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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">W1</font>, <font color="Olive" title="b2">W2</font> being   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">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="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Olive" title="b2">W2</font> );<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">W1</font> being   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">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="c3">W</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Olive" title="b1">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font> );<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E2:46_1_1"/><span class="lab"><font color="Green" title="E35">A1</font></span>: 
 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="c4">X</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp_5.html#K3" title="VECTSP_5:func.3">Subspaces</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></span>
<a NAME="E3:46_1_1"/>
( <font color="Maroon" title="c3">W</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p1">(<span class="default"><a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font> &amp;  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp_5.html#K3" title="VECTSP_5:func.3">Subspaces</a> <font color="Maroon" title="c2">V</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_5.html#D3" target="_self" title="VECTSP_5:def.3">Def3</a>, <a class="ref" href="vectsp_5.html#T20" target="_self" title="VECTSP_5:th.20">Th25</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">X</font> = <font color="Maroon" title="c4">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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_1_1"><span class="lab"><font color="Green" title="E35">A1</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">R</font> being   <a href="relset_1.html#NM2" title="RELSET_1:NM.2">Relation</a> of <font color="Maroon" title="c5">X</font><span class="kw"> such that </span><br/><a NAME="E6:46_1_1"/><span class="lab"><font color="Green" title="E36">A2</font></span>: 
 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="c5">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="c6">R</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="relset_1.html#S2" title="RELSET_1:sch.2">RELSET_1:sch 2</a>();<br/></span>
<span class="kw">defpred </span><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> ] <span class="kw">means </span><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="c6">R</font>;<br/>
<div><span class="lab"><font color="Green" title="E37">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c7">x</font>] implies <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font> )</span><br/><span class="kw">assume </span><a NAME="E1:46_1_1_1"/>
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c7">x</font>] )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font></span><br/><a NAME="E2:46_1_1_1"/><span class="kw">then </span>
(  ex <font color="Olive" title="b1">W1</font>, <font color="Olive" title="b2">W2</font> being   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font> st <br/>( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W1</font> &amp; <font color="Maroon" title="c8">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="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Olive" title="b2">W2</font> ) &amp;  ex <font color="Olive" title="b1">W3</font>, <font color="Olive" title="b2">W4</font> being   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font> st <br/>( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W3</font> &amp; <font color="Maroon" title="c7">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="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Olive" title="b2">W4</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:46_1_1"><span class="lab"><font color="Green" title="E36">A2</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:46_1_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T25" title="VECTSP_4:th.25">VECTSP_4:25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E8:46_1_1"/><span class="lab"><font color="Green" title="E38">A4</font></span>: 
 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="c5">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="c5">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="c5">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="c5">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="46_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">X</font> &amp; (  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="c5">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="c7">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="c7">Y</font> &amp;  not <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>] ) implies  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="c5">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="c5">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="c7">Y</font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:46_1_1_2"/><span class="lab"><font color="Green" title="E39">A5</font></span>: 
<font color="Maroon" title="c7">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">X</font>
 <span class="kw">and </span><br/><a NAME="E2:46_1_1_2"/><span class="lab"><font color="Green" title="E40">A6</font></span>: 
 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="c5">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="c7">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="c7">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="c6">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="c6">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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="c5">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="c5">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="c7">Y</font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:46_1_1_2_1_1"/>
( <font color="Maroon" title="c7">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="c7">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="46_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:46_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E41">A7</font></span>: 
<font color="Maroon" title="c7">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y9</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">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="c5">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="c7">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">y9</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="c6">R</font></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c8">y</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">X</font>;<br/><span class="kw">take </span><font color="Maroon" title="c9">y9</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">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="c7">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="b1">x</font>,<font color="Maroon" title="c9">y9</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="c6">R</font></span><br/><span class="kw">let </span><font color="Maroon" title="c10">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font> implies <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c9">y9</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="c6">R</font> )</span><br/><span class="kw">assume </span><a NAME="E2:46_1_1_2_1_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c9">y9</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="c6">R</font></span><br/><span class="kw">hence </span><a NAME="E3:46_1_1_2_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="c10">x</font>,<font color="Maroon" title="c9">y9</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="c6">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E41">A7</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:46_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E41">A8</font></span>: 
<font color="Maroon" title="c7">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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="c5">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="c5">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="c7">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="c6">R</font></span><br/><div class="add"><span class="kw">defpred </span><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> ] <span class="kw">means </span> ex <font color="Olive" title="b1">W1</font> being   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">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="E2:46_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E42">A9</font></span>: 
 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="c7">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="46_1_1_2_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font> implies  ex <font color="Olive" title="b1">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="Maroon" title="c8">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:46_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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="Maroon" title="c8">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">W1</font> being   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E3:46_1_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E43">A10</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">W1</font>
 <span class="kw">and </span><br/><a NAME="E4:46_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c3">W</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c9">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_1_1"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E1:46_1_1_2"><span class="lab"><font color="Green" title="E39">A5</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">y</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">W1</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c10">y</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c10">y</font>]</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c9">W1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">W1</font> &amp; <font color="Maroon" title="c10">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="c9">W1</font> )</span><br/>

<span class="kw">thus </span><a NAME="E6:46_1_1_2_1_1_2_1"/>
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">W1</font> &amp; <font color="Maroon" title="c10">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="c9">W1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E43">A10</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">consider </span><font color="Maroon" title="c8">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E4:46_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E43">A11</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">Y</font>
 <span class="kw">and </span><br/><a NAME="E5:46_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E44">A12</font></span>: 
 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="c7">Y</font> holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="classes1.html#S1" title="CLASSES1:sch.1">CLASSES1:sch 1</a>(<a class="txt" href="#E2:46_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E42">A9</font></span></a>);<br/></span><span class="kw">set </span><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="c8">f</font></span>)</span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_2_1_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">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="c8">f</font></span>)</span> implies <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="c2">V</font> )</span><br/><span class="kw">assume </span><a NAME="E1:46_1_1_2_1_1_2_2"/>
<font color="Maroon" title="c10">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="c8">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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="c2">V</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">Y9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:46_1_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E45">A13</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y9</font>
 <span class="kw">and </span><br/><a NAME="E4:46_1_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E46">A14</font></span>: 
<font color="Maroon" title="c11">Y9</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="c8">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c12">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:46_1_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E47">A15</font></span>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">f</font>
 <span class="kw">and </span><br/><a NAME="E7:46_1_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E48">A16</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Y9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46_1_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E46">A14</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c13">W1</font> being   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E9:46_1_1_2_1_1_2_2"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">W1</font>
 <span class="kw">and </span><br/><a NAME="E10:46_1_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E49">A17</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">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="c13">W1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E43">A11</font></span></a>, <a class="txt" href="#E5:46_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E44">A12</font></span></a>, <a class="txt" href="#E6:46_1_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E47">A15</font></span></a></span>;<br/><a NAME="E11:46_1_1_2_1_1_2_2"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c13">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="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#D2" title="VECTSP_4:def.2">VECTSP_4:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E12:46_1_1_2_1_1_2_2"/>
<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="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46_1_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E45">A13</font></span></a>, <a class="txt" href="#E7:46_1_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E48">A16</font></span></a>, <a class="txt" href="#E10:46_1_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E49">A17</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">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="c8">f</font></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><a NAME="E8:46_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E45">A18</font></span>: 
<font color="Maroon" title="c10">Z</font> is  <a href="vectsp_4.html#V1" title="VECTSP_4:attr.1">linearly-closed</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_2_1_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:46_1_1_2_1_1_2_3"/>
 for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">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">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="c10">Z</font> holds <br/><font color="Olive" title="b1">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="vectsp_4.html#D1" title="VECTSP_4:def.1">VECTSP_4:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font><br/>  for <font color="Olive">b<sub>2</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>  holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font> or <font color="Olive">b<sub>1</sub></font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_2_1_1_2_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">v1</font>, <font color="Maroon" title="c12">v2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font> &amp; <font color="Maroon" title="c12">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font> implies <font color="Maroon" title="c11">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c12">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:46_1_1_2_1_1_2_3_1"/><span class="lab"><font color="Green" title="E46">A19</font></span>: 
<font color="Maroon" title="c11">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font>
 <span class="kw">and </span><br/><a NAME="E2:46_1_1_2_1_1_2_3_1"/><span class="lab"><font color="Green" title="E47">A20</font></span>: 
<font color="Maroon" title="c12">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c12">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c13">Y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:46_1_1_2_1_1_2_3_1"/><span class="lab"><font color="Green" title="E48">A21</font></span>: 
<font color="Maroon" title="c11">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Y1</font>
 <span class="kw">and </span><br/><a NAME="E5:46_1_1_2_1_1_2_3_1"/><span class="lab"><font color="Green" title="E49">A22</font></span>: 
<font color="Maroon" title="c13">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="c8">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_1_1_2_1_1_2_3_1"><span class="lab"><font color="Green" title="E46">A19</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:46_1_1_2_1_1_2_3_1"/><span class="lab"><font color="Green" title="E50">A23</font></span>: 
<font color="Maroon" title="c14">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">f</font>
 <span class="kw">and </span><br/><a NAME="E8:46_1_1_2_1_1_2_3_1"/><span class="lab"><font color="Green" title="E51">A24</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Y1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46_1_1_2_1_1_2_3_1"><span class="lab"><font color="Green" title="E49">A22</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">Y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E10:46_1_1_2_1_1_2_3_1"/><span class="lab"><font color="Green" title="E52">A25</font></span>: 
<font color="Maroon" title="c12">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Y2</font>
 <span class="kw">and </span><br/><a NAME="E11:46_1_1_2_1_1_2_3_1"/><span class="lab"><font color="Green" title="E53">A26</font></span>: 
<font color="Maroon" title="c15">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="c8">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_1_1_2_1_1_2_3_1"><span class="lab"><font color="Green" title="E47">A20</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c16">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E13:46_1_1_2_1_1_2_3_1"/><span class="lab"><font color="Green" title="E54">A27</font></span>: 
<font color="Maroon" title="c16">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">f</font>
 <span class="kw">and </span><br/><a NAME="E14:46_1_1_2_1_1_2_3_1"/><span class="lab"><font color="Green" title="E55">A28</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Y2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:46_1_1_2_1_1_2_3_1"><span class="lab"><font color="Green" title="E53">A26</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c17">W1</font> being   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E16:46_1_1_2_1_1_2_3_1"/><span class="lab"><font color="Green" title="E56">A29</font></span>: 
<font color="Maroon" title="c14">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">W1</font>
 <span class="kw">and </span><br/><a NAME="E17:46_1_1_2_1_1_2_3_1"/><span class="lab"><font color="Green" title="E57">A30</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E43">A11</font></span></a>, <a class="txt" href="#E5:46_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E44">A12</font></span></a>, <a class="txt" href="#E7:46_1_1_2_1_1_2_3_1"><span class="lab"><font color="Green" title="E50">A23</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c18">W2</font> being   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E19:46_1_1_2_1_1_2_3_1"/><span class="lab"><font color="Green" title="E58">A31</font></span>: 
<font color="Maroon" title="c16">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">W2</font>
 <span class="kw">and </span><br/><a NAME="E20:46_1_1_2_1_1_2_3_1"/><span class="lab"><font color="Green" title="E59">A32</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">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="c18">W2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E43">A11</font></span></a>, <a class="txt" href="#E5:46_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E44">A12</font></span></a>, <a class="txt" href="#E13:46_1_1_2_1_1_2_3_1"><span class="lab"><font color="Green" title="E54">A27</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">y1</font> = <font color="Maroon" title="c14">y1</font>, <font color="Maroon" title="c20">y2</font> = <font color="Maroon" title="c16">y2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_1_1_2"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="txt" href="#E4:46_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E43">A11</font></span></a>, <a class="txt" href="#E7:46_1_1_2_1_1_2_3_1"><span class="lab"><font color="Green" title="E50">A23</font></span></a>, <a class="txt" href="#E13:46_1_1_2_1_1_2_3_1"><span class="lab"><font color="Green" title="E54">A27</font></span></a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_2_1_1_2_3_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_2_1_1_2_3_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:46_1_1_2_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="c19">y1</font>,<font color="Maroon" title="c20">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="c6">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="c20">y2</font>,<font color="Maroon" title="c19">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="c6">R</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_1_1_2"><span class="lab"><font color="Green" title="E40">A6</font></span></a>, <a class="txt" href="#E4:46_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E43">A11</font></span></a>, <a class="txt" href="#E7:46_1_1_2_1_1_2_3_1"><span class="lab"><font color="Green" title="E50">A23</font></span></a>, <a class="txt" href="#E13:46_1_1_2_1_1_2_3_1"><span class="lab"><font color="Green" title="E54">A27</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_2_1_1_2_3_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:46_1_1_2_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="c19">y1</font>,<font color="Maroon" title="c20">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="c6">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c12">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font></span><br/><div class="add"><a NAME="E2:46_1_1_2_1_1_2_3_1_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">W3</font>, <font color="Olive" title="b2">W4</font> being   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font> st <br/>( <font color="Maroon" title="c19">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W3</font> &amp; <font color="Maroon" title="c20">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="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Olive" title="b2">W4</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:46_1_1"><span class="lab"><font color="Green" title="E36">A2</font></span></a></span>;<br/><a NAME="E3:46_1_1_2_1_1_2_3_1_1_1_1"/><span class="kw">then </span>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c17">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="c18">W2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:46_1_1_2_1_1_2_3_1"><span class="lab"><font color="Green" title="E56">A29</font></span></a>, <a class="txt" href="#E19:46_1_1_2_1_1_2_3_1"><span class="lab"><font color="Green" title="E58">A31</font></span></a>, <a class="ref" href="vectsp_4.html#D2" title="VECTSP_4:def.2">VECTSP_4:def 2</a></span>;<br/><a NAME="E4:46_1_1_2_1_1_2_3_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A33</font></span>: 
<font color="Maroon" title="c11">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c18">W2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46_1_1_2_1_1_2_3_1"><span class="lab"><font color="Green" title="E48">A21</font></span></a>, <a class="txt" href="#E8:46_1_1_2_1_1_2_3_1"><span class="lab"><font color="Green" title="E51">A24</font></span></a>, <a class="txt" href="#E17:46_1_1_2_1_1_2_3_1"><span class="lab"><font color="Green" title="E57">A30</font></span></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/><a NAME="E5:46_1_1_2_1_1_2_3_1_1_1_1"/>
<font color="Maroon" title="c12">v2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c18">W2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:46_1_1_2_1_1_2_3_1"><span class="lab"><font color="Green" title="E52">A25</font></span></a>, <a class="txt" href="#E14:46_1_1_2_1_1_2_3_1"><span class="lab"><font color="Green" title="E55">A28</font></span></a>, <a class="txt" href="#E20:46_1_1_2_1_1_2_3_1"><span class="lab"><font color="Green" title="E59">A32</font></span></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/><a NAME="E6:46_1_1_2_1_1_2_3_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c12">v2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c18">W2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46_1_1_2_1_1_2_3_1_1_1_1"><span class="lab"><font color="Green" title="E60">A33</font></span></a>, <a class="ref" href="vectsp_4.html#T20" title="VECTSP_4:th.20">VECTSP_4:20</a></span>;<br/><a NAME="E7:46_1_1_2_1_1_2_3_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A34</font></span>: 
<font color="Maroon" title="c11">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c12">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="c18">W2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/><a NAME="E8:46_1_1_2_1_1_2_3_1_1_1_1"/>
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">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="c8">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:46_1_1_2_1_1_2_3_1"><span class="lab"><font color="Green" title="E54">A27</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E9:46_1_1_2_1_1_2_3_1_1_1_1"/>
<font color="Maroon" title="c11">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c12">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:46_1_1_2_1_1_2_3_1"><span class="lab"><font color="Green" title="E59">A32</font></span></a>, <a class="txt" href="#E7:46_1_1_2_1_1_2_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A34</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_2_1_1_2_3_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:46_1_1_2_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="c20">y2</font>,<font color="Maroon" title="c19">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="c6">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c12">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font></span><br/><div class="add"><a NAME="E2:46_1_1_2_1_1_2_3_1_1_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">W3</font>, <font color="Olive" title="b2">W4</font> being   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font> st <br/>( <font color="Maroon" title="c20">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W3</font> &amp; <font color="Maroon" title="c19">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="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Olive" title="b2">W4</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:46_1_1"><span class="lab"><font color="Green" title="E36">A2</font></span></a></span>;<br/><a NAME="E3:46_1_1_2_1_1_2_3_1_1_1_2"/><span class="kw">then </span>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c18">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="c17">W1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:46_1_1_2_1_1_2_3_1"><span class="lab"><font color="Green" title="E56">A29</font></span></a>, <a class="txt" href="#E19:46_1_1_2_1_1_2_3_1"><span class="lab"><font color="Green" title="E58">A31</font></span></a>, <a class="ref" href="vectsp_4.html#D2" title="VECTSP_4:def.2">VECTSP_4:def 2</a></span>;<br/><a NAME="E4:46_1_1_2_1_1_2_3_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A35</font></span>: 
<font color="Maroon" title="c12">v2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c17">W1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:46_1_1_2_1_1_2_3_1"><span class="lab"><font color="Green" title="E52">A25</font></span></a>, <a class="txt" href="#E14:46_1_1_2_1_1_2_3_1"><span class="lab"><font color="Green" title="E55">A28</font></span></a>, <a class="txt" href="#E20:46_1_1_2_1_1_2_3_1"><span class="lab"><font color="Green" title="E59">A32</font></span></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/><a NAME="E5:46_1_1_2_1_1_2_3_1_1_1_2"/>
<font color="Maroon" title="c11">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c17">W1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46_1_1_2_1_1_2_3_1"><span class="lab"><font color="Green" title="E48">A21</font></span></a>, <a class="txt" href="#E8:46_1_1_2_1_1_2_3_1"><span class="lab"><font color="Green" title="E51">A24</font></span></a>, <a class="txt" href="#E17:46_1_1_2_1_1_2_3_1"><span class="lab"><font color="Green" title="E57">A30</font></span></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/><a NAME="E6:46_1_1_2_1_1_2_3_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c12">v2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c17">W1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46_1_1_2_1_1_2_3_1_1_1_2"><span class="lab"><font color="Green" title="E60">A35</font></span></a>, <a class="ref" href="vectsp_4.html#T20" title="VECTSP_4:th.20">VECTSP_4:20</a></span>;<br/><a NAME="E7:46_1_1_2_1_1_2_3_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A36</font></span>: 
<font color="Maroon" title="c11">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c12">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="c17">W1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/><a NAME="E8:46_1_1_2_1_1_2_3_1_1_1_2"/>
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">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="c8">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:46_1_1_2_1_1_2_3_1"><span class="lab"><font color="Green" title="E50">A23</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E9:46_1_1_2_1_1_2_3_1_1_1_2"/>
<font color="Maroon" title="c11">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c12">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:46_1_1_2_1_1_2_3_1"><span class="lab"><font color="Green" title="E57">A30</font></span></a>, <a class="txt" href="#E7:46_1_1_2_1_1_2_3_1_1_1_2"><span class="lab"><font color="Green" title="E61">A36</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E23:46_1_1_2_1_1_2_3_1"/>
<font color="Maroon" title="c11">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c12">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">let </span><font color="Maroon" title="c11">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font> or <font color="Maroon" title="c11">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c12">v1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font> or <font color="Maroon" title="c11">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c12">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font> )</span><br/>



<span class="kw">assume </span><a NAME="E2:46_1_1_2_1_1_2_3"/>
<font color="Maroon" title="c12">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c12">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">Y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:46_1_1_2_1_1_2_3"/><span class="lab"><font color="Green" title="E46">A37</font></span>: 
<font color="Maroon" title="c12">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Y1</font>
 <span class="kw">and </span><br/><a NAME="E5:46_1_1_2_1_1_2_3"/><span class="lab"><font color="Green" title="E47">A38</font></span>: 
<font color="Maroon" title="c13">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="c8">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:46_1_1_2_1_1_2_3"/><span class="lab"><font color="Green" title="E48">A39</font></span>: 
<font color="Maroon" title="c14">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">f</font>
 <span class="kw">and </span><br/><a NAME="E8:46_1_1_2_1_1_2_3"/><span class="lab"><font color="Green" title="E49">A40</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Y1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46_1_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E47">A38</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">W1</font> being   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E10:46_1_1_2_1_1_2_3"/>
<font color="Maroon" title="c14">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">W1</font>
 <span class="kw">and </span><br/><a NAME="E11:46_1_1_2_1_1_2_3"/><span class="lab"><font color="Green" title="E50">A41</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">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="c15">W1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E43">A11</font></span></a>, <a class="txt" href="#E5:46_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E44">A12</font></span></a>, <a class="txt" href="#E7:46_1_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E48">A39</font></span></a></span>;<br/>
<a NAME="E12:46_1_1_2_1_1_2_3"/>
<font color="Maroon" title="c12">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c15">W1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46_1_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E46">A37</font></span></a>, <a class="txt" href="#E8:46_1_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E49">A40</font></span></a>, <a class="txt" href="#E11:46_1_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E50">A41</font></span></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E13:46_1_1_2_1_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c11">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c12">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c15">W1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T21" title="VECTSP_4:th.21">VECTSP_4:21</a></span>;<br/>
<a NAME="E14:46_1_1_2_1_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A42</font></span>: 
<font color="Maroon" title="c11">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c12">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="c15">W1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E15:46_1_1_2_1_1_2_3"/>
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">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="c8">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:46_1_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E48">A39</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:46_1_1_2_1_1_2_3"/>
<font color="Maroon" title="c11">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c12">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:46_1_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E50">A41</font></span></a>, <a class="txt" href="#E14:46_1_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E51">A42</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">set </span><font color="Maroon" title="c11">z</font> =  the    <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="c8">f</font>;<br/><a NAME="E9:46_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E46">A43</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="txt" href="#E4:46_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E43">A11</font></span></a>, <a class="ref" href="relat_1.html#T42" title="RELAT_1:th.42">RELAT_1:42</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">z1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E11:46_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E47">A44</font></span>: 
<font color="Maroon" title="c12">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">f</font>
 <span class="kw">and </span><br/><a NAME="E12:46_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E48">A45</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <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="c8">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><a NAME="E13:46_1_1_2_1_1_2"/>
 ex <font color="Olive" title="b1">W3</font> being   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font> st <br/>( <font color="Maroon" title="c12">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W3</font> &amp; <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">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="Olive" title="b1">W3</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E43">A11</font></span></a>, <a class="txt" href="#E5:46_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E44">A12</font></span></a>, <a class="txt" href="#E11:46_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E47">A44</font></span></a></span>;<br/><a NAME="E14:46_1_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:46_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E46">A43</font></span></a>, <a class="txt" href="#E12:46_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E48">A45</font></span></a>, <a class="ref" href="orders_1.html#T6" title="ORDERS_1:th.6">ORDERS_1:6</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">E</font> being   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E16:46_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E49">A46</font></span>: 
<font color="Maroon" title="c10">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:46_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="ref" href="vectsp_4.html#T34" title="VECTSP_4:th.34">VECTSP_4:34</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_2_1_1_2_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">u</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c14">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c13">E</font> implies <font color="Maroon" title="c14">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font> ) &amp; ( <font color="Maroon" title="c14">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font> implies <font color="Maroon" title="c14">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c13">E</font> ) )</span><br/><span class="kw">thus </span><a NAME="E1:46_1_1_2_1_1_2_4"/>
( <font color="Maroon" title="c14">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c13">E</font> implies <font color="Maroon" title="c14">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font> implies <font color="Maroon" title="c14">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c13">E</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_2_1_1_2_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:46_1_1_2_1_1_2_4_1"/><span class="lab"><font color="Green" title="E50">A47</font></span>: 
<font color="Maroon" title="c14">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c13">E</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font></span><br/>

<a NAME="E2:46_1_1_2_1_1_2_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A48</font></span>: 
<font color="Maroon" title="c14">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_5.html#T3" target="_self" title="VECTSP_5:th.3">Th7</a></span>;<br/>
<a NAME="E3:46_1_1_2_1_1_2_4_1"/>
<font color="Maroon" title="c14">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c13">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_1_1_2_1_1_2_4_1"><span class="lab"><font color="Green" title="E50">A47</font></span></a>, <a class="ref" href="vectsp_5.html#T3" target="_self" title="VECTSP_5:th.3">Th7</a></span>;<br/>
<a NAME="E4:46_1_1_2_1_1_2_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:46_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E49">A46</font></span></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">Y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:46_1_1_2_1_1_2_4_1"/><span class="lab"><font color="Green" title="E52">A49</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E7:46_1_1_2_1_1_2_4_1"/><span class="lab"><font color="Green" title="E53">A50</font></span>: 
<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="c8">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c16">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E9:46_1_1_2_1_1_2_4_1"/><span class="lab"><font color="Green" title="E54">A51</font></span>: 
<font color="Maroon" title="c16">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">f</font>
 <span class="kw">and </span><br/><a NAME="E10:46_1_1_2_1_1_2_4_1"/><span class="lab"><font color="Green" title="E55">A52</font></span>: 
<font color="Maroon" title="c8">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:46_1_1_2_1_1_2_4_1"><span class="lab"><font color="Green" title="E53">A50</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E11:46_1_1_2_1_1_2_4_1"/><span class="lab"><font color="Green" title="E56">A53</font></span>: 
 ex <font color="Olive" title="b1">W2</font> being   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">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="c3">W</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Olive" title="b1">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_1_1"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E1:46_1_1_2"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="txt" href="#E4:46_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E43">A11</font></span></a>, <a class="txt" href="#E9:46_1_1_2_1_1_2_4_1"><span class="lab"><font color="Green" title="E54">A51</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c17">W1</font> being   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E13:46_1_1_2_1_1_2_4_1"/><span class="lab"><font color="Green" title="E57">A54</font></span>: 
<font color="Maroon" title="c16">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">W1</font>
 <span class="kw">and </span><br/><a NAME="E14:46_1_1_2_1_1_2_4_1"/><span class="lab"><font color="Green" title="E58">A55</font></span>: 
<font color="Maroon" title="c8">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E43">A11</font></span></a>, <a class="txt" href="#E5:46_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E44">A12</font></span></a>, <a class="txt" href="#E9:46_1_1_2_1_1_2_4_1"><span class="lab"><font color="Green" title="E54">A51</font></span></a></span>;<br/>
<a NAME="E15:46_1_1_2_1_1_2_4_1"/>
<font color="Maroon" title="c14">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c17">W1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:46_1_1_2_1_1_2_4_1"><span class="lab"><font color="Green" title="E52">A49</font></span></a>, <a class="txt" href="#E10:46_1_1_2_1_1_2_4_1"><span class="lab"><font color="Green" title="E55">A52</font></span></a>, <a class="txt" href="#E14:46_1_1_2_1_1_2_4_1"><span class="lab"><font color="Green" title="E58">A55</font></span></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:46_1_1_2_1_1_2_4_1"/>
<font color="Maroon" title="c14">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:46_1_1_2_1_1_2_4_1"><span class="lab"><font color="Green" title="E57">A54</font></span></a>, <a class="txt" href="#E2:46_1_1_2_1_1_2_4_1"><span class="lab"><font color="Green" title="E51">A48</font></span></a>, <a class="txt" href="#E11:46_1_1_2_1_1_2_4_1"><span class="lab"><font color="Green" title="E56">A53</font></span></a>, <a class="ref" href="vectsp_5.html#T3" target="_self" title="VECTSP_5:th.3">Th7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E2:46_1_1_2_1_1_2_4"/>
<font color="Maroon" title="c14">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c13">E</font></span><br/><a NAME="E3:46_1_1_2_1_1_2_4"/><span class="kw">then </span>
<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="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/><a NAME="E4:46_1_1_2_1_1_2_4"/><span class="kw">then </span>
<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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#D3" title="VECTSP_4:def.3">VECTSP_4:def 3</a></span>;<br/><a NAME="E5:46_1_1_2_1_1_2_4"/><span class="kw">then </span>
<font color="Maroon" title="c14">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:46_1_1_2_1_1_2_4"/>
<font color="Maroon" title="c14">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c13">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T17" title="VECTSP_4:th.17">VECTSP_4:17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E18:46_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A56</font></span>: 
<font color="Maroon" title="c3">W</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c13">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T30" title="VECTSP_4:th.30">VECTSP_4:30</a></span>;<br/><a NAME="E19:46_1_1_2_1_1_2"/>
<font color="Maroon" title="c13">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp_5.html#K3" title="VECTSP_5:func.3">Subspaces</a> <font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_5.html#D3" target="_self" title="VECTSP_5:def.3">Def3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">y9</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="c5">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_1_1"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E18:46_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E50">A56</font></span></a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c15">y</font> = <font color="Maroon" title="c14">y9</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">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="c7">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="b1">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="c6">R</font></span><br/><span class="kw">let </span><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="c5">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font> implies <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="c6">R</font> )</span><br/><span class="kw">assume </span><a NAME="E21:46_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E51">A57</font></span>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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="c6">R</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">W1</font> being   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E23:46_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E52">A58</font></span>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">W1</font>
 <span class="kw">and </span><br/><a NAME="E24:46_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E53">A59</font></span>: 
<font color="Maroon" title="c8">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E44">A12</font></span></a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_2_1_1_2_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c18">u</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c17">W1</font> implies <font color="Maroon" title="c18">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c13">E</font> )</span><br/><span class="kw">assume </span><a NAME="E1:46_1_1_2_1_1_2_5"/>
<font color="Maroon" title="c18">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c17">W1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c13">E</font></span><br/><a NAME="E2:46_1_1_2_1_1_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A60</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/><a NAME="E3:46_1_1_2_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="c8">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E43">A11</font></span></a>, <a class="txt" href="#E21:46_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E51">A57</font></span></a>, <a class="txt" href="#E24:46_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E53">A59</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><a NAME="E4:46_1_1_2_1_1_2_5"/><span class="kw">then </span>
<font color="Maroon" title="c18">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_1_1_2_1_1_2_5"><span class="lab"><font color="Green" title="E54">A60</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E5:46_1_1_2_1_1_2_5"/>
<font color="Maroon" title="c18">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c13">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:46_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E49">A46</font></span></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E26:46_1_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c17">W1</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c13">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T28" title="VECTSP_4:th.28">VECTSP_4:28</a></span>;<br/><span class="kw">hence </span><a NAME="E27:46_1_1_2_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="c6">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:46_1_1"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="txt" href="#E23:46_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E52">A58</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:46_1_1_2"/>
 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="c5">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="c5">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="c7">Y</font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">y</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E39">A61</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font>, <font color="Maroon" title="c9">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c9">z</font>] implies <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">z</font>] )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:46_1_1_3"/><span class="lab"><font color="Green" title="E40">A62</font></span>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>]
 <span class="kw">and </span><br/><a NAME="E2:46_1_1_3"/><span class="lab"><font color="Green" title="E41">A63</font></span>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c9">z</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">z</font>]</span><br/><span class="kw">consider </span><font color="Maroon" title="c10">W1</font>, <font color="Maroon" title="c11">W2</font> being   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E4:46_1_1_3"/><span class="lab"><font color="Green" title="E42">A64</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">W1</font>
 <span class="kw">and </span><br/><a NAME="E5:46_1_1_3"/><span class="lab"><font color="Green" title="E43">A65</font></span>: 
( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">W2</font> &amp; <font color="Maroon" title="c10">W1</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c11">W2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:46_1_1"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="txt" href="#E1:46_1_1_3"><span class="lab"><font color="Green" title="E40">A62</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c12">W3</font>, <font color="Maroon" title="c13">W4</font> being   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E7:46_1_1_3"/><span class="lab"><font color="Green" title="E44">A66</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">W3</font>
 <span class="kw">and </span><br/><a NAME="E8:46_1_1_3"/><span class="lab"><font color="Green" title="E45">A67</font></span>: 
<font color="Maroon" title="c9">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">W4</font>
 <span class="kw">and </span><br/><a NAME="E9:46_1_1_3"/><span class="lab"><font color="Green" title="E46">A68</font></span>: 
<font color="Maroon" title="c12">W3</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c13">W4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:46_1_1"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="txt" href="#E2:46_1_1_3"><span class="lab"><font color="Green" title="E41">A63</font></span></a></span>;<br/><a NAME="E10:46_1_1_3"/>
<font color="Maroon" title="c10">W1</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c13">W4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46_1_1_3"><span class="lab"><font color="Green" title="E43">A65</font></span></a>, <a class="txt" href="#E7:46_1_1_3"><span class="lab"><font color="Green" title="E44">A66</font></span></a>, <a class="txt" href="#E9:46_1_1_3"><span class="lab"><font color="Green" title="E46">A68</font></span></a>, <a class="ref" href="vectsp_4.html#T26" title="VECTSP_4:th.26">VECTSP_4:26</a></span>;<br/><span class="kw">hence </span><a NAME="E11:46_1_1_3"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">z</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:46_1_1"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="txt" href="#E4:46_1_1_3"><span class="lab"><font color="Green" title="E42">A64</font></span></a>, <a class="txt" href="#E8:46_1_1_3"><span class="lab"><font color="Green" title="E45">A67</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E40">A69</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c7">x</font>]</span><br/><span class="kw">consider </span><font color="Maroon" title="c8">W1</font> being   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E2:46_1_1_4"/><span class="lab"><font color="Green" title="E41">A70</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">W1</font>
 <span class="kw">and </span><br/><a NAME="E3:46_1_1_4"/>
<font color="Maroon" title="c3">W</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c8">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_1_1"><span class="lab"><font color="Green" title="E35">A1</font></span></a></span>;<br/><a NAME="E4:46_1_1_4"/>
<font color="Maroon" title="c8">W1</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c8">W1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T24" title="VECTSP_4:th.24">VECTSP_4:24</a></span>;<br/><span class="kw">hence </span><a NAME="E5:46_1_1_4"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c7">x</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:46_1_1"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="txt" href="#E2:46_1_1_4"><span class="lab"><font color="Green" title="E41">A70</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c7">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">X</font><span class="kw"> such that </span><br/><a NAME="E12:46_1_1"/><span class="lab"><font color="Green" title="E41">A71</font></span>: 
 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="c5">X</font>  st <font color="Maroon" title="c7">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="c7">x</font>,<font color="Olive" title="b1">y</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="orders_1.html#S1" title="ORDERS_1:sch.1">ORDERS_1:sch 1</a>(<a class="txt" href="#E10:46_1_1"><span class="lab"><font color="Green" title="E40">A69</font></span></a>, <a class="txt" href="#E7:46_1_1"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="txt" href="#E9:46_1_1"><span class="lab"><font color="Green" title="E39">A61</font></span></a>, <a class="txt" href="#E8:46_1_1"><span class="lab"><font color="Green" title="E38">A4</font></span></a>);<br/></span>
<span class="kw">consider </span><font color="Maroon" title="c8">L</font> being   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E14:46_1_1"/><span class="lab"><font color="Green" title="E42">A72</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L</font>
 <span class="kw">and </span><br/><a NAME="E15:46_1_1"/><span class="lab"><font color="Green" title="E43">A73</font></span>: 
<font color="Maroon" title="c3">W</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c8">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_1_1"><span class="lab"><font color="Green" title="E35">A1</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c8">L</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">V</font> <a href="vectsp_5.html#R1" title="VECTSP_5:pred.1">is_the_direct_sum_of</a> <font color="Maroon" title="c8">L</font>,<font color="Maroon" title="c3">W</font></span><br/>

<span class="kw">thus </span><a NAME="E16:46_1_1"/>
 <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">VectSpStr</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c2">V</font>, the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c2">V</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c3">W</font>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="vectsp_5.html#D4" target="_self" title="VECTSP_5:def.4">VECTSP_5:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">L</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:46_1_1_5"/>
 not  <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">VectSpStr</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c2">V</font>, the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c2">V</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c3">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E3:46_1_1_5"/><span class="lab"><font color="Green" title="E44">A74</font></span>: 
 for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font>  holds <br/>(  not <font color="Olive" title="b1">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c8">L</font> or  not <font color="Olive" title="b2">v2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W</font> or <font color="Maroon" title="c9">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#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">v2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="vectsp_5.html#E56"><span class="lab"><font color="Green" title="E34">Lm16</font></span></a></span>;<br/>
<a NAME="E4:46_1_1_5"/>
( <font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">v</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T4" title="RLVECT_1:th.4">RLVECT_1:4</a>, <a class="ref" href="vectsp_4.html#T17" title="VECTSP_4:th.17">VECTSP_4:17</a></span>;<br/>
<a NAME="E5:46_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A75</font></span>: 
 not <font color="Maroon" title="c9">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c8">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46_1_1_5"><span class="lab"><font color="Green" title="E44">A74</font></span></a></span>;<br/>
<span class="kw">set </span><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="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c9">v</font></span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font> : verum </span> } </span> ;<br/>
<a NAME="E6:46_1_1_5"/><span class="lab"><font color="Green" title="E46">A76</font></span>: 
<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c9">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="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c9">v</font></span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font> : verum </span> } </span> 
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_5_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">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="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c9">v</font></span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font> : verum </span> } </span>  implies <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font> )</span><br/><span class="kw">assume </span><a NAME="E1:46_1_1_5_1"/>
<font color="Maroon" title="c11">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="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c9">v</font></span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font> : verum </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font></span><br/><a NAME="E2:46_1_1_5_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font> st <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c9">v</font>
 ;<br/><span class="kw">hence </span><a NAME="E3:46_1_1_5_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">A</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c9">v</font></span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font> : verum </span> } </span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E9:46_1_1_5"/>
<font color="Maroon" title="c11">A</font> is  <a href="vectsp_4.html#V1" title="VECTSP_4:attr.1">linearly-closed</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_5_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:46_1_1_5_2"/>
 for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">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="c11">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="c11">A</font> holds <br/><font color="Olive" title="b1">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">A</font>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="vectsp_4.html#D1" title="VECTSP_4:def.1">VECTSP_4:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font><br/>  for <font color="Olive">b<sub>2</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>  holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">A</font> or <font color="Olive">b<sub>1</sub></font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">A</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_5_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">v1</font>, <font color="Maroon" title="c13">v2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">A</font> &amp; <font color="Maroon" title="c13">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">A</font> implies <font color="Maroon" title="c12">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c13">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">A</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:46_1_1_5_2_1"/>
<font color="Maroon" title="c12">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c13">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">A</font> or <font color="Maroon" title="c12">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c13">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">A</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">a1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font><span class="kw"> such that </span><br/><a NAME="E3:46_1_1_5_2_1"/><span class="lab"><font color="Green" title="E47">A77</font></span>: 
<font color="Maroon" title="c12">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a1</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c9">v</font>
 ;<br/>
<span class="kw">assume </span><a NAME="E4:46_1_1_5_2_1"/>
<font color="Maroon" title="c13">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c13">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">A</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">a2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font><span class="kw"> such that </span><br/><a NAME="E6:46_1_1_5_2_1"/><span class="lab"><font color="Green" title="E48">A78</font></span>: 
<font color="Maroon" title="c13">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">a2</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c9">v</font>
 ;<br/>
<a NAME="E7:46_1_1_5_2_1"/>
<font color="Maroon" title="c12">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c13">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">a1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c15">a2</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c9">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46_1_1_5_2_1"><span class="lab"><font color="Green" title="E47">A77</font></span></a>, <a class="txt" href="#E6:46_1_1_5_2_1"><span class="lab"><font color="Green" title="E48">A78</font></span></a>, <a class="ref" href="vectsp_1.html#D15" title="VECTSP_1:def.15">VECTSP_1:def 15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:46_1_1_5_2_1"/>
<font color="Maroon" title="c12">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c13">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">let </span><font color="Maroon" title="c12">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">A</font> or <font color="Maroon" title="c12">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">A</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c13">v1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c13">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">A</font> or <font color="Maroon" title="c12">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c13">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">A</font> )</span><br/>



<span class="kw">assume </span><a NAME="E2:46_1_1_5_2"/>
<font color="Maroon" title="c13">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c13">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">A</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">a1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font><span class="kw"> such that </span><br/><a NAME="E4:46_1_1_5_2"/><span class="lab"><font color="Green" title="E47">A79</font></span>: 
<font color="Maroon" title="c13">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a1</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c9">v</font>
 ;<br/>
<a NAME="E5:46_1_1_5_2"/>
<font color="Maroon" title="c12">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c13">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c14">a1</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c9">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46_1_1_5_2"><span class="lab"><font color="Green" title="E47">A79</font></span></a>, <a class="ref" href="vectsp_1.html#D16" title="VECTSP_1:def.16">VECTSP_1:def 16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:46_1_1_5_2"/>
<font color="Maroon" title="c12">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c13">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">Z</font> being   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E11:46_1_1_5"/><span class="lab"><font color="Green" title="E47">A80</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:46_1_1_5"><span class="lab"><font color="Green" title="E46">A76</font></span></a>, <a class="ref" href="vectsp_4.html#T34" title="VECTSP_4:th.34">VECTSP_4:34</a></span>;<br/>
<a NAME="E12:46_1_1_5"/><span class="lab"><font color="Green" title="E48">A81</font></span>: 
 not <font color="Maroon" title="c9">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c8">L</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c3">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46_1_1_5"><span class="lab"><font color="Green" title="E44">A74</font></span></a>, <a class="ref" href="vectsp_5.html#T1" target="_self" title="VECTSP_5:th.1">Th5</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_5_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">u</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c13">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c12">Z</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c8">L</font></span>)</span> implies <font color="Maroon" title="c13">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font> ) &amp; ( <font color="Maroon" title="c13">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font> implies <font color="Maroon" title="c13">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c12">Z</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c8">L</font></span>)</span> ) )</span><br/><span class="kw">thus </span><a NAME="E1:46_1_1_5_3"/>
( <font color="Maroon" title="c13">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c12">Z</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c8">L</font></span>)</span> implies <font color="Maroon" title="c13">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font> implies <font color="Maroon" title="c13">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c12">Z</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c8">L</font></span>)</span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_5_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:46_1_1_5_3_1"/><span class="lab"><font color="Green" title="E49">A82</font></span>: 
<font color="Maroon" title="c13">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c12">Z</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c8">L</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font></span><br/>

<a NAME="E2:46_1_1_5_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c12">Z</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_5.html#T3" target="_self" title="VECTSP_5:th.3">Th7</a></span>;<br/>
<a NAME="E3:46_1_1_5_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:46_1_1_5"><span class="lab"><font color="Green" title="E47">A80</font></span></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font><span class="kw"> such that </span><br/><a NAME="E5:46_1_1_5_3_1"/><span class="lab"><font color="Green" title="E50">A83</font></span>: 
<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c9">v</font>
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_5_3_1_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:46_1_1_5_3_1_1"/>
<font color="Maroon" title="c13">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c8">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_1_1_5_3_1"><span class="lab"><font color="Green" title="E49">A82</font></span></a>, <a class="ref" href="vectsp_5.html#T3" target="_self" title="VECTSP_5:th.3">Th7</a></span>;<br/><a NAME="E2:46_1_1_5_3_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c9">v</font></span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c8">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46_1_1_5_3_1"><span class="lab"><font color="Green" title="E50">A83</font></span></a>, <a class="ref" href="vectsp_4.html#T21" title="VECTSP_4:th.21">VECTSP_4:21</a></span>;<br/><a NAME="E3:46_1_1_5_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A84</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c14">a</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c9">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c8">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D16" title="VECTSP_1:def.16">VECTSP_1:def 16</a></span>;<br/><span class="kw">assume </span><a NAME="E4:46_1_1_5_3_1_1"/>
<font color="Maroon" title="c14">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E5:46_1_1_5_3_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c9">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c8">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46_1_1_5_3_1_1"><span class="lab"><font color="Green" title="E51">A84</font></span></a>, <a class="ref" href="vectsp_1.html#D10" title="VECTSP_1:def.10">VECTSP_1:def 10</a></span>;<br/><a NAME="E6:46_1_1_5_3_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c9">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c8">L</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="vectsp_5.html#E2"><span class="lab"><font color="Green" title="E2">Lm1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:46_1_1_5_3_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:46_1_1_5"><span class="lab"><font color="Green" title="E48">A81</font></span></a>, <a class="ref" href="vectsp_1.html#D17" title="VECTSP_1:def.17">VECTSP_1:def 17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E7:46_1_1_5_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46_1_1_5_3_1"><span class="lab"><font color="Green" title="E50">A83</font></span></a>, <a class="ref" href="vectsp_1.html#T14" title="VECTSP_1:th.14">VECTSP_1:14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:46_1_1_5_3_1"/>
<font color="Maroon" title="c13">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T17" title="VECTSP_4:th.17">VECTSP_4:17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E2:46_1_1_5_3"/>
<font color="Maroon" title="c13">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c12">Z</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c8">L</font></span>)</span></span><br/><a NAME="E3:46_1_1_5_3"/><span class="kw">then </span>
<font color="Maroon" title="c13">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="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/><a NAME="E4:46_1_1_5_3"/><span class="kw">then </span>
<font color="Maroon" title="c13">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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#D3" title="VECTSP_4:def.3">VECTSP_4:def 3</a></span>;<br/><a NAME="E5:46_1_1_5_3"/><span class="kw">then </span>
<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:46_1_1_5_3"/>
<font color="Maroon" title="c13">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c12">Z</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c8">L</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T17" title="VECTSP_4:th.17">VECTSP_4:17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E14:46_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A85</font></span>: 
<font color="Maroon" title="c12">Z</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T30" title="VECTSP_4:th.30">VECTSP_4:30</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_5_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">u</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c13">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Z</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c3">W</font> implies <font color="Maroon" title="c13">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font> ) &amp; ( <font color="Maroon" title="c13">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font> implies <font color="Maroon" title="c13">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Z</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c3">W</font> ) )</span><br/><span class="kw">thus </span><a NAME="E1:46_1_1_5_4"/>
( <font color="Maroon" title="c13">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Z</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c3">W</font> implies <font color="Maroon" title="c13">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font> implies <font color="Maroon" title="c13">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Z</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c3">W</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_5_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:46_1_1_5_4_1"/><span class="lab"><font color="Green" title="E50">A86</font></span>: 
<font color="Maroon" title="c13">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Z</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c3">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font></span><br/>

<a NAME="E2:46_1_1_5_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c12">Z</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c8">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_5.html#T3" target="_self" title="VECTSP_5:th.3">Th7</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">v1</font>, <font color="Maroon" title="c15">v2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E4:46_1_1_5_4_1"/><span class="lab"><font color="Green" title="E51">A87</font></span>: 
<font color="Maroon" title="c14">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c12">Z</font>
 <span class="kw">and </span><br/><a NAME="E5:46_1_1_5_4_1"/><span class="lab"><font color="Green" title="E52">A88</font></span>: 
<font color="Maroon" title="c15">v2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c8">L</font>
 <span class="kw">and </span><br/><a NAME="E6:46_1_1_5_4_1"/><span class="lab"><font color="Green" title="E53">A89</font></span>: 
<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c15">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_5.html#T1" target="_self" title="VECTSP_5:th.1">Th5</a></span>;<br/>
<a NAME="E7:46_1_1_5_4_1"/><span class="lab"><font color="Green" title="E54">A90</font></span>: 
<font color="Maroon" title="c13">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_1_1_5_4_1"><span class="lab"><font color="Green" title="E50">A86</font></span></a>, <a class="ref" href="vectsp_5.html#T3" target="_self" title="VECTSP_5:th.3">Th7</a></span>;<br/>
<a NAME="E8:46_1_1_5_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A91</font></span>: 
<font color="Maroon" title="c13">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c8">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_5.html#T2" target="_self" title="VECTSP_5:th.2">Th6</a></span>;<br/>
<a NAME="E9:46_1_1_5_4_1"/>
( <font color="Maroon" title="c14">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">u</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c15">v2</font> &amp; <font color="Maroon" title="c15">v2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c8">L</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46_1_1_5_4_1"><span class="lab"><font color="Green" title="E52">A88</font></span></a>, <a class="txt" href="#E6:46_1_1_5_4_1"><span class="lab"><font color="Green" title="E53">A89</font></span></a>, <a class="ref" href="vectsp_5.html#T2" target="_self" title="VECTSP_5:th.2">Th6</a>, <a class="ref" href="vectsp_2.html#T2" title="VECTSP_2:th.2">VECTSP_2:2</a></span>;<br/>
<a NAME="E10:46_1_1_5_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c8">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:46_1_1_5_4_1"><span class="lab"><font color="Green" title="E55">A91</font></span></a>, <a class="ref" href="vectsp_4.html#T23" title="VECTSP_4:th.23">VECTSP_4:23</a></span>;<br/>
<a NAME="E11:46_1_1_5_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c12">Z</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c8">L</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46_1_1_5_4_1"><span class="lab"><font color="Green" title="E51">A87</font></span></a>, <a class="ref" href="vectsp_5.html#T3" target="_self" title="VECTSP_5:th.3">Th7</a></span>;<br/>
<a NAME="E12:46_1_1_5_4_1"/><span class="kw">then </span>
<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 <span class="p1">(<span class="default"><a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:46_1_1_5"><span class="lab"><font color="Green" title="E49">A85</font></span></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E13:46_1_1_5_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#D3" title="VECTSP_4:def.3">VECTSP_4:def 3</a></span>;<br/>
<a NAME="E14:46_1_1_5_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E15:46_1_1_5_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">u</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:46_1_1_5_4_1"><span class="lab"><font color="Green" title="E53">A89</font></span></a>, <a class="ref" href="rlvect_1.html#T4" title="RLVECT_1:th.4">RLVECT_1:4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:46_1_1_5_4_1"/>
<font color="Maroon" title="c13">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:46_1_1"><span class="lab"><font color="Green" title="E43">A73</font></span></a>, <a class="txt" href="#E5:46_1_1_5_4_1"><span class="lab"><font color="Green" title="E52">A88</font></span></a>, <a class="txt" href="#E7:46_1_1_5_4_1"><span class="lab"><font color="Green" title="E54">A90</font></span></a>, <a class="ref" href="vectsp_5.html#T3" target="_self" title="VECTSP_5:th.3">Th7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E2:46_1_1_5_4"/>
<font color="Maroon" title="c13">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Z</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c3">W</font></span><br/><a NAME="E3:46_1_1_5_4"/><span class="kw">then </span>
<font color="Maroon" title="c13">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="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/><a NAME="E4:46_1_1_5_4"/><span class="kw">then </span>
<font color="Maroon" title="c13">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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#D3" title="VECTSP_4:def.3">VECTSP_4:def 3</a></span>;<br/><a NAME="E5:46_1_1_5_4"/><span class="kw">then </span>
<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:46_1_1_5_4"/>
<font color="Maroon" title="c13">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Z</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T17" title="VECTSP_4:th.17">VECTSP_4:17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E16:46_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A92</font></span>: 
<font color="Maroon" title="c3">W</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Z</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T30" title="VECTSP_4:th.30">VECTSP_4:30</a></span>;<br/>
<a NAME="E17:46_1_1_5"/>
<font color="Maroon" title="c12">Z</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c8">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp_5.html#K3" title="VECTSP_5:func.3">Subspaces</a> <font color="Maroon" title="c2">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_5.html#D3" target="_self" title="VECTSP_5:def.3">Def3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">x1</font> = <font color="Maroon" title="c12">Z</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c8">L</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_1_1"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E16:46_1_1_5"><span class="lab"><font color="Green" title="E50">A92</font></span></a></span>;<br/>
<a NAME="E19:46_1_1_5"/>
<font color="Maroon" title="c8">L</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c12">Z</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c8">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_5.html#T7" target="_self" title="VECTSP_5:th.7">Th11</a></span>;<br/>
<a NAME="E20:46_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A93</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c13">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="c6">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:46_1_1"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="txt" href="#E14:46_1_1"><span class="lab"><font color="Green" title="E42">A72</font></span></a></span>;<br/>
<a NAME="E21:46_1_1_5"/>
<font color="Maroon" title="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:46_1_1_5"><span class="lab"><font color="Green" title="E46">A76</font></span></a>, <a class="ref" href="vectsp_1.html#D17" title="VECTSP_1:def.17">VECTSP_1:def 17</a></span>;<br/>
<a NAME="E22:46_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c9">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c12">Z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:46_1_1_5"><span class="lab"><font color="Green" title="E47">A80</font></span></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E23:46_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c12">Z</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c8">L</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46_1_1_5"><span class="lab"><font color="Green" title="E45">A75</font></span></a>, <a class="ref" href="vectsp_5.html#T2" target="_self" title="VECTSP_5:th.2">Th6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E24:46_1_1_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:46_1_1"><span class="lab"><font color="Green" title="E41">A71</font></span></a>, <a class="txt" href="#E14:46_1_1"><span class="lab"><font color="Green" title="E42">A72</font></span></a>, <a class="txt" href="#E20:46_1_1_5"><span class="lab"><font color="Green" title="E51">A93</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">thus </span><a NAME="E17:46_1_1"/>
<font color="Maroon" title="c8">L</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:46_1_1"><span class="lab"><font color="Green" title="E43">A73</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
