E1:47_3_1"><i><font color="Green" title="E44">A6</font></i></a></i>;<br/><a NAME="E4:47_3_1"/>
 ex <font color="Olive" title="b1">W3</font>, <font color="Olive" title="b2">W4</font> being <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">strict</a>  <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W3</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">W4</font> &amp; <font color="Olive" title="b1">W3</font> is    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Olive" title="b2">W4</font> )
 <b>by </b><i><a class="txt" href="rlsub_2.html#E7:47"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="rlsub_2.html#E2:47_3_1"><i><font color="Green" title="E45">A7</font></i></a></i>;<br/><b>hence </b><a NAME="E5:47_3_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E3:47_3_1"><i><font color="Green" title="E46">A8</font></i></a>, <a class="ref" href="rlsub_1.html#T34" title="RLSUB_1:th.34">RLSUB_1:34</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:47_3"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>] holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E10:47"/><i><font color="Green" title="E44">A9</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">z</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_4"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font>, <font color="Maroon" title="c8">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:47_4_1"/><i><font color="Green" title="E45">A10</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font>
 <b>and </b><br/><a NAME="E2:47_4_1"/><i><font color="Green" title="E46">A11</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font>
 ;<br/><b>consider </b><font color="Maroon" title="c9">W1</font>, <font color="Maroon" title="c10">W2</font> being  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">strict</a>  <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E4:47_4_1"/><i><font color="Green" title="E47">A12</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">W1</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">W2</font> &amp; <font color="Maroon" title="c9">W1</font> is    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c10">W2</font> )
 <b>by </b><i><a class="txt" href="rlsub_2.html#E7:47"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="rlsub_2.html#E1:47_4_1"><i><font color="Green" title="E45">A10</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c11">W3</font>, <font color="Maroon" title="c12">W4</font> being  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">strict</a>  <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E6:47_4_1"/><i><font color="Green" title="E48">A13</font></i>: 
( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">W3</font> &amp; <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">W4</font> &amp; <font color="Maroon" title="c11">W3</font> is    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c12">W4</font> )
 <b>by </b><i><a class="txt" href="rlsub_2.html#E7:47"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="rlsub_2.html#E2:47_4_1"><i><font color="Green" title="E46">A11</font></i></a></i>;<br/><a NAME="E7:47_4_1"/>
<font color="Maroon" title="c9">W1</font> is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">strict</a>  <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c12">W4</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E4:47_4_1"><i><font color="Green" title="E47">A12</font></i></a>, <a class="txt" href="rlsub_2.html#E6:47_4_1"><i><font color="Green" title="E48">A13</font></i></a>, <a class="ref" href="rlsub_1.html#T35" title="RLSUB_1:th.35">RLSUB_1:35</a></i>;<br/><b>hence </b><a NAME="E8:47_4_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E7:47"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="rlsub_2.html#E4:47_4_1"><i><font color="Green" title="E47">A12</font></i></a>, <a class="txt" href="rlsub_2.html#E6:47_4_1"><i><font color="Green" title="E48">A13</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:47_4"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">z</font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E11:47"/><i><font color="Green" title="E45">A14</font></i>: 
for <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font> &amp; ( for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font> &amp; <font color="Olive" title="b3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font> &amp;  not <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b3">y</font>,<font color="Olive" title="b2">x</font>] ) holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font> st <br/>for <font color="Olive" title="b3">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b3">x</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_5"><b>proof </b></a><div class="add">

<a NAME="E1:47_5"/>
for <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font> &amp; ( for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font> &amp; <font color="Olive" title="b3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font> holds <br/><span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b3">y</font>,<font color="Olive" title="b2">x</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font> ) holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font> st <br/>for <font color="Olive" title="b3">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font> holds <br/><span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b3">x</font>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:47_5_1"/><i><font color="Green" title="E46">A15</font></i>: 
<font color="Maroon" title="c6">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font>
 <b>and </b><br/><a NAME="E2:47_5_1"/><i><font color="Green" title="E47">A16</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font> holds <br/><span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_5_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_5_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_5_1_1_1"/>
( <font color="Maroon" title="c6">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c6">Y</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>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_5_1_1_1_1"><b>suppose </b></a><a NAME="E1:47_5_1_1_1_1"/><i><font color="Green" title="E48">A17</font></i>: 
<font color="Maroon" title="c6">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c7">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font>;<br/><b>take </b><font color="Maroon" title="c8">y'</font> = <font color="Maroon" title="c7">y</font>;<br/><b>let </b><font color="Maroon" title="c9">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font>;<br/><b>assume </b><a NAME="E3:47_5_1_1_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font>
 ;<br/><b>hence </b><a NAME="E4:47_5_1_1_1_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c8">y'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E1:47_5_1_1_1_1"><i><font color="Green" title="E48">A17</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_5_1_1_1_2"><b>suppose </b></a><a NAME="E1:47_5_1_1_1_2"/><i><font color="Green" title="E48">A18</font></i>: 
<font color="Maroon" title="c6">Y</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> 
 ;<br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">W1</font> being <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">strict</a>  <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W1</font> &amp; $2 <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">W1</font> );<br/><a NAME="E2:47_5_1_1_1_2"/><i><font color="Green" title="E49">A19</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y1</font>, <font color="Olive" title="b3">y2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font> &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y1</font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">y2</font>] holds <br/><font color="Olive" title="b2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y2</font>
 ;<br/><a NAME="E3:47_5_1_1_1_2"/><i><font color="Green" title="E50">A20</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_5_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:47_5_1_1_1_2_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">W1</font> being  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">strict</a>  <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E3:47_5_1_1_1_2_1"/><i><font color="Green" title="E51">A21</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">W1</font>
 <b>and </b><br/><a NAME="E4:47_5_1_1_1_2_1"/>
<font color="Maroon" title="c2">W</font> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <font color="Maroon" title="c8">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlsub_1.html#K1" title="RLSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E2:47"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="rlsub_2.html#E1:47_5_1"><i><font color="Green" title="E46">A15</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">y</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">W1</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon" title="c9">y</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c8">W1</font>
;<br/>

<b>thus </b><a NAME="E6:47_5_1_1_1_2_1"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">W1</font> &amp; <font color="Maroon" title="c9">y</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="c8">W1</font> )
 <b>by </b><i><a class="txt" href="rlsub_2.html#E3:47_5_1_1_1_2_1"><i><font color="Green" title="E51">A21</font></i></a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon" title="c7">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:47_5_1_1_1_2"/><i><font color="Green" title="E51">A22</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Y</font>
 <b>and </b><br/><a NAME="E6:47_5_1_1_1_2"/><i><font color="Green" title="E52">A23</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font> holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_1.html#S2" title="FUNCT_1:sch.2">FUNCT_1:sch 2</a>(<a class="txt" href="rlsub_2.html#E2:47_5_1_1_1_2"><i><font color="Green" title="E49">A19</font></i></a>, <a class="txt" href="rlsub_2.html#E3:47_5_1_1_1_2"><i><font color="Green" title="E50">A20</font></i></a>);<br/></i><b>set </b><font color="Maroon" title="c8">Z</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">f</font></span>)</span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_5_1_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:47_5_1_1_1_2_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">f</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c10">Y'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:47_5_1_1_1_2_2"/><i><font color="Green" title="E53">A24</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Y'</font>
 <b>and </b><br/><a NAME="E4:47_5_1_1_1_2_2"/><i><font color="Green" title="E54">A25</font></i>: 
<font color="Maroon" title="c10">Y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">f</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:47_5_1_1_1_2_2"/><i><font color="Green" title="E55">A26</font></i>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">f</font>
 <b>and </b><br/><a NAME="E7:47_5_1_1_1_2_2"/><i><font color="Green" title="E56">A27</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Y'</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E4:47_5_1_1_1_2_2"><i><font color="Green" title="E54">A25</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">W1</font> being  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">strict</a>  <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E9:47_5_1_1_1_2_2"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">W1</font>
 <b>and </b><br/><a NAME="E10:47_5_1_1_1_2_2"/><i><font color="Green" title="E57">A28</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</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="c12">W1</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E5:47_5_1_1_1_2"><i><font color="Green" title="E51">A22</font></i></a>, <a class="txt" href="rlsub_2.html#E6:47_5_1_1_1_2"><i><font color="Green" title="E52">A23</font></i></a>, <a class="txt" href="rlsub_2.html#E6:47_5_1_1_1_2_2"><i><font color="Green" title="E55">A26</font></i></a></i>;<br/><a NAME="E11:47_5_1_1_1_2_2"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">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="c1">V</font> &amp; <font color="Maroon" title="c9">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="c12">W1</font> )
 <b>by </b><i><a class="txt" href="rlsub_2.html#E3:47_5_1_1_1_2_2"><i><font color="Green" title="E53">A24</font></i></a>, <a class="txt" href="rlsub_2.html#E7:47_5_1_1_1_2_2"><i><font color="Green" title="E56">A27</font></i></a>, <a class="txt" href="rlsub_2.html#E10:47_5_1_1_1_2_2"><i><font color="Green" title="E57">A28</font></i></a>, <a class="ref" href="rlsub_1.html#D2" title="RLSUB_1:def.2">RLSUB_1:def 2</a></i>;<br/><b>hence </b><a NAME="E12:47_5_1_1_1_2_2"/>
<font color="Maroon" title="c9">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="c1">V</font>
 ;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c9">Z</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">f</font></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E9:47_5_1_1_1_2"/><i><font color="Green" title="E53">A29</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">f</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="rlsub_2.html#E1:47_5_1_1_1_2"><i><font color="Green" title="E48">A18</font></i></a>, <a class="txt" href="rlsub_2.html#E5:47_5_1_1_1_2"><i><font color="Green" title="E51">A22</font></i></a>, <a class="ref" href="relat_1.html#T65" title="RELAT_1:th.65">RELAT_1:65</a></i>;<br/><b>consider </b><font color="Maroon" title="c10">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">f</font>;<br/><b>consider </b><font color="Maroon" title="c11">z1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E12:47_5_1_1_1_2"/><i><font color="Green" title="E54">A30</font></i>: 
<font color="Maroon" title="c11">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">f</font>
 <b>and </b><br/><a NAME="E13:47_5_1_1_1_2"/><i><font color="Green" title="E55">A31</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">z</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E9:47_5_1_1_1_2"><i><font color="Green" title="E53">A29</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">W3</font> being  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">strict</a>  <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E15:47_5_1_1_1_2"/><i><font color="Green" title="E56">A32</font></i>: 
( <font color="Maroon" title="c11">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">W3</font> &amp; <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">z1</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="c12">W3</font> )
 <b>by </b><i><a class="txt" href="rlsub_2.html#E5:47_5_1_1_1_2"><i><font color="Green" title="E51">A22</font></i></a>, <a class="txt" href="rlsub_2.html#E6:47_5_1_1_1_2"><i><font color="Green" title="E52">A23</font></i></a>, <a class="txt" href="rlsub_2.html#E12:47_5_1_1_1_2"><i><font color="Green" title="E54">A30</font></i></a></i>;<br/><a NAME="E16:47_5_1_1_1_2"/><i><font color="Green" title="E57">A33</font></i>: 
<font color="Maroon" title="c9">Z</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="rlsub_2.html#E9:47_5_1_1_1_2"><i><font color="Green" title="E53">A29</font></i></a>, <a class="txt" href="rlsub_2.html#E13:47_5_1_1_1_2"><i><font color="Green" title="E55">A31</font></i></a>, <a class="txt" href="rlsub_2.html#E15:47_5_1_1_1_2"><i><font color="Green" title="E56">A32</font></i></a>, <a class="ref" href="orders_1.html#T91" title="ORDERS_1:th.91">ORDERS_1:91</a></i>;<br/><a NAME="E17:47_5_1_1_1_2"/>
<font color="Maroon" title="c9">Z</font> is <a href="rlsub_1.html#V1" title="RLSUB_1:attr.1">lineary-closed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_5_1_1_1_2_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:47_5_1_1_1_2_3"/>
for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Olive" title="b2">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> holds <br/><font color="Olive" title="b1">v1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Olive" title="b2">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlsub_1.html#D1" title="RLSUB_1:def.1">RLSUB_1:def 1</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_5_1_1_1_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">v1</font>, <font color="Maroon" title="c14">v2</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:47_5_1_1_1_2_3_1"/><i><font color="Green" title="E58">A34</font></i>: 
<font color="Maroon" title="c13">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font>
 <b>and </b><br/><a NAME="E2:47_5_1_1_1_2_3_1"/><i><font color="Green" title="E59">A35</font></i>: 
<font color="Maroon" title="c14">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c15">Y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:47_5_1_1_1_2_3_1"/><i><font color="Green" title="E60">A36</font></i>: 
<font color="Maroon" title="c13">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Y1</font>
 <b>and </b><br/><a NAME="E5:47_5_1_1_1_2_3_1"/><i><font color="Green" title="E61">A37</font></i>: 
<font color="Maroon" title="c15">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">f</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E1:47_5_1_1_1_2_3_1"><i><font color="Green" title="E58">A34</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">Y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:47_5_1_1_1_2_3_1"/><i><font color="Green" title="E62">A38</font></i>: 
<font color="Maroon" title="c14">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Y2</font>
 <b>and </b><br/><a NAME="E8:47_5_1_1_1_2_3_1"/><i><font color="Green" title="E63">A39</font></i>: 
<font color="Maroon" title="c16">Y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">f</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E2:47_5_1_1_1_2_3_1"><i><font color="Green" title="E59">A35</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:47_5_1_1_1_2_3_1"/><i><font color="Green" title="E64">A40</font></i>: 
<font color="Maroon" title="c17">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">f</font>
 <b>and </b><br/><a NAME="E11:47_5_1_1_1_2_3_1"/><i><font color="Green" title="E65">A41</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Y1</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E5:47_5_1_1_1_2_3_1"><i><font color="Green" title="E61">A37</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E13:47_5_1_1_1_2_3_1"/><i><font color="Green" title="E66">A42</font></i>: 
<font color="Maroon" title="c18">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">f</font>
 <b>and </b><br/><a NAME="E14:47_5_1_1_1_2_3_1"/><i><font color="Green" title="E67">A43</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">Y2</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E8:47_5_1_1_1_2_3_1"><i><font color="Green" title="E63">A39</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c19">W1</font> being  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">strict</a>  <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E16:47_5_1_1_1_2_3_1"/><i><font color="Green" title="E68">A44</font></i>: 
<font color="Maroon" title="c17">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">W1</font>
 <b>and </b><br/><a NAME="E17:47_5_1_1_1_2_3_1"/><i><font color="Green" title="E69">A45</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">y1</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="c19">W1</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E5:47_5_1_1_1_2"><i><font color="Green" title="E51">A22</font></i></a>, <a class="txt" href="rlsub_2.html#E6:47_5_1_1_1_2"><i><font color="Green" title="E52">A23</font></i></a>, <a class="txt" href="rlsub_2.html#E10:47_5_1_1_1_2_3_1"><i><font color="Green" title="E64">A40</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c20">W2</font> being  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">strict</a>  <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E19:47_5_1_1_1_2_3_1"/><i><font color="Green" title="E70">A46</font></i>: 
<font color="Maroon" title="c18">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">W2</font>
 <b>and </b><br/><a NAME="E20:47_5_1_1_1_2_3_1"/><i><font color="Green" title="E71">A47</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y2</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="c20">W2</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E5:47_5_1_1_1_2"><i><font color="Green" title="E51">A22</font></i></a>, <a class="txt" href="rlsub_2.html#E6:47_5_1_1_1_2"><i><font color="Green" title="E52">A23</font></i></a>, <a class="txt" href="rlsub_2.html#E13:47_5_1_1_1_2_3_1"><i><font color="Green" title="E66">A42</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c21">y1</font> = <font color="Maroon" title="c17">y1</font>, <font color="Maroon" title="c22">y2</font> = <font color="Maroon" title="c18">y2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font> <b>by </b><i><a class="txt" href="rlsub_2.html#E1:47_5_1"><i><font color="Green" title="E46">A15</font></i></a>, <a class="txt" href="rlsub_2.html#E5:47_5_1_1_1_2"><i><font color="Green" title="E51">A22</font></i></a>, <a class="txt" href="rlsub_2.html#E10:47_5_1_1_1_2_3_1"><i><font color="Green" title="E64">A40</font></i></a>, <a class="txt" href="rlsub_2.html#E13:47_5_1_1_1_2_3_1"><i><font color="Green" title="E66">A42</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_5_1_1_1_2_3_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_5_1_1_1_2_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_5_1_1_1_2_3_1_1_1"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c21">y1</font>,<font color="Maroon" title="c22">y2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c22">y2</font>,<font color="Maroon" title="c21">y1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font> )
 </span><b>by </b><i><a class="txt" href="rlsub_2.html#E2:47_5_1"><i><font color="Green" title="E47">A16</font></i></a>, <a class="txt" href="rlsub_2.html#E5:47_5_1_1_1_2"><i><font color="Green" title="E51">A22</font></i></a>, <a class="txt" href="rlsub_2.html#E10:47_5_1_1_1_2_3_1"><i><font color="Green" title="E64">A40</font></i></a>, <a class="txt" href="rlsub_2.html#E13:47_5_1_1_1_2_3_1"><i><font color="Green" title="E66">A42</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_5_1_1_1_2_3_1_1_1_1"><b>suppose </b></a><a NAME="E1:47_5_1_1_1_2_3_1_1_1_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c21">y1</font>,<font color="Maroon" title="c22">y2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font>
 ;<br/><div class="add"><a NAME="E2:47_5_1_1_1_2_3_1_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">W3</font>, <font color="Olive" title="b2">W4</font> being <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">strict</a>  <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Maroon" title="c21">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W3</font> &amp; <font color="Maroon" title="c22">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">W4</font> &amp; <font color="Olive" title="b1">W3</font> is    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Olive" title="b2">W4</font> )
 <b>by </b><i><a class="txt" href="rlsub_2.html#E7:47"><i><font color="Green" title="E41">A3</font></i></a></i>;<br/><a NAME="E3:47_5_1_1_1_2_3_1_1_1_1"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c19">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="c20">W2</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E16:47_5_1_1_1_2_3_1"><i><font color="Green" title="E68">A44</font></i></a>, <a class="txt" href="rlsub_2.html#E19:47_5_1_1_1_2_3_1"><i><font color="Green" title="E70">A46</font></i></a>, <a class="ref" href="rlsub_1.html#D2" title="RLSUB_1:def.2">RLSUB_1:def 2</a></i>;<br/><a NAME="E4:47_5_1_1_1_2_3_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c13">v1</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c20">W2</font> &amp; <font color="Maroon" title="c14">v2</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c20">W2</font> )
 <b>by </b><i><a class="txt" href="rlsub_2.html#E4:47_5_1_1_1_2_3_1"><i><font color="Green" title="E60">A36</font></i></a>, <a class="txt" href="rlsub_2.html#E7:47_5_1_1_1_2_3_1"><i><font color="Green" title="E62">A38</font></i></a>, <a class="txt" href="rlsub_2.html#E11:47_5_1_1_1_2_3_1"><i><font color="Green" title="E65">A41</font></i></a>, <a class="txt" href="rlsub_2.html#E14:47_5_1_1_1_2_3_1"><i><font color="Green" title="E67">A43</font></i></a>, <a class="txt" href="rlsub_2.html#E17:47_5_1_1_1_2_3_1"><i><font color="Green" title="E69">A45</font></i></a>, <a class="txt" href="rlsub_2.html#E20:47_5_1_1_1_2_3_1"><i><font color="Green" title="E71">A47</font></i></a>, <a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/><a NAME="E5:47_5_1_1_1_2_3_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c13">v1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c14">v2</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c20">W2</font>
 <b>by </b><i><a class="ref" href="rlsub_1.html#T28" title="RLSUB_1:th.28">RLSUB_1:28</a></i>;<br/><a NAME="E6:47_5_1_1_1_2_3_1_1_1_1"/><b>then </b><i><font color="Green" title="E72">A48</font></i>: 
<font color="Maroon" title="c13">v1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c14">v2</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="c20">W2</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/><a NAME="E7:47_5_1_1_1_2_3_1_1_1_1"/>
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">f</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E13:47_5_1_1_1_2_3_1"><i><font color="Green" title="E66">A42</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E8:47_5_1_1_1_2_3_1_1_1_1"/>
<font color="Maroon" title="c13">v1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c14">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E20:47_5_1_1_1_2_3_1"><i><font color="Green" title="E71">A47</font></i></a>, <a class="txt" href="rlsub_2.html#E6:47_5_1_1_1_2_3_1_1_1_1"><i><font color="Green" title="E72">A48</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_5_1_1_1_2_3_1_1_1_2"><b>suppose </b></a><a NAME="E1:47_5_1_1_1_2_3_1_1_1_2"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c22">y2</font>,<font color="Maroon" title="c21">y1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font>
 ;<br/><div class="add"><a NAME="E2:47_5_1_1_1_2_3_1_1_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">W3</font>, <font color="Olive" title="b2">W4</font> being <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">strict</a>  <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Maroon" title="c22">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W3</font> &amp; <font color="Maroon" title="c21">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">W4</font> &amp; <font color="Olive" title="b1">W3</font> is    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Olive" title="b2">W4</font> )
 <b>by </b><i><a class="txt" href="rlsub_2.html#E7:47"><i><font color="Green" title="E41">A3</font></i></a></i>;<br/><a NAME="E3:47_5_1_1_1_2_3_1_1_1_2"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c20">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="c19">W1</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E16:47_5_1_1_1_2_3_1"><i><font color="Green" title="E68">A44</font></i></a>, <a class="txt" href="rlsub_2.html#E19:47_5_1_1_1_2_3_1"><i><font color="Green" title="E70">A46</font></i></a>, <a class="ref" href="rlsub_1.html#D2" title="RLSUB_1:def.2">RLSUB_1:def 2</a></i>;<br/><a NAME="E4:47_5_1_1_1_2_3_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c13">v1</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c19">W1</font> &amp; <font color="Maroon" title="c14">v2</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c19">W1</font> )
 <b>by </b><i><a class="txt" href="rlsub_2.html#E4:47_5_1_1_1_2_3_1"><i><font color="Green" title="E60">A36</font></i></a>, <a class="txt" href="rlsub_2.html#E7:47_5_1_1_1_2_3_1"><i><font color="Green" title="E62">A38</font></i></a>, <a class="txt" href="rlsub_2.html#E11:47_5_1_1_1_2_3_1"><i><font color="Green" title="E65">A41</font></i></a>, <a class="txt" href="rlsub_2.html#E14:47_5_1_1_1_2_3_1"><i><font color="Green" title="E67">A43</font></i></a>, <a class="txt" href="rlsub_2.html#E17:47_5_1_1_1_2_3_1"><i><font color="Green" title="E69">A45</font></i></a>, <a class="txt" href="rlsub_2.html#E20:47_5_1_1_1_2_3_1"><i><font color="Green" title="E71">A47</font></i></a>, <a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/><a NAME="E5:47_5_1_1_1_2_3_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c13">v1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c14">v2</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c19">W1</font>
 <b>by </b><i><a class="ref" href="rlsub_1.html#T28" title="RLSUB_1:th.28">RLSUB_1:28</a></i>;<br/><a NAME="E6:47_5_1_1_1_2_3_1_1_1_2"/><b>then </b><i><font color="Green" title="E72">A49</font></i>: 
<font color="Maroon" title="c13">v1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c14">v2</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="c19">W1</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/><a NAME="E7:47_5_1_1_1_2_3_1_1_1_2"/>
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">f</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E10:47_5_1_1_1_2_3_1"><i><font color="Green" title="E64">A40</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E8:47_5_1_1_1_2_3_1_1_1_2"/>
<font color="Maroon" title="c13">v1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c14">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E17:47_5_1_1_1_2_3_1"><i><font color="Green" title="E69">A45</font></i></a>, <a class="txt" href="rlsub_2.html#E6:47_5_1_1_1_2_3_1_1_1_2"><i><font color="Green" title="E72">A49</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E23:47_5_1_1_1_2_3_1"/>
<font color="Maroon" title="c13">v1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c14">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font>
 ;<br/>


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

<b>let </b><font color="Maroon" title="c13">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c14">v1</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>;<br/>



<b>assume </b><a NAME="E2:47_5_1_1_1_2_3"/>
<font color="Maroon" title="c14">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c15">Y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:47_5_1_1_1_2_3"/><i><font color="Green" title="E58">A50</font></i>: 
<font color="Maroon" title="c14">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Y1</font>
 <b>and </b><br/><a NAME="E5:47_5_1_1_1_2_3"/><i><font color="Green" title="E59">A51</font></i>: 
<font color="Maroon" title="c15">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">f</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:47_5_1_1_1_2_3"/><i><font color="Green" title="E60">A52</font></i>: 
<font color="Maroon" title="c16">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">f</font>
 <b>and </b><br/><a NAME="E8:47_5_1_1_1_2_3"/><i><font color="Green" title="E61">A53</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Y1</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E5:47_5_1_1_1_2_3"><i><font color="Green" title="E59">A51</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">W1</font> being  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">strict</a>  <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E10:47_5_1_1_1_2_3"/>
<font color="Maroon" title="c16">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">W1</font>
 <b>and </b><br/><a NAME="E11:47_5_1_1_1_2_3"/><i><font color="Green" title="E62">A54</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">y1</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="c17">W1</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E5:47_5_1_1_1_2"><i><font color="Green" title="E51">A22</font></i></a>, <a class="txt" href="rlsub_2.html#E6:47_5_1_1_1_2"><i><font color="Green" title="E52">A23</font></i></a>, <a class="txt" href="rlsub_2.html#E7:47_5_1_1_1_2_3"><i><font color="Green" title="E60">A52</font></i></a></i>;<br/>
<a NAME="E12:47_5_1_1_1_2_3"/>
<font color="Maroon" title="c14">v1</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c17">W1</font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E4:47_5_1_1_1_2_3"><i><font color="Green" title="E58">A50</font></i></a>, <a class="txt" href="rlsub_2.html#E8:47_5_1_1_1_2_3"><i><font color="Green" title="E61">A53</font></i></a>, <a class="txt" href="rlsub_2.html#E11:47_5_1_1_1_2_3"><i><font color="Green" title="E62">A54</font></i></a>, <a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E13:47_5_1_1_1_2_3"/><b>then </b>
<font color="Maroon" title="c13">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c14">v1</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c17">W1</font>
 
<b>by </b><i><a class="ref" href="rlsub_1.html#T29" title="RLSUB_1:th.29">RLSUB_1:29</a></i>;<br/>
<a NAME="E14:47_5_1_1_1_2_3"/><b>then </b><i><font color="Green" title="E63">A55</font></i>: 
<font color="Maroon" title="c13">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c14">v1</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="c17">W1</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E15:47_5_1_1_1_2_3"/>
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">f</font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E7:47_5_1_1_1_2_3"><i><font color="Green" title="E60">A52</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E16:47_5_1_1_1_2_3"/>
<font color="Maroon" title="c13">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c14">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E11:47_5_1_1_1_2_3"><i><font color="Green" title="E62">A54</font></i></a>, <a class="txt" href="rlsub_2.html#E14:47_5_1_1_1_2_3"><i><font color="Green" title="E63">A55</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c13">E</font> being  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">strict</a>  <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E19:47_5_1_1_1_2"/><i><font color="Green" title="E58">A56</font></i>: 
<font color="Maroon" title="c9">Z</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="c13">E</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E16:47_5_1_1_1_2"><i><font color="Green" title="E57">A33</font></i></a>, <a class="ref" href="rlsub_1.html#T43" title="RLSUB_1:th.43">RLSUB_1:43</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_5_1_1_1_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">u</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>;<br/><b>thus </b><a NAME="E1:47_5_1_1_1_2_4"/>
( <font color="Maroon" title="c14">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">W</font> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <font color="Maroon" title="c13">E</font> implies <font color="Maroon" title="c14">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a>  <a href="rlsub_1.html#K1" title="RLSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_5_1_1_1_2_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:47_5_1_1_1_2_4_1"/><i><font color="Green" title="E59">A57</font></i>: 
<font color="Maroon" title="c14">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">W</font> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <font color="Maroon" title="c13">E</font>
 ;<br/>

<a NAME="E2:47_5_1_1_1_2_4_1"/><b>then </b>
<font color="Maroon" title="c14">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c13">E</font>
 
<b>by </b><i><a class="ref" href="rlsub_2.html#T7" target="_self" title="RLSUB_2:th.7">Th7</a></i>;<br/>
<a NAME="E3:47_5_1_1_1_2_4_1"/><b>then </b>
<font color="Maroon" title="c14">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E19:47_5_1_1_1_2"><i><font color="Green" title="E58">A56</font></i></a>, <a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">Y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:47_5_1_1_1_2_4_1"/><i><font color="Green" title="E60">A58</font></i>: 
<font color="Maroon" title="c14">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Y1</font>
 <b>and </b><br/><a NAME="E6:47_5_1_1_1_2_4_1"/><i><font color="Green" title="E61">A59</font></i>: 
<font color="Maroon" title="c15">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">f</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:47_5_1_1_1_2_4_1"/><i><font color="Green" title="E62">A60</font></i>: 
<font color="Maroon" title="c16">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">f</font>
 <b>and </b><br/><a NAME="E9:47_5_1_1_1_2_4_1"/><i><font color="Green" title="E63">A61</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Y1</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E6:47_5_1_1_1_2_4_1"><i><font color="Green" title="E61">A59</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">W1</font> being  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">strict</a>  <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E11:47_5_1_1_1_2_4_1"/><i><font color="Green" title="E64">A62</font></i>: 
<font color="Maroon" title="c16">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">W1</font>
 <b>and </b><br/><a NAME="E12:47_5_1_1_1_2_4_1"/><i><font color="Green" title="E65">A63</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">y1</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="c17">W1</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E5:47_5_1_1_1_2"><i><font color="Green" title="E51">A22</font></i></a>, <a class="txt" href="rlsub_2.html#E6:47_5_1_1_1_2"><i><font color="Green" title="E52">A23</font></i></a>, <a class="txt" href="rlsub_2.html#E8:47_5_1_1_1_2_4_1"><i><font color="Green" title="E62">A60</font></i></a></i>;<br/>
<a NAME="E13:47_5_1_1_1_2_4_1"/><i><font color="Green" title="E66">A64</font></i>: 
( <font color="Maroon" title="c14">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c17">W1</font> &amp; <font color="Maroon" title="c14">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">W</font> )
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E1:47_5_1_1_1_2_4_1"><i><font color="Green" title="E59">A57</font></i></a>, <a class="txt" href="rlsub_2.html#E5:47_5_1_1_1_2_4_1"><i><font color="Green" title="E60">A58</font></i></a>, <a class="txt" href="rlsub_2.html#E9:47_5_1_1_1_2_4_1"><i><font color="Green" title="E63">A61</font></i></a>, <a class="txt" href="rlsub_2.html#E12:47_5_1_1_1_2_4_1"><i><font color="Green" title="E65">A63</font></i></a>, <a class="ref" href="rlsub_2.html#T7" target="_self" title="RLSUB_2:th.7">Th7</a>, <a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E14:47_5_1_1_1_2_4_1"/>
 ex <font color="Olive" title="b1">W2</font> being <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">strict</a>  <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Maroon" title="c16">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W2</font> &amp; <font color="Maroon" title="c2">W</font> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <font color="Olive" title="b1">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlsub_1.html#K1" title="RLSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font> )
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E2:47"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="rlsub_2.html#E1:47_5_1"><i><font color="Green" title="E46">A15</font></i></a>, <a class="txt" href="rlsub_2.html#E5:47_5_1_1_1_2"><i><font color="Green" title="E51">A22</font></i></a>, <a class="txt" href="rlsub_2.html#E8:47_5_1_1_1_2_4_1"><i><font color="Green" title="E62">A60</font></i></a></i>;<br/>
<b>hence </b><a NAME="E15:47_5_1_1_1_2_4_1"/>
<font color="Maroon" title="c14">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a>  <a href="rlsub_1.html#K1" title="RLSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E11:47_5_1_1_1_2_4_1"><i><font color="Green" title="E64">A62</font></i></a>, <a class="txt" href="rlsub_2.html#E13:47_5_1_1_1_2_4_1"><i><font color="Green" title="E66">A64</font></i></a>, <a class="ref" href="rlsub_2.html#T7" target="_self" title="RLSUB_2:th.7">Th7</a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:47_5_1_1_1_2_4"/>
<font color="Maroon" title="c14">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a>  <a href="rlsub_1.html#K1" title="RLSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font>
 ;<br/><a NAME="E3:47_5_1_1_1_2_4"/><b>then </b>
<font color="Maroon" title="c14">u</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 <span class="p1">(<span class="default"><a href="rlsub_1.html#K1" title="RLSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/><a NAME="E4:47_5_1_1_1_2_4"/><b>then </b>
<font color="Maroon" title="c14">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="rlsub_1.html#D3" title="RLSUB_1:def.3">RLSUB_1:def 3</a></i>;<br/><a NAME="E5:47_5_1_1_1_2_4"/><b>then </b>
<font color="Maroon" title="c14">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:47_5_1_1_1_2_4"/>
<font color="Maroon" title="c14">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">W</font> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <font color="Maroon" title="c13">E</font>
 <b>by </b><i><a class="ref" href="rlsub_1.html#T25" title="RLSUB_1:th.25">RLSUB_1:25</a></i>;<br/></div><b>end;</b></div><a NAME="E21:47_5_1_1_1_2"/><b>then </b><i><font color="Green" title="E59">A65</font></i>: 
<font color="Maroon" title="c2">W</font> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <font color="Maroon" title="c13">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlsub_1.html#K1" title="RLSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="ref" href="rlsub_1.html#T39" title="RLSUB_1:th.39">RLSUB_1:39</a></i>;<br/><a NAME="E22:47_5_1_1_1_2"/>
<font color="Maroon" title="c13">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlsub_2.html#K3" title="RLSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="ref" href="rlsub_2.html#D3" target="_self" title="RLSUB_2:def.3">Def3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c14">y'</font> = <font color="Maroon" title="c13">E</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font> <b>by </b><i><a class="txt" href="rlsub_2.html#E2:47"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="rlsub_2.html#E21:47_5_1_1_1_2"><i><font color="Green" title="E59">A65</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c15">y</font> = <font color="Maroon" title="c14">y'</font>;<br/><b>let </b><font color="Maroon" title="c16">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font>;<br/><b>assume </b><a NAME="E24:47_5_1_1_1_2"/><i><font color="Green" title="E60">A66</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c17">W1</font> being  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">strict</a>  <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E26:47_5_1_1_1_2"/><i><font color="Green" title="E61">A67</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">W1</font>
 <b>and </b><br/><a NAME="E27:47_5_1_1_1_2"/><i><font color="Green" title="E62">A68</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</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="c17">W1</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E6:47_5_1_1_1_2"><i><font color="Green" title="E52">A23</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_5_1_1_1_2_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">u</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>;<br/><b>assume </b><a NAME="E1:47_5_1_1_1_2_5"/>
<font color="Maroon" title="c18">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c17">W1</font>
 ;<br/><a NAME="E2:47_5_1_1_1_2_5"/><b>then </b><i><font color="Green" title="E63">A69</font></i>: 
<font color="Maroon" title="c18">u</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="c17">W1</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/><a NAME="E3:47_5_1_1_1_2_5"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c17">W1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">f</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E5:47_5_1_1_1_2"><i><font color="Green" title="E51">A22</font></i></a>, <a class="txt" href="rlsub_2.html#E24:47_5_1_1_1_2"><i><font color="Green" title="E60">A66</font></i></a>, <a class="txt" href="rlsub_2.html#E27:47_5_1_1_1_2"><i><font color="Green" title="E62">A68</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:47_5_1_1_1_2_5"/><b>then </b>
<font color="Maroon" title="c18">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E2:47_5_1_1_1_2_5"><i><font color="Green" title="E63">A69</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E5:47_5_1_1_1_2_5"/>
<font color="Maroon" title="c18">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c13">E</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E19:47_5_1_1_1_2"><i><font color="Green" title="E58">A56</font></i></a>, <a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/></div><b>end;</b></div><a NAME="E29:47_5_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c17">W1</font> is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">strict</a>  <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c13">E</font>
 <b>by </b><i><a class="ref" href="rlsub_1.html#T37" title="RLSUB_1:th.37">RLSUB_1:37</a></i>;<br/><b>hence </b><a NAME="E30:47_5_1_1_1_2"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c16">x</font>,<font color="Maroon" title="c15">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E7:47"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="rlsub_2.html#E26:47_5_1_1_1_2"><i><font color="Green" title="E61">A67</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:47_5_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font> st <br/>for <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font> holds <br/><span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:47_5"/>
for <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font> &amp; ( for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font> &amp; <font color="Olive" title="b3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font> &amp;  not <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b3">y</font>,<font color="Olive" title="b2">x</font>] ) holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font> st <br/>for <font color="Olive" title="b3">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b3">x</font>,<font color="Olive" title="b2">y</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c6">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font><b> such that </b><br/><a NAME="E13:47"/><i><font color="Green" title="E46">A70</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font>  st <font color="Maroon" title="c6">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">y</font> holds <br/> not <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c6">x</font>,<font color="Olive" title="b1">y</font>]
 <b>from </b><i><a class="ref" href="orders_1.html#S1" title="ORDERS_1:sch.1">ORDERS_1:sch 1</a>(<a class="txt" href="rlsub_2.html#E8:47"><i><font color="Green" title="E42">A4</font></i></a>, <a class="txt" href="rlsub_2.html#E9:47"><i><font color="Green" title="E43">A5</font></i></a>, <a class="txt" href="rlsub_2.html#E10:47"><i><font color="Green" title="E44">A9</font></i></a>, <a class="txt" href="rlsub_2.html#E11:47"><i><font color="Green" title="E45">A14</font></i></a>);<br/></i>
<b>consider </b><font color="Maroon" title="c7">L</font> being  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">strict</a>  <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E15:47"/><i><font color="Green" title="E47">A71</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">L</font>
 <b>and </b><br/><a NAME="E16:47"/><i><font color="Green" title="E48">A72</font></i>: 
<font color="Maroon" title="c2">W</font> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <font color="Maroon" title="c7">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlsub_1.html#K1" title="RLSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E2:47"><i><font color="Green" title="E39">A1</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">L</font>
;<br/>

<b>thus </b><a NAME="E17:47"/>
 <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">Zero</a> of <font color="Maroon" title="c1">V</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">V</font>,the <a href="rlvect_1.html#U2" title="RLVECT_1:sel.2">Mult</a> of <font color="Maroon" title="c1">V</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">L</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c2">W</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlsub_2.html#D4" target="_self" title="RLSUB_2:def.4">RLSUB_2:def 4</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:47_6"/>
not  <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">Zero</a> of <font color="Maroon" title="c1">V</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">V</font>,the <a href="rlvect_1.html#U2" title="RLVECT_1:sel.2">Mult</a> of <font color="Maroon" title="c1">V</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">L</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c2">W</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">v</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E3:47_6"/><i><font color="Green" title="E49">A73</font></i>: 
for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  holds <br/>(  not <font color="Olive" title="b1">v1</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c7">L</font> or  not <font color="Olive" title="b2">v2</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">W</font> or <font color="Maroon" title="c8">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">v1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Olive" title="b2">v2</font> )
 <b>by </b><i><a class="txt" href="rlsub_2.html#E56"><i><font color="Green" title="E36">Lm13</font></i></a></i>;<br/>
<a NAME="E4:47_6"/><i><font color="Green" title="E50">A74</font></i>: 
not <font color="Maroon" title="c8">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c7">L</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c2">W</font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E3:47_6"><i><font color="Green" title="E49">A73</font></i></a>, <a class="ref" href="rlsub_2.html#T5" target="_self" title="RLSUB_2:th.5">Th5</a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">A</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c8">v</font></span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : verum </span>}</span> ;<br/>
<a NAME="E5:47_6"/><i><font color="Green" title="E51">A75</font></i>: 
1 <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c8">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c8">v</font></span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : verum </span>}</span> 
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:47_6_1"/>
<font color="Maroon" title="c10">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"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c8">v</font></span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : verum </span>}</span> 
 ;<br/><a NAME="E2:47_6_1"/><b>then </b>
 ex <font color="Olive" title="b1">a</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c8">v</font>
 ;<br/><b>hence </b><a NAME="E3:47_6_1"/>
<font color="Maroon" title="c10">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="c1">V</font>
 ;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c10">A</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c8">v</font></span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : verum </span>}</span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E8:47_6"/>
<font color="Maroon" title="c10">A</font> is <a href="rlsub_1.html#V1" title="RLSUB_1:attr.1">lineary-closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_6_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:47_6_2"/>
for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font> &amp; <font color="Olive" title="b2">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font> holds <br/><font color="Olive" title="b1">v1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Olive" title="b2">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlsub_1.html#D1" title="RLSUB_1:def.1">RLSUB_1:def 1</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_6_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">v1</font>, <font color="Maroon" title="c12">v2</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>;<br/>

<b>assume </b><a NAME="E1:47_6_2_1"/>
<font color="Maroon" title="c11">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c13">a1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:47_6_2_1"/><i><font color="Green" title="E52">A76</font></i>: 
<font color="Maroon" title="c11">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c8">v</font>
 ;<br/>
<b>assume </b><a NAME="E4:47_6_2_1"/>
<font color="Maroon" title="c12">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c14">a2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E6:47_6_2_1"/><i><font color="Green" title="E53">A77</font></i>: 
<font color="Maroon" title="c12">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c8">v</font>
 ;<br/>
<a NAME="E7:47_6_2_1"/>
<font color="Maroon" title="c11">v1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c12">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">a1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c14">a2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c8">v</font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E3:47_6_2_1"><i><font color="Green" title="E52">A76</font></i></a>, <a class="txt" href="rlsub_2.html#E6:47_6_2_1"><i><font color="Green" title="E53">A77</font></i></a>, <a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>;<br/>
<b>hence </b><a NAME="E8:47_6_2_1"/>
<font color="Maroon" title="c11">v1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c12">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font>
 ;<br/>


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

<b>let </b><font color="Maroon" title="c11">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c12">v1</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>;<br/>



<b>assume </b><a NAME="E2:47_6_2"/>
<font color="Maroon" title="c12">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c13">a1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:47_6_2"/><i><font color="Green" title="E52">A78</font></i>: 
<font color="Maroon" title="c12">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c8">v</font>
 ;<br/>
<a NAME="E5:47_6_2"/>
<font color="Maroon" title="c11">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c12">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c13">a1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c8">v</font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E4:47_6_2"><i><font color="Green" title="E52">A78</font></i></a>, <a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>;<br/>
<b>hence </b><a NAME="E6:47_6_2"/>
<font color="Maroon" title="c11">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c12">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font>
 ;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c11">Z</font> being  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">strict</a>  <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E10:47_6"/><i><font color="Green" title="E52">A79</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">A</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E5:47_6"><i><font color="Green" title="E51">A75</font></i></a>, <a class="ref" href="rlsub_1.html#T43" title="RLSUB_1:th.43">RLSUB_1:43</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_6_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">u</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>;<br/><b>thus </b><a NAME="E1:47_6_3"/>
( <font color="Maroon" title="c12">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c11">Z</font> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c7">L</font></span>)</span> implies <font color="Maroon" title="c12">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a>  <a href="rlsub_1.html#K1" title="RLSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_6_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:47_6_3_1"/><i><font color="Green" title="E53">A80</font></i>: 
<font color="Maroon" title="c12">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c11">Z</font> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c7">L</font></span>)</span>
 ;<br/>

<a NAME="E2:47_6_3_1"/><b>then </b>
<font color="Maroon" title="c12">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c11">Z</font>
 
<b>by </b><i><a class="ref" href="rlsub_2.html#T7" target="_self" title="RLSUB_2:th.7">Th7</a></i>;<br/>
<a NAME="E3:47_6_3_1"/><b>then </b>
<font color="Maroon" title="c12">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E10:47_6"><i><font color="Green" title="E52">A79</font></i></a>, <a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">a</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E5:47_6_3_1"/><i><font color="Green" title="E54">A81</font></i>: 
<font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c8">v</font>
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_6_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_6_3_1_1"/><i><font color="Green" title="E55">A82</font></i>: 
<font color="Maroon" title="c13">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:47_6_3_1_1"/>
<font color="Maroon" title="c12">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">W</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c7">L</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E1:47_6_3_1"><i><font color="Green" title="E53">A80</font></i></a>, <a class="ref" href="rlsub_2.html#T7" target="_self" title="RLSUB_2:th.7">Th7</a></i>;<br/><a NAME="E3:47_6_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c8">v</font></span>)</span> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">W</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c7">L</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E5:47_6_3_1"><i><font color="Green" title="E54">A81</font></i></a>, <a class="ref" href="rlsub_1.html#T29" title="RLSUB_1:th.29">RLSUB_1:29</a></i>;<br/><a NAME="E4:47_6_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c13">a</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c8">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">W</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c7">L</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>;<br/><a NAME="E5:47_6_3_1_1"/><b>then </b>
1 <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c8">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">W</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c7">L</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E1:47_6_3_1_1"><i><font color="Green" title="E55">A82</font></i></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></i>;<br/><a NAME="E6:47_6_3_1_1"/><b>then </b>
1 <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c8">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c7">L</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c2">W</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E10"><i><font color="Green" title="E6">Lm1</font></i></a></i>;<br/><b>hence </b><a NAME="E7:47_6_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="rlsub_2.html#E4:47_6"><i><font color="Green" title="E50">A74</font></i></a>, <a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:47_6_3_1"/><b>then </b>
<font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">V</font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E5:47_6_3_1"><i><font color="Green" title="E54">A81</font></i></a>, <a class="ref" href="rlvect_1.html#T23" title="RLVECT_1:th.23">RLVECT_1:23</a></i>;<br/>
<b>hence </b><a NAME="E8:47_6_3_1"/>
<font color="Maroon" title="c12">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a>  <a href="rlsub_1.html#K1" title="RLSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="ref" href="rlsub_1.html#T25" title="RLSUB_1:th.25">RLSUB_1:25</a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:47_6_3"/>
<font color="Maroon" title="c12">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a>  <a href="rlsub_1.html#K1" title="RLSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font>
 ;<br/><a NAME="E3:47_6_3"/><b>then </b>
<font color="Maroon" title="c12">u</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 <span class="p1">(<span class="default"><a href="rlsub_1.html#K1" title="RLSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/><a NAME="E4:47_6_3"/><b>then </b>
<font color="Maroon" title="c12">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="rlsub_1.html#D3" title="RLSUB_1:def.3">RLSUB_1:def 3</a></i>;<br/><a NAME="E5:47_6_3"/><b>then </b>
<font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:47_6_3"/>
<font color="Maroon" title="c12">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c11">Z</font> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c7">L</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlsub_1.html#T25" title="RLSUB_1:th.25">RLSUB_1:25</a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:47_6"/><b>then </b><i><font color="Green" title="E53">A83</font></i>: 
<font color="Maroon" title="c11">Z</font> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlsub_1.html#K1" title="RLSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font>
 
<b>by </b><i><a class="ref" href="rlsub_1.html#T39" title="RLSUB_1:th.39">RLSUB_1:39</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_6_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">u</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>;<br/><b>thus </b><a NAME="E1:47_6_4"/>
( <font color="Maroon" title="c12">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">Z</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <font color="Maroon" title="c2">W</font> implies <font color="Maroon" title="c12">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a>  <a href="rlsub_1.html#K1" title="RLSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_6_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:47_6_4_1"/><i><font color="Green" title="E54">A84</font></i>: 
<font color="Maroon" title="c12">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">Z</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <font color="Maroon" title="c2">W</font>
 ;<br/>

<a NAME="E2:47_6_4_1"/><b>then </b>
<font color="Maroon" title="c12">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c11">Z</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c7">L</font>
 
<b>by </b><i><a class="ref" href="rlsub_2.html#T7" target="_self" title="RLSUB_2:th.7">Th7</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">v1</font>, <font color="Maroon" title="c14">v2</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E4:47_6_4_1"/><i><font color="Green" title="E55">A85</font></i>: 
<font color="Maroon" title="c13">v1</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c11">Z</font>
 <b>and </b><br/><a NAME="E5:47_6_4_1"/><i><font color="Green" title="E56">A86</font></i>: 
<font color="Maroon" title="c14">v2</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c7">L</font>
 <b>and </b><br/><a NAME="E6:47_6_4_1"/><i><font color="Green" title="E57">A87</font></i>: 
<font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">v1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c14">v2</font>
 <b>by </b><i><a class="ref" href="rlsub_2.html#T5" target="_self" title="RLSUB_2:th.5">Th5</a></i>;<br/>
<a NAME="E7:47_6_4_1"/><i><font color="Green" title="E58">A88</font></i>: 
<font color="Maroon" title="c13">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">u</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c14">v2</font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E6:47_6_4_1"><i><font color="Green" title="E57">A87</font></i></a>, <a class="txt" href="rlsub_2.html#E57"><i><font color="Green" title="E37">Lm14</font></i></a></i>;<br/>
<a NAME="E8:47_6_4_1"/><i><font color="Green" title="E59">A89</font></i>: 
<font color="Maroon" title="c12">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">W</font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E1:47_6_4_1"><i><font color="Green" title="E54">A84</font></i></a>, <a class="ref" href="rlsub_2.html#T7" target="_self" title="RLSUB_2:th.7">Th7</a></i>;<br/>
<a NAME="E9:47_6_4_1"/><b>then </b>
( <font color="Maroon" title="c12">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">W</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c7">L</font> &amp; <font color="Maroon" title="c14">v2</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">W</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c7">L</font> )
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E5:47_6_4_1"><i><font color="Green" title="E56">A86</font></i></a>, <a class="ref" href="rlsub_2.html#T6" target="_self" title="RLSUB_2:th.6">Th6</a></i>;<br/>
<a NAME="E10:47_6_4_1"/><b>then </b>
<font color="Maroon" title="c13">v1</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">W</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c7">L</font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E7:47_6_4_1"><i><font color="Green" title="E58">A88</font></i></a>, <a class="ref" href="rlsub_1.html#T31" title="RLSUB_1:th.31">RLSUB_1:31</a></i>;<br/>
<a NAME="E11:47_6_4_1"/><b>then </b>
<font color="Maroon" title="c13">v1</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c11">Z</font> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c7">L</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E4:47_6_4_1"><i><font color="Green" title="E55">A85</font></i></a>, <a class="ref" href="rlsub_2.html#T7" target="_self" title="RLSUB_2:th.7">Th7</a></i>;<br/>
<a NAME="E12:47_6_4_1"/><b>then </b>
<font color="Maroon" title="c13">v1</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 <span class="p1">(<span class="default"><a href="rlsub_1.html#K1" title="RLSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E12:47_6"><i><font color="Green" title="E53">A83</font></i></a>, <a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E13:47_6_4_1"/><b>then </b>
<font color="Maroon" title="c13">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="rlsub_1.html#D3" title="RLSUB_1:def.3">RLSUB_1:def 3</a></i>;<br/>
<a NAME="E14:47_6_4_1"/><b>then </b>
<font color="Maroon" title="c13">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">V</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E15:47_6_4_1"/><b>then </b>
<font color="Maroon" title="c14">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">u</font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E6:47_6_4_1"><i><font color="Green" title="E57">A87</font></i></a>, <a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>;<br/>
<b>hence </b><a NAME="E16:47_6_4_1"/>
<font color="Maroon" title="c12">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a>  <a href="rlsub_1.html#K1" title="RLSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E16:47"><i><font color="Green" title="E48">A72</font></i></a>, <a class="txt" href="rlsub_2.html#E5:47_6_4_1"><i><font color="Green" title="E56">A86</font></i></a>, <a class="txt" href="rlsub_2.html#E8:47_6_4_1"><i><font color="Green" title="E59">A89</font></i></a>, <a class="ref" href="rlsub_2.html#T7" target="_self" title="RLSUB_2:th.7">Th7</a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:47_6_4"/>
<font color="Maroon" title="c12">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a>  <a href="rlsub_1.html#K1" title="RLSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font>
 ;<br/><a NAME="E3:47_6_4"/><b>then </b>
<font color="Maroon" title="c12">u</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 <span class="p1">(<span class="default"><a href="rlsub_1.html#K1" title="RLSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/><a NAME="E4:47_6_4"/><b>then </b>
<font color="Maroon" title="c12">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="rlsub_1.html#D3" title="RLSUB_1:def.3">RLSUB_1:def 3</a></i>;<br/><a NAME="E5:47_6_4"/><b>then </b>
<font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:47_6_4"/>
<font color="Maroon" title="c12">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">Z</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <font color="Maroon" title="c2">W</font>
 <b>by </b><i><a class="ref" href="rlsub_1.html#T25" title="RLSUB_1:th.25">RLSUB_1:25</a></i>;<br/></div><b>end;</b></div>
<a NAME="E14:47_6"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">Z</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <font color="Maroon" title="c2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlsub_1.html#K1" title="RLSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font>
 
<b>by </b><i><a class="ref" href="rlsub_1.html#T39" title="RLSUB_1:th.39">RLSUB_1:39</a></i>;<br/>
<a NAME="E15:47_6"/><b>then </b>
( <font color="Maroon" title="c2">W</font> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">Z</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlsub_1.html#K1" title="RLSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font> &amp; <font color="Maroon" title="c11">Z</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c7">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlsub_2.html#K3" title="RLSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font> )
 
<b>by </b><i><a class="ref" href="rlsub_2.html#D3" target="_self" title="RLSUB_2:def.3">Def3</a>, <a class="ref" href="rlsub_2.html#T18" target="_self" title="RLSUB_2:th.18">Th18</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">x1</font> = <font color="Maroon" title="c11">Z</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c7">L</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font> <b>by </b><i><a class="txt" href="rlsub_2.html#E2:47"><i><font color="Green" title="E39">A1</font></i></a></i>;<br/>
<a NAME="E17:47_6"/>
( <font color="Maroon" title="c8">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c8">v</font> &amp; <font color="Maroon" title="c8">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">V</font></span>)</span> &amp;  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">V</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">W</font> &amp;  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">V</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c7">L</font> )
 
<b>by </b><i><a class="ref" href="rlsub_1.html#T25" title="RLSUB_1:th.25">RLSUB_1:25</a>, <a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>;<br/>
<a NAME="E18:47_6"/><b>then </b><i><font color="Green" title="E54">A90</font></i>: 
not <font color="Maroon" title="c8">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c7">L</font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E3:47_6"><i><font color="Green" title="E49">A73</font></i></a></i>;<br/>
<a NAME="E19:47_6"/>
<font color="Maroon" title="c8">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E5:47_6"><i><font color="Green" title="E51">A75</font></i></a>, <a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>;<br/>
<a NAME="E20:47_6"/><b>then </b>
<font color="Maroon" title="c8">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c11">Z</font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E10:47_6"><i><font color="Green" title="E52">A79</font></i></a>, <a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E21:47_6"/><b>then </b><i><font color="Green" title="E55">A91</font></i>: 
<font color="Maroon" title="c11">Z</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c7">L</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">L</font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E18:47_6"><i><font color="Green" title="E54">A90</font></i></a>, <a class="ref" href="rlsub_2.html#T6" target="_self" title="RLSUB_2:th.6">Th6</a></i>;<br/>
<a NAME="E22:47_6"/>
<font color="Maroon" title="c7">L</font> is    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c11">Z</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c7">L</font>
 
<b>by </b><i><a class="ref" href="rlsub_2.html#T11" target="_self" title="RLSUB_2:th.11">Th11</a></i>;<br/>
<a NAME="E23:47_6"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c12">x1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E7:47"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="rlsub_2.html#E15:47"><i><font color="Green" title="E47">A71</font></i></a></i>;<br/>
<b>hence </b><a NAME="E24:47_6"/>
contradiction
 <b>by </b><i><a class="txt" href="rlsub_2.html#E13:47"><i><font color="Green" title="E46">A70</font></i></a>, <a class="txt" href="rlsub_2.html#E15:47"><i><font color="Green" title="E47">A71</font></i></a>, <a class="txt" href="rlsub_2.html#E21:47_6"><i><font color="Green" title="E55">A91</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E18:47"/>
<font color="Maroon" title="c7">L</font> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <font color="Maroon" title="c2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlsub_1.html#K1" title="RLSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E16:47"><i><font color="Green" title="E48">A72</font></i></a>, <a class="ref" href="rlsub_2.html#T18" target="_self" title="RLSUB_2:th.18">Th18</a></i>;<br/>


</div>
