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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">W1</font> being <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">W1</font> &amp; <font color="Maroon">W</font> <a href="vectsp_5.html#K2" target="_self">/\</a> <font color="Olive">W1</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font> );<br/>
<b>consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:53_1_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_5.html#K3" target="_self">Subspaces</a> <font color="Maroon">V</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E3:53_1_1"/>
( <font color="Maroon">W</font> <a href="vectsp_5.html#K2" target="_self">/\</a> <span class="p1">(<span class="default"><a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font> &amp;  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_5.html#K3" target="_self">Subspaces</a> <font color="Maroon">V</font> )
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i/>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">W1</font>, <font color="Olive">W2</font> being <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">W1</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">W2</font> &amp; <font color="Olive">W1</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Olive">W2</font> );<br/>
<b>consider </b><font color="Maroon">R</font> being   <a href="relset_1.html#NM3">Relation</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E6:53_1_1"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> iff <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] )
 <b>from </b><i><a class="ref" href="relset_1.html#S2">RELSET_1:sch 2</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>;<br/>
<div><i><font color="Green">E33</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/><b>consider </b><font color="Maroon">W1</font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E2:53_1_1_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">W</font> <a href="vectsp_5.html#K2" target="_self">/\</a> <font color="Maroon">W1</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font> )
 <b>by </b><i/>;<br/><a NAME="E3:53_1_1_1"/>
<font color="Maroon">W1</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">W1</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T32">VECTSP_4:32</a></i>;<br/><b>hence </b><a NAME="E4:53_1_1_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">x</font>]
 <b>by </b><i><a class="ref" href="vectsp_5.html#D2" target="_self">Def2</a>, </i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E35</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:53_1_1_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>]
 <b>and </b><br/><a NAME="E2:53_1_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">y</font>,<font color="Maroon">x</font>]
 ;<br/><a NAME="E3:53_1_1_2"/><i><font color="Green">E54</font></i>: 
 ex <font color="Olive">W1</font>, <font color="Olive">W2</font> being <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">W1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">W2</font> &amp; <font color="Olive">W1</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Olive">W2</font> )
 <b>by </b><i><a class="ref" href="vectsp_5.html#D2" target="_self">Def2</a>, </i>;<br/><a NAME="E4:53_1_1_2"/>
 ex <font color="Olive">W3</font>, <font color="Olive">W4</font> being <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">W3</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">W4</font> &amp; <font color="Olive">W3</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Olive">W4</font> )
 <b>by </b><i><a class="ref" href="vectsp_5.html#D2" target="_self">Def2</a>, </i>;<br/><b>hence </b><a NAME="E5:53_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#T5" target="_self">Th5</a>, <a class="ref" href="vectsp_4.html#T33">VECTSP_4:33</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E55</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>, <font color="Maroon">z</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:53_1_1_3"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>]
 <b>and </b><br/><a NAME="E2:53_1_1_3"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">y</font>,<font color="Maroon">z</font>]
 ;<br/><b>consider </b><font color="Maroon">W1</font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font>, <font color="Maroon">W2</font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E4:53_1_1_3"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> &amp; <font color="Maroon">W1</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">W2</font> )
 <b>by </b><i><a class="ref" href="vectsp_5.html#D2" target="_self">Def2</a>, </i>;<br/><b>consider </b><font color="Maroon">W3</font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font>, <font color="Maroon">W4</font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E6:53_1_1_3"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W3</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W4</font> &amp; <font color="Maroon">W3</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">W4</font> )
 <b>by </b><i><a class="ref" href="vectsp_5.html#D2" target="_self">Def2</a>, <a class="ref" href="vectsp_5.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E7:53_1_1_3"/>
<font color="Maroon">W1</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">W4</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#T6" target="_self">Th6</a>, <a class="ref" href="vectsp_5.html#T7" target="_self">Th7</a>, <a class="ref" href="vectsp_4.html#T34">VECTSP_4:34</a></i>;<br/><b>hence </b><a NAME="E8:53_1_1_3"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">z</font>]
 <b>by </b><i><a class="ref" href="vectsp_5.html#D2" target="_self">Def2</a>, <a class="ref" href="vectsp_5.html#T6" target="_self">Th6</a>, <a class="ref" href="vectsp_5.html#T7" target="_self">Th7</a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:53_1_1"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; ( for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp;  not <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">y</font>,<font color="Olive">x</font>] ) holds <br/> ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> st <br/>for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:53_1_1_4"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E2:53_1_1_4"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; not <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> holds <br/><span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">x</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_1_1_4_1_1"/>
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:53_1_1_4_1_1_1"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/><b>take </b><font color="Maroon">y'</font> = <font color="Maroon">y</font>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E3:53_1_1_4_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/><b>hence </b><a NAME="E4:53_1_1_4_1_1_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y'</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#T7" target="_self">Th7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:53_1_1_4_1_1_2"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">W1</font> being <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">W1</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">W1</font> );<br/><a NAME="E2:53_1_1_4_1_1_2"/><i><font color="Green">E111</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">x</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">x</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 ;<br/><a NAME="E3:53_1_1_4_1_1_2"/><i><font color="Green">E112</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_4_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:53_1_1_4_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<b>then consider </b><font color="Maroon">W1</font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:53_1_1_4_1_1_2_1"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font>
 <b>and </b><br/><a NAME="E4:53_1_1_4_1_1_2_1"/>
<font color="Maroon">W</font> <a href="vectsp_5.html#K2" target="_self">/\</a> <font color="Maroon">W1</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="vectsp_5.html#T5" target="_self">Th5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> as    <a href="hidden.html#M1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

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

<b>thus </b><a NAME="E6:53_1_1_4_1_1_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> )
 <b>by </b><i><a class="ref" href="vectsp_5.html#T10" target="_self">Th10</a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:53_1_1_4_1_1_2"/><i><font color="Green">E114</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E6:53_1_1_4_1_1_2"/><i><font color="Green">E115</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(<a class="txt" href="vectsp_5.html#E11"><i><font color="Green">Lemma48</font></i></a>, <a class="txt" href="vectsp_5.html#E12"><i><font color="Green">Lemma49</font></i></a>);<br/></i><b>set </b><font color="Maroon">Z</font> =  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_4_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:53_1_1_4_1_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">Y'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:53_1_1_4_1_1_2_2"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font>
 <b>and </b><br/><a NAME="E4:53_1_1_4_1_1_2_2"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">Y'</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:53_1_1_4_1_1_2_2"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E7:53_1_1_4_1_1_2_2"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y'</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon">W1</font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E9:53_1_1_4_1_1_2_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font>
 <b>and </b><br/><a NAME="E10:53_1_1_4_1_1_2_2"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#T10" target="_self">Th10</a>, , </i>;<br/><a NAME="E11:53_1_1_4_1_1_2_2"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> )
 <b>by </b><i>, <a class="txt" href="vectsp_5.html#E9:53_1_1"><i><font color="Green">E55</font></i></a>, , <a class="ref" href="vectsp_4.html#D2">VECTSP_4:def 2</a></i>;<br/><b>hence </b><a NAME="E12:53_1_1_4_1_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 ;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">Z</font> =  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E9:53_1_1_4_1_1_2"/><i><font color="Green">E122</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="vectsp_5.html#T7" target="_self">Th7</a>, <a class="ref" href="vectsp_5.html#T10" target="_self">Th10</a>, <a class="ref" href="relat_1.html#T65">RELAT_1:65</a></i>;<br/><b>consider </b><font color="Maroon">z</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>;<br/><b>consider </b><font color="Maroon">z1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:53_1_1_4_1_1_2"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">z1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E13:53_1_1_4_1_1_2"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon">W3</font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E15:53_1_1_4_1_1_2"/><i><font color="Green">E126</font></i>: 
( <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W3</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W3</font> )
 <b>by </b><i><a class="ref" href="vectsp_5.html#T10" target="_self">Th10</a>, , </i>;<br/><a NAME="E16:53_1_1_4_1_1_2"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, , <a class="txt" href="vectsp_5.html#E9:53_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="orders_1.html#T91">ORDERS_1:91</a></i>;<br/><a NAME="E17:53_1_1_4_1_1_2"/>
<font color="Maroon">Z</font> is <a href="vectsp_4.html#V1">lineary-closed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_4_1_1_2_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:53_1_1_4_1_1_2_3"/>
for <font color="Olive">v1</font>, <font color="Olive">v2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>  st <font color="Olive">v1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Olive">v2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> holds <br/><font color="Olive">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">v2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_4.html#D1">VECTSP_4:def 1</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_4_1_1_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">v2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:53_1_1_4_1_1_2_3_1"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E2:53_1_1_4_1_1_2_3_1"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/>

<b>consider </b><font color="Maroon">Y1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:53_1_1_4_1_1_2_3_1"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y1</font>
 <b>and </b><br/><a NAME="E5:53_1_1_4_1_1_2_3_1"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">Y1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">Y2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:53_1_1_4_1_1_2_3_1"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y2</font>
 <b>and </b><br/><a NAME="E8:53_1_1_4_1_1_2_3_1"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">Y2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:53_1_1_4_1_1_2_3_1"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E11:53_1_1_4_1_1_2_3_1"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y1</font>
 <b>by </b><i><a class="txt" href="vectsp_5.html#E10:53_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">y2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E13:53_1_1_4_1_1_2_3_1"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">y2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E14:53_1_1_4_1_1_2_3_1"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y2</font>
 <b>by </b><i><a class="txt" href="vectsp_5.html#E2:53_1_1_4"><i><font color="Green">E62</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">W1</font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E16:53_1_1_4_1_1_2_3_1"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font>
 <b>and </b><br/><a NAME="E17:53_1_1_4_1_1_2_3_1"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#T10" target="_self">Th10</a>, , <a class="ref" href="vectsp_5.html#T11" target="_self">Th11</a></i>;<br/>
<b>consider </b><font color="Maroon">W2</font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E19:53_1_1_4_1_1_2_3_1"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font>
 <b>and </b><br/><a NAME="E20:53_1_1_4_1_1_2_3_1"/><i><font color="Green">E143</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#T10" target="_self">Th10</a>, , <a class="ref" href="vectsp_5.html#T13" target="_self">Th13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y1</font>, <font color="Maroon">y2</font> = <font color="Maroon">y2</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="vectsp_5.html#T5" target="_self">Th5</a>, <a class="ref" href="vectsp_5.html#T10" target="_self">Th10</a>, <a class="ref" href="vectsp_5.html#T11" target="_self">Th11</a>, <a class="ref" href="vectsp_5.html#T13" target="_self">Th13</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_4_1_1_2_3_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_4_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="E1:53_1_1_4_1_1_2_3_1_1_1"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> or <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y2</font>,<font color="Maroon">y1</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 </span><b>by </b><i><a class="ref" href="vectsp_5.html#T6" target="_self">Th6</a>, <a class="ref" href="vectsp_5.html#T10" target="_self">Th10</a>, <a class="ref" href="vectsp_5.html#T11" target="_self">Th11</a>, <a class="ref" href="vectsp_5.html#T13" target="_self">Th13</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_4_1_1_2_3_1_1_1_1"><b>suppose </b></a><a NAME="E1:53_1_1_4_1_1_2_3_1_1_1_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 ;<br/><div class="add"><a NAME="E2:53_1_1_4_1_1_2_3_1_1_1_1"/><b>then </b>
 ex <font color="Olive">W3</font>, <font color="Olive">W4</font> being <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">W3</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <font color="Olive">W4</font> &amp; <font color="Olive">W3</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Olive">W4</font> )
 <b>by </b><i><a class="ref" href="vectsp_5.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E3:53_1_1_4_1_1_2_3_1_1_1_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font>
 <b>by </b><i>, , <a class="ref" href="vectsp_4.html#D2">VECTSP_4:def 2</a></i>;<br/><a NAME="E4:53_1_1_4_1_1_2_3_1_1_1_1"/><b>then </b>
( <font color="Maroon">v1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> &amp; <font color="Maroon">v2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> )
 <b>by </b><i>, <a class="txt" href="vectsp_5.html#E1:53_1_1_4"><i><font color="Green">E61</font></i></a>, <a class="ref" href="vectsp_5.html#T12" target="_self">Th12</a>, <a class="txt" href="vectsp_5.html#E19"><i><font color="Green">Lemma66</font></i></a>, <a class="txt" href="vectsp_5.html#E20"><i><font color="Green">Lemma68</font></i></a>, , <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E5:53_1_1_4_1_1_2_3_1_1_1_1"/><b>then </b>
<font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T28">VECTSP_4:28</a></i>;<br/><a NAME="E6:53_1_1_4_1_1_2_3_1_1_1_1"/><b>then </b><i><font color="Green">E144</font></i>: 
<font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E7:53_1_1_4_1_1_2_3_1_1_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#T13" target="_self">Th13</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E8:53_1_1_4_1_1_2_3_1_1_1_1"/>
<font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i>, <a class="txt" href="vectsp_5.html#E21"><i><font color="Green">Lemma71</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_4_1_1_2_3_1_1_1_2"><b>suppose </b></a><a NAME="E1:53_1_1_4_1_1_2_3_1_1_1_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y2</font>,<font color="Maroon">y1</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 ;<br/><div class="add"><a NAME="E2:53_1_1_4_1_1_2_3_1_1_1_2"/><b>then </b>
 ex <font color="Olive">W3</font>, <font color="Olive">W4</font> being <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <font color="Olive">W3</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">W4</font> &amp; <font color="Olive">W3</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Olive">W4</font> )
 <b>by </b><i><a class="ref" href="vectsp_5.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E3:53_1_1_4_1_1_2_3_1_1_1_2"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font>
 <b>by </b><i>, , <a class="ref" href="vectsp_4.html#D2">VECTSP_4:def 2</a></i>;<br/><a NAME="E4:53_1_1_4_1_1_2_3_1_1_1_2"/><b>then </b>
( <font color="Maroon">v1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">v2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> )
 <b>by </b><i>, <a class="txt" href="vectsp_5.html#E1:53_1_1_4"><i><font color="Green">E61</font></i></a>, <a class="ref" href="vectsp_5.html#T12" target="_self">Th12</a>, <a class="txt" href="vectsp_5.html#E19"><i><font color="Green">Lemma66</font></i></a>, <a class="txt" href="vectsp_5.html#E20"><i><font color="Green">Lemma68</font></i></a>, , <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E5:53_1_1_4_1_1_2_3_1_1_1_2"/><b>then </b>
<font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T28">VECTSP_4:28</a></i>;<br/><a NAME="E6:53_1_1_4_1_1_2_3_1_1_1_2"/><b>then </b><i><font color="Green">E145</font></i>: 
<font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E7:53_1_1_4_1_1_2_3_1_1_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#T11" target="_self">Th11</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E8:53_1_1_4_1_1_2_3_1_1_1_2"/>
<font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="txt" href="vectsp_5.html#E20"><i><font color="Green">Lemma68</font></i></a>, <a class="txt" href="vectsp_5.html#E21"><i><font color="Green">Lemma71</font></i></a>, <a class="ref" href="tarski.html#D4">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="E23:53_1_1_4_1_1_2_3_1"/>
<font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/>


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

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F</font>;<br/><b>let </b><font color="Maroon">v1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/>



<b>assume </b><a NAME="E2:53_1_1_4_1_1_2_3"/>
<font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/>

<b>then consider </b><font color="Maroon">Y1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:53_1_1_4_1_1_2_3"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y1</font>
 <b>and </b><br/><a NAME="E5:53_1_1_4_1_1_2_3"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">Y1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:53_1_1_4_1_1_2_3"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E8:53_1_1_4_1_1_2_3"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y1</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">W1</font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E10:53_1_1_4_1_1_2_3"/>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font>
 <b>and </b><br/><a NAME="E11:53_1_1_4_1_1_2_3"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#T10" target="_self">Th10</a>, , </i>;<br/>
<a NAME="E12:53_1_1_4_1_1_2_3"/>
<font color="Maroon">v1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font>
 
<b>by </b><i>, <a class="txt" href="vectsp_5.html#E10:53_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="vectsp_5.html#E1:53_1_1_4"><i><font color="Green">E61</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E13:53_1_1_4_1_1_2_3"/><b>then </b>
<font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T29">VECTSP_4:29</a></i>;<br/>
<a NAME="E14:53_1_1_4_1_1_2_3"/><b>then </b><i><font color="Green">E152</font></i>: 
<font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E15:53_1_1_4_1_1_2_3"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E16:53_1_1_4_1_1_2_3"/>
<font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="txt" href="vectsp_5.html#E1:53_1_1_4"><i><font color="Green">E61</font></i></a>, <a class="txt" href="vectsp_5.html#E2:53_1_1_4"><i><font color="Green">E62</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">E</font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E19:53_1_1_4_1_1_2"/><i><font color="Green">E154</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">E</font>
 <b>by </b><i>, <a class="ref" href="vectsp_4.html#T42">VECTSP_4:42</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_4_1_1_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/><b>thus </b><a NAME="E1:53_1_1_4_1_1_2_4"/>
( <font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> <a href="vectsp_5.html#K2" target="_self">/\</a> <font color="Maroon">E</font> implies <font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_4_1_1_2_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:53_1_1_4_1_1_2_4_1"/><i><font color="Green">E155</font></i>: 
<font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> <a href="vectsp_5.html#K2" target="_self">/\</a> <font color="Maroon">E</font>
 ;<br/>

<a NAME="E2:53_1_1_4_1_1_2_4_1"/><b>then </b>
<font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">E</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:53_1_1_4_1_1_2_4_1"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">Y1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:53_1_1_4_1_1_2_4_1"/><i><font color="Green">E156</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y1</font>
 <b>and </b><br/><a NAME="E6:53_1_1_4_1_1_2_4_1"/><i><font color="Green">E157</font></i>: 
<font color="Maroon">Y1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:53_1_1_4_1_1_2_4_1"/><i><font color="Green">E158</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E9:53_1_1_4_1_1_2_4_1"/><i><font color="Green">E159</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y1</font>
 <b>by </b><i><a class="txt" href="vectsp_5.html#E10:53_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">W1</font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E11:53_1_1_4_1_1_2_4_1"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font>
 <b>and </b><br/><a NAME="E12:53_1_1_4_1_1_2_4_1"/><i><font color="Green">E161</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#T10" target="_self">Th10</a>, , <a class="txt" href="vectsp_5.html#E1:53_1_1_4"><i><font color="Green">E61</font></i></a></i>;<br/>
<a NAME="E13:53_1_1_4_1_1_2_4_1"/><i><font color="Green">E162</font></i>: 
( <font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> )
 
<b>by </b><i>, , <a class="txt" href="vectsp_5.html#E2:53_1_1_4"><i><font color="Green">E62</font></i></a>, <a class="ref" href="vectsp_5.html#T12" target="_self">Th12</a>, , <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E14:53_1_1_4_1_1_2_4_1"/>
 ex <font color="Olive">W2</font> being <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">W2</font> &amp; <font color="Maroon">W</font> <a href="vectsp_5.html#K2" target="_self">/\</a> <font color="Olive">W2</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font> )
 
<b>by </b><i>, <a class="ref" href="vectsp_5.html#T5" target="_self">Th5</a>, <a class="ref" href="vectsp_5.html#T10" target="_self">Th10</a>, <a class="txt" href="vectsp_5.html#E1:53_1_1_4"><i><font color="Green">E61</font></i></a></i>;<br/>
<b>hence </b><a NAME="E15:53_1_1_4_1_1_2_4_1"/>
<font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#T11" target="_self">Th11</a>, <a class="ref" href="vectsp_5.html#T13" target="_self">Th13</a>, </i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:53_1_1_4_1_1_2_4"/>
<font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font>
 ;<br/><a NAME="E3:53_1_1_4_1_1_2_4"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E4:53_1_1_4_1_1_2_4"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D3">VECTSP_4:def 3</a></i>;<br/><a NAME="E5:53_1_1_4_1_1_2_4"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:53_1_1_4_1_1_2_4"/>
<font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> <a href="vectsp_5.html#K2" target="_self">/\</a> <font color="Maroon">E</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T25">VECTSP_4:25</a></i>;<br/></div><b>end;</b></div><a NAME="E21:53_1_1_4_1_1_2"/><b>then </b><i><font color="Green">E163</font></i>: 
<font color="Maroon">W</font> <a href="vectsp_5.html#K2" target="_self">/\</a> <font color="Maroon">E</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T38">VECTSP_4:38</a></i>;<br/><a NAME="E22:53_1_1_4_1_1_2"/>
<font color="Maroon">E</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_5.html#K3" target="_self">Subspaces</a> <font color="Maroon">V</font>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">E</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> <b>by </b><i>, </i>;<br/><b>take </b><font color="Maroon">y</font> = <font color="Maroon">y'</font>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E24:53_1_1_4_1_1_2"/><i><font color="Green">E164</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/><b>then consider </b><font color="Maroon">W1</font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E26:53_1_1_4_1_1_2"/><i><font color="Green">E165</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font>
 <b>and </b><br/><a NAME="E27:53_1_1_4_1_1_2"/><i><font color="Green">E166</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_4_1_1_2_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/><b>assume </b><a NAME="E1:53_1_1_4_1_1_2_5"/>
<font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font>
 ;<br/><a NAME="E2:53_1_1_4_1_1_2_5"/><b>then </b><i><font color="Green">E167</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E3:53_1_1_4_1_1_2_5"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#T10" target="_self">Th10</a>, , <a class="txt" href="vectsp_5.html#E1:53_1_1_4"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:53_1_1_4_1_1_2_5"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="txt" href="vectsp_5.html#E2:53_1_1_4"><i><font color="Green">E62</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E5:53_1_1_4_1_1_2_5"/>
<font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">E</font>
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/></div><b>end;</b></div><a NAME="E29:53_1_1_4_1_1_2"/><b>then </b>
<font color="Maroon">W1</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">E</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T36">VECTSP_4:36</a></i>;<br/><b>hence </b><a NAME="E30:53_1_1_4_1_1_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#D2" target="_self">Def2</a>, <a class="txt" href="vectsp_5.html#E10:53_1_1"><i><font color="Green">E60</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="E4:53_1_1_4"/>
 ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> st <br/>for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E12:53_1_1"/><i><font color="Green">E168</font></i>: 
for <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> holds <br/> not <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>]
 <b>from </b><i><a class="ref" href="orders_1.html#S1">ORDERS_1:sch 1</a>(, , , );<br/></i>
<b>consider </b><font color="Maroon">L</font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E14:53_1_1"/><i><font color="Green">E170</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E15:53_1_1"/><i><font color="Green">E171</font></i>: 
<font color="Maroon">W</font> <a href="vectsp_5.html#K2" target="_self">/\</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font>
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">L</font>
;<br/>

<b>thus </b><a NAME="E16:53_1_1"/>
 <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">V</font>,the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">V</font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">W</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_5.html#D4" target="_self">VECTSP_5:def 4</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:53_1_1_5"/>
not  <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">V</font>,the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">V</font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">W</font>
 ;<br/>

<b>then consider </b><font color="Maroon">v</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:53_1_1_5"/><i><font color="Green">E172</font></i>: 
for <font color="Olive">v1</font>, <font color="Olive">v2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>  holds <br/>(  not <font color="Olive">v1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">L</font> or  not <font color="Olive">v2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> or <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">v2</font> )
 <b>by </b><i/>;<br/>
<a NAME="E4:53_1_1_5"/><i><font color="Green">E173</font></i>: 
not <font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">L</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="txt" href="vectsp_5.html#E11"><i><font color="Green">Lemma48</font></i></a>, </i>;<br/>
<b>set </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F</font> : verum </span>}</span> ;<br/>
<a NAME="E5:53_1_1_5"/><i><font color="Green">E174</font></i>: 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F</font> : verum </span>}</span> 
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:53_1_1_5_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F</font> : verum </span>}</span> 
 ;<br/><a NAME="E2:53_1_1_5_1"/><b>then </b>
 ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F</font> st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font>
 ;<br/><b>hence </b><a NAME="E3:53_1_1_5_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 ;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F</font> : verum </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E8:53_1_1_5"/>
<font color="Maroon">A</font> is <a href="vectsp_4.html#V1">lineary-closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_5_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:53_1_1_5_2"/>
for <font color="Olive">v1</font>, <font color="Olive">v2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>  st <font color="Olive">v1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">v2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Olive">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">v2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_4.html#D1">VECTSP_4:def 1</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_5_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">v2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/>

<b>assume </b><a NAME="E1:53_1_1_5_2_1"/>
<font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon">a1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F</font><b> such that </b><br/><a NAME="E3:53_1_1_5_2_1"/><i><font color="Green">E176</font></i>: 
<font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font>
 ;<br/>
<b>assume </b><a NAME="E4:53_1_1_5_2_1"/>
<font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon">a2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F</font><b> such that </b><br/><a NAME="E6:53_1_1_5_2_1"/><i><font color="Green">E178</font></i>: 
<font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font>
 ;<br/>
<a NAME="E7:53_1_1_5_2_1"/>
<font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">a2</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font>
 
<b>by </b><i>, , <a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>;<br/>
<b>hence </b><a NAME="E8:53_1_1_5_2_1"/>
<font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>


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

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F</font>;<br/><b>let </b><font color="Maroon">v1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/>



<b>assume </b><a NAME="E2:53_1_1_5_2"/>
<font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon">a1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F</font><b> such that </b><br/><a NAME="E4:53_1_1_5_2"/><i><font color="Green">E179</font></i>: 
<font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font>
 ;<br/>
<a NAME="E5:53_1_1_5_2"/>
<font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Maroon">a1</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font>
 
<b>by </b><i>, <a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>;<br/>
<b>hence </b><a NAME="E6:53_1_1_5_2"/>
<font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">Z</font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E10:53_1_1_5"/><i><font color="Green">E180</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#T10" target="_self">Th10</a>, <a class="ref" href="vectsp_4.html#T42">VECTSP_4:42</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_5_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/><b>thus </b><a NAME="E1:53_1_1_5_3"/>
( <font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">Z</font> <a href="vectsp_5.html#K2" target="_self">/\</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">L</font></span>)</span> implies <font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_5_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:53_1_1_5_3_1"/><i><font color="Green">E181</font></i>: 
<font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">Z</font> <a href="vectsp_5.html#K2" target="_self">/\</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">L</font></span>)</span>
 ;<br/>

<a NAME="E2:53_1_1_5_3_1"/><b>then </b>
<font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">Z</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:53_1_1_5_3_1"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F</font><b> such that </b><br/><a NAME="E5:53_1_1_5_3_1"/><i><font color="Green">E182</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font>
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_5_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_1_1_5_3_1_1"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font>
 ;<br/><a NAME="E2:53_1_1_5_3_1_1"/>
<font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">L</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:53_1_1_5_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="vectsp_4.html#T29">VECTSP_4:29</a></i>;<br/><a NAME="E4:53_1_1_5_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">a</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>;<br/><a NAME="E5:53_1_1_5_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="vectsp_1.html#D22">VECTSP_1:def 22</a></i>;<br/><a NAME="E6:53_1_1_5_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">L</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">W</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:53_1_1_5_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="vectsp_5.html#E12"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:53_1_1_5_3_1"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 
<b>by </b><i>, <a class="ref" href="vectsp_1.html#T59">VECTSP_1:59</a></i>;<br/>
<b>hence </b><a NAME="E8:53_1_1_5_3_1"/>
<font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T25">VECTSP_4:25</a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:53_1_1_5_3"/>
<font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font>
 ;<br/><a NAME="E3:53_1_1_5_3"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E4:53_1_1_5_3"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D3">VECTSP_4:def 3</a></i>;<br/><a NAME="E5:53_1_1_5_3"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:53_1_1_5_3"/>
<font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">Z</font> <a href="vectsp_5.html#K2" target="_self">/\</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">L</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T25">VECTSP_4:25</a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:53_1_1_5"/><b>then </b><i><font color="Green">E184</font></i>: 
<font color="Maroon">Z</font> <a href="vectsp_5.html#K2" target="_self">/\</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T38">VECTSP_4:38</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_5_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/><b>thus </b><a NAME="E1:53_1_1_5_4"/>
( <font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <span class="p1">(<span class="default"><font color="Maroon">Z</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">L</font></span>)</span> <a href="vectsp_5.html#K2" target="_self">/\</a> <font color="Maroon">W</font> implies <font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_5_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:53_1_1_5_4_1"/><i><font color="Green">E185</font></i>: 
<font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <span class="p1">(<span class="default"><font color="Maroon">Z</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">L</font></span>)</span> <a href="vectsp_5.html#K2" target="_self">/\</a> <font color="Maroon">W</font>
 ;<br/>

<a NAME="E2:53_1_1_5_4_1"/><b>then </b>
<font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">Z</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">v1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">v2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E4:53_1_1_5_4_1"/><i><font color="Green">E186</font></i>: 
<font color="Maroon">v1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E5:53_1_1_5_4_1"/><i><font color="Green">E187</font></i>: 
<font color="Maroon">v2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E6:53_1_1_5_4_1"/><i><font color="Green">E188</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font>
 <b>by </b><i/>;<br/>
<a NAME="E7:53_1_1_5_4_1"/><i><font color="Green">E189</font></i>: 
<font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v2</font>
 
<b>by </b><i>, <a class="ref" href="vectsp_2.html#T22">VECTSP_2:22</a></i>;<br/>
<a NAME="E8:53_1_1_5_4_1"/><i><font color="Green">E190</font></i>: 
<font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:53_1_1_5_4_1"/><b>then </b>
( <font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">L</font> &amp; <font color="Maroon">v2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">L</font> )
 
<b>by </b><i><a class="txt" href="vectsp_5.html#E9:53_1_1"><i><font color="Green">E55</font></i></a>, </i>;<br/>
<a NAME="E10:53_1_1_5_4_1"/><b>then </b>
<font color="Maroon">v1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">L</font>
 
<b>by </b><i>, <a class="ref" href="vectsp_4.html#T31">VECTSP_4:31</a></i>;<br/>
<a NAME="E11:53_1_1_5_4_1"/><b>then </b>
<font color="Maroon">v1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">Z</font> <a href="vectsp_5.html#K2" target="_self">/\</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">L</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E12:53_1_1_5_4_1"/><b>then </b>
<font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E13:53_1_1_5_4_1"/><b>then </b>
<font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#D3">VECTSP_4:def 3</a></i>;<br/>
<a NAME="E14:53_1_1_5_4_1"/><b>then </b>
<font color="Maroon">v1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E15:53_1_1_5_4_1"/><b>then </b>
<font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font>
 
<b>by </b><i>, <a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>;<br/>
<b>hence </b><a NAME="E16:53_1_1_5_4_1"/>
<font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#T7" target="_self">Th7</a>, <a class="txt" href="vectsp_5.html#E9:53_1_1"><i><font color="Green">E55</font></i></a>, , </i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:53_1_1_5_4"/>
<font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font>
 ;<br/><a NAME="E3:53_1_1_5_4"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E4:53_1_1_5_4"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D3">VECTSP_4:def 3</a></i>;<br/><a NAME="E5:53_1_1_5_4"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:53_1_1_5_4"/>
<font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <span class="p1">(<span class="default"><font color="Maroon">Z</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">L</font></span>)</span> <a href="vectsp_5.html#K2" target="_self">/\</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T25">VECTSP_4:25</a></i>;<br/></div><b>end;</b></div>
<a NAME="E14:53_1_1_5"/><b>then </b>
( <font color="Maroon">W</font> <a href="vectsp_5.html#K2" target="_self">/\</a> <span class="p1">(<span class="default"><font color="Maroon">Z</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font> &amp; <font color="Maroon">Z</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">L</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_5.html#K3" target="_self">Subspaces</a> <font color="Maroon">V</font> )
 
<b>by </b><i>, <a class="ref" href="vectsp_4.html#T38">VECTSP_4:38</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">Z</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">L</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<a NAME="E16:53_1_1_5"/>
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> &amp;  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> &amp;  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">L</font> )
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a>, <a class="ref" href="vectsp_4.html#T25">VECTSP_4:25</a></i>;<br/>
<a NAME="E17:53_1_1_5"/><b>then </b><i><font color="Green">E191</font></i>: 
not <font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="txt" href="vectsp_5.html#E11"><i><font color="Green">Lemma48</font></i></a></i>;<br/>
<a NAME="E18:53_1_1_5"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="vectsp_5.html#T10" target="_self">Th10</a>, <a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>;<br/>
<a NAME="E19:53_1_1_5"/><b>then </b>
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">Z</font>
 
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E20:53_1_1_5"/><b>then </b><i><font color="Green">E192</font></i>: 
<font color="Maroon">Z</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">L</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">L</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E21:53_1_1_5"/>
<font color="Maroon">L</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">Z</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<a NAME="E22:53_1_1_5"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x1</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="vectsp_5.html#D2" target="_self">Def2</a>, <a class="ref" href="vectsp_5.html#T6" target="_self">Th6</a></i>;<br/>
<b>hence </b><a NAME="E23:53_1_1_5"/>
contradiction
 <b>by </b><i><a class="ref" href="vectsp_5.html#T5" target="_self">Th5</a>, <a class="ref" href="vectsp_5.html#T6" target="_self">Th6</a>, </i>;<br/>


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

<b>thus </b><a NAME="E17:53_1_1"/>
<font color="Maroon">L</font> <a href="vectsp_5.html#K2" target="_self">/\</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#T7" target="_self">Th7</a></i>;<br/>


</div>
