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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">W1</font> being <a href="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/>
<b>consider </b><font color="Maroon" title="c4">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:46_1_1"/><i><font color="Green" title="E35">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="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>1</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E3:46_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> )
 
<b>by </b><i><a class="ref" href="vectsp_5.html#D3" target="_self" title="VECTSP_5:def.3">Def3</a>, <a class="ref" href="vectsp_5.html#T25" target="_self" title="VECTSP_5:th.25">Th25</a></i>;<br/>
<b>then reconsider </b><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>  <b>by </b><i><a class="txt" href="#E2:46_1_1"><i><font color="Green" title="E35">A1</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">W1</font>, <font color="Olive" title="b2">W2</font> being <a href="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/>
<b>consider </b><font color="Maroon" title="c6">R</font> being   <a href="relset_1.html#NM3" title="RELSET_1:NM.3">Relation</a> of <font color="Maroon" title="c5">X</font><b> such that </b><br/><a NAME="E6:46_1_1"/><i><font color="Green" title="E36">A2</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="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>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 <b>from </b><i><a class="ref" href="relset_1.html#S2" title="RELSET_1:sch.2">RELSET_1:sch 2</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">$1,$2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">R</font>;<br/>
<div><i><font color="Green" title="E37">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_1"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/><b>consider </b><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><b> such that </b><br/><a NAME="E2:46_1_1_1"/><i><font color="Green" title="E38">A4</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">W1</font> &amp; <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> )
 <b>by </b><i><a class="txt" href="#E2:46_1_1"><i><font color="Green" title="E35">A1</font></i></a></i>;<br/><a NAME="E3:46_1_1_1"/>
<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>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T32" title="VECTSP_4:th.32">VECTSP_4:32</a></i>;<br/><b>hence </b><a NAME="E4:46_1_1_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c7">x</font>]
 <b>by </b><i><a class="txt" href="#E6:46_1_1"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="#E2:46_1_1_1"><i><font color="Green" title="E38">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E38">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_2"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/><b>assume </b><b>that </b><br/><a NAME="E1:46_1_1_2"/><i><font color="Green" title="E39">A6</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>]
 <b>and </b><br/><a NAME="E2:46_1_1_2"/><i><font color="Green" title="E40">A7</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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="E3:46_1_1_2"/><i><font color="Green" title="E41">A8</font></i>: 
 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> )
 <b>by </b><i><a class="txt" href="#E6:46_1_1"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="#E1:46_1_1_2"><i><font color="Green" title="E39">A6</font></i></a></i>;<br/><a NAME="E4:46_1_1_2"/>
 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> )
 <b>by </b><i><a class="txt" href="#E6:46_1_1"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="#E2:46_1_1_2"><i><font color="Green" title="E40">A7</font></i></a></i>;<br/><b>hence </b><a NAME="E5:46_1_1_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font>
 <b>by </b><i><a class="txt" href="#E3:46_1_1_2"><i><font color="Green" title="E41">A8</font></i></a>, <a class="ref" href="vectsp_4.html#T33" title="VECTSP_4:th.33">VECTSP_4:33</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E39">A9</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_3"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/><b>assume </b><b>that </b><br/><a NAME="E1:46_1_1_3"/><i><font color="Green" title="E40">A10</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>]
 <b>and </b><br/><a NAME="E2:46_1_1_3"/><i><font color="Green" title="E41">A11</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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/><b>consider </b><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><b> such that </b><br/><a NAME="E4:46_1_1_3"/><i><font color="Green" title="E42">A12</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">W1</font> &amp; <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> )
 <b>by </b><i><a class="txt" href="#E6:46_1_1"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="#E1:46_1_1_3"><i><font color="Green" title="E40">A10</font></i></a></i>;<br/><b>consider </b><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><b> such that </b><br/><a NAME="E6:46_1_1_3"/><i><font color="Green" title="E43">A13</font></i>: 
( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">W3</font> &amp; <font color="Maroon" title="c9">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">W4</font> &amp; <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> )
 <b>by </b><i><a class="txt" href="#E6:46_1_1"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="#E2:46_1_1_3"><i><font color="Green" title="E41">A11</font></i></a></i>;<br/><a NAME="E7:46_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>
 <b>by </b><i><a class="txt" href="#E4:46_1_1_3"><i><font color="Green" title="E42">A12</font></i></a>, <a class="txt" href="#E6:46_1_1_3"><i><font color="Green" title="E43">A13</font></i></a>, <a class="ref" href="vectsp_4.html#T34" title="VECTSP_4:th.34">VECTSP_4:34</a></i>;<br/><b>hence </b><a NAME="E8:46_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>]
 <b>by </b><i><a class="txt" href="#E6:46_1_1"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="#E4:46_1_1_3"><i><font color="Green" title="E42">A12</font></i></a>, <a class="txt" href="#E6:46_1_1_3"><i><font color="Green" title="E43">A13</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E10:46_1_1"/><i><font color="Green" title="E40">A14</font></i>: 
for <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="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_4"><b>proof </b></a><div class="add">

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

<b>assume </b><b>that </b><br/><a NAME="E1:46_1_1_4"/><i><font color="Green" title="E41">A15</font></i>: 
<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>
 <b>and </b><br/><a NAME="E2:46_1_1_4"/><i><font color="Green" title="E42">A16</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="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:()"><i><font color="Red">::  thesis: </font></i></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_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_1_1_4_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_4_1_1_1"><b>suppose </b></a><a NAME="E1:46_1_1_4_1_1_1"/><i><font color="Green" title="E43">A17</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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"><b>consider </b><font color="Maroon" title="c8">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">X</font>;<br/><b>take </b><font color="Maroon" title="c9">y'</font> = <font color="Maroon" title="c8">y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">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">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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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">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/><b>assume </b><a NAME="E3:46_1_1_4_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:()"><i><font color="Red">::  thesis: </font></i></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">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/><b>hence </b><a NAME="E4:46_1_1_4_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">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>
 <b>by </b><i><a class="txt" href="#E1:46_1_1_4_1_1_1"><i><font color="Green" title="E43">A17</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:46_1_1_4_1_1_2"/><i><font color="Green" title="E43">A18</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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"><b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">W1</font> being <a href="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_4_1_1_2"/><i><font color="Green" title="E44">A20</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="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_4_1_1_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:46_1_1_4_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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>then consider </b><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><b> such that </b><br/><a NAME="E3:46_1_1_4_1_1_2_1"/><i><font color="Green" title="E45">A21</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">W1</font>
 <b>and </b><br/><a NAME="E4:46_1_1_4_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>
 <b>by </b><i><a class="txt" href="#E2:46_1_1"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="#E1:46_1_1_4"><i><font color="Green" title="E41">A15</font></i></a></i>;<br/>
<b>reconsider </b><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/>
<b>take </b>
<font color="Maroon" title="c10">y</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>take </b>
<font color="Maroon" title="c9">W1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>thus </b><a NAME="E6:46_1_1_4_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> )
 <b>by </b><i><a class="txt" href="#E3:46_1_1_4_1_1_2_1"><i><font color="Green" title="E45">A21</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon" title="c8">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E4:46_1_1_4_1_1_2"/><i><font color="Green" title="E45">A22</font></i>: 
 <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>
 <b>and </b><br/><a NAME="E5:46_1_1_4_1_1_2"/><i><font color="Green" title="E46">A23</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="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>]
 <b>from </b><i><a class="ref" href="classes1.html#S1" title="CLASSES1:sch.1">CLASSES1:sch 1</a>(<a class="txt" href="#E2:46_1_1_4_1_1_2"><i><font color="Green" title="E44">A20</font></i></a>);<br/></i><b>set </b><font color="Maroon" title="c9">Z</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">f</font></span>)</span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_4_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/><b>assume </b><a NAME="E1:46_1_1_4_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:()"><i><font color="Red">::  thesis: </font></i></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/><b>then consider </b><font color="Maroon" title="c11">Y'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:46_1_1_4_1_1_2_2"/><i><font color="Green" title="E47">A24</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y'</font>
 <b>and </b><br/><a NAME="E4:46_1_1_4_1_1_2_2"/><i><font color="Green" title="E48">A25</font></i>: 
<font color="Maroon" title="c11">Y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">f</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:46_1_1_4_1_1_2_2"/><i><font color="Green" title="E49">A26</font></i>: 
<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>
 <b>and </b><br/><a NAME="E7:46_1_1_4_1_1_2_2"/><i><font color="Green" title="E50">A27</font></i>: 
<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">Y'</font>
 <b>by </b><i><a class="txt" href="#E4:46_1_1_4_1_1_2_2"><i><font color="Green" title="E48">A25</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="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><b> such that </b><br/><a NAME="E9:46_1_1_4_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>
 <b>and </b><br/><a NAME="E10:46_1_1_4_1_1_2_2"/><i><font color="Green" title="E51">A28</font></i>: 
<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>
 <b>by </b><i><a class="txt" href="#E4:46_1_1_4_1_1_2"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="#E5:46_1_1_4_1_1_2"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="#E6:46_1_1_4_1_1_2_2"><i><font color="Green" title="E49">A26</font></i></a></i>;<br/><a NAME="E11:46_1_1_4_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> &amp; <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="c13">W1</font> )
 <b>by </b><i><a class="txt" href="#E3:46_1_1_4_1_1_2_2"><i><font color="Green" title="E47">A24</font></i></a>, <a class="txt" href="#E7:46_1_1_4_1_1_2_2"><i><font color="Green" title="E50">A27</font></i></a>, <a class="txt" href="#E10:46_1_1_4_1_1_2_2"><i><font color="Green" title="E51">A28</font></i></a>, <a class="ref" href="vectsp_4.html#D2" title="VECTSP_4:def.2">VECTSP_4:def 2</a></i>;<br/><b>hence </b><a NAME="E12:46_1_1_4_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>then reconsider </b><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> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E8:46_1_1_4_1_1_2"/><i><font color="Green" title="E47">A29</font></i>: 
 <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> 
 <b>by </b><i><a class="txt" href="#E1:46_1_1_4_1_1_2"><i><font color="Green" title="E43">A18</font></i></a>, <a class="txt" href="#E4:46_1_1_4_1_1_2"><i><font color="Green" title="E45">A22</font></i></a>, <a class="ref" href="relat_1.html#T65" title="RELAT_1:th.65">RELAT_1:65</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">f</font>;<br/><b>consider </b><font color="Maroon" title="c12">z1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:46_1_1_4_1_1_2"/><i><font color="Green" title="E48">A30</font></i>: 
<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>
 <b>and </b><br/><a NAME="E12:46_1_1_4_1_1_2"/><i><font color="Green" title="E49">A31</font></i>: 
<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> <font color="Maroon" title="c11">z</font>
 <b>by </b><i><a class="txt" href="#E8:46_1_1_4_1_1_2"><i><font color="Green" title="E47">A29</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c13">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><b> such that </b><br/><a NAME="E14:46_1_1_4_1_1_2"/><i><font color="Green" title="E50">A32</font></i>: 
( <font color="Maroon" title="c12">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">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="Maroon" title="c13">W3</font> )
 <b>by </b><i><a class="txt" href="#E4:46_1_1_4_1_1_2"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="#E5:46_1_1_4_1_1_2"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="#E11:46_1_1_4_1_1_2"><i><font color="Green" title="E48">A30</font></i></a></i>;<br/><a NAME="E15:46_1_1_4_1_1_2"/><i><font color="Green" title="E51">A33</font></i>: 
<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> 
 <b>by </b><i><a class="txt" href="#E8:46_1_1_4_1_1_2"><i><font color="Green" title="E47">A29</font></i></a>, <a class="txt" href="#E12:46_1_1_4_1_1_2"><i><font color="Green" title="E49">A31</font></i></a>, <a class="txt" href="#E14:46_1_1_4_1_1_2"><i><font color="Green" title="E50">A32</font></i></a>, <a class="ref" href="orders_1.html#T91" title="ORDERS_1:th.91">ORDERS_1:91</a></i>;<br/><a NAME="E16:46_1_1_4_1_1_2"/>
<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_4_1_1_2_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:46_1_1_4_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>
  <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></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_4_1_1_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">v1</font>, <font color="Maroon" title="c15">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c14">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="c15">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="c14">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c15">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:46_1_1_4_1_1_2_3_1"/><i><font color="Green" title="E52">A34</font></i>: 
<font color="Maroon" title="c14">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font>
 <b>and </b><br/><a NAME="E2:46_1_1_4_1_1_2_3_1"/><i><font color="Green" title="E53">A35</font></i>: 
<font color="Maroon" title="c15">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font></span><br/>

<b>consider </b><font color="Maroon" title="c16">Y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:46_1_1_4_1_1_2_3_1"/><i><font color="Green" title="E54">A36</font></i>: 
<font color="Maroon" title="c14">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Y1</font>
 <b>and </b><br/><a NAME="E5:46_1_1_4_1_1_2_3_1"/><i><font color="Green" title="E55">A37</font></i>: 
<font color="Maroon" title="c16">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">f</font>
 <b>by </b><i><a class="txt" href="#E1:46_1_1_4_1_1_2_3_1"><i><font color="Green" title="E52">A34</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">Y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:46_1_1_4_1_1_2_3_1"/><i><font color="Green" title="E56">A38</font></i>: 
<font color="Maroon" title="c15">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">Y2</font>
 <b>and </b><br/><a NAME="E8:46_1_1_4_1_1_2_3_1"/><i><font color="Green" title="E57">A39</font></i>: 
<font color="Maroon" title="c17">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>
 <b>by </b><i><a class="txt" href="#E2:46_1_1_4_1_1_2_3_1"><i><font color="Green" title="E53">A35</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:46_1_1_4_1_1_2_3_1"/><i><font color="Green" title="E58">A40</font></i>: 
<font color="Maroon" title="c18">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>
 <b>and </b><br/><a NAME="E11:46_1_1_4_1_1_2_3_1"/><i><font color="Green" title="E59">A41</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">Y1</font>
 <b>by </b><i><a class="txt" href="#E5:46_1_1_4_1_1_2_3_1"><i><font color="Green" title="E55">A37</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c19">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E13:46_1_1_4_1_1_2_3_1"/><i><font color="Green" title="E60">A42</font></i>: 
<font color="Maroon" title="c19">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>
 <b>and </b><br/><a NAME="E14:46_1_1_4_1_1_2_3_1"/><i><font color="Green" title="E61">A43</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">Y2</font>
 <b>by </b><i><a class="txt" href="#E8:46_1_1_4_1_1_2_3_1"><i><font color="Green" title="E57">A39</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c20">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><b> such that </b><br/><a NAME="E16:46_1_1_4_1_1_2_3_1"/><i><font color="Green" title="E62">A44</font></i>: 
<font color="Maroon" title="c18">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">W1</font>
 <b>and </b><br/><a NAME="E17:46_1_1_4_1_1_2_3_1"/><i><font color="Green" title="E63">A45</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">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="c20">W1</font>
 <b>by </b><i><a class="txt" href="#E4:46_1_1_4_1_1_2"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="#E5:46_1_1_4_1_1_2"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="#E10:46_1_1_4_1_1_2_3_1"><i><font color="Green" title="E58">A40</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c21">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><b> such that </b><br/><a NAME="E19:46_1_1_4_1_1_2_3_1"/><i><font color="Green" title="E64">A46</font></i>: 
<font color="Maroon" title="c19">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">W2</font>
 <b>and </b><br/><a NAME="E20:46_1_1_4_1_1_2_3_1"/><i><font color="Green" title="E65">A47</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">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="c21">W2</font>
 <b>by </b><i><a class="txt" href="#E4:46_1_1_4_1_1_2"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="#E5:46_1_1_4_1_1_2"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="#E13:46_1_1_4_1_1_2_3_1"><i><font color="Green" title="E60">A42</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c22">y1</font> = <font color="Maroon" title="c18">y1</font>, <font color="Maroon" title="c23">y2</font> = <font color="Maroon" title="c19">y2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">X</font> <b>by </b><i><a class="txt" href="#E1:46_1_1_4"><i><font color="Green" title="E41">A15</font></i></a>, <a class="txt" href="#E4:46_1_1_4_1_1_2"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="#E10:46_1_1_4_1_1_2_3_1"><i><font color="Green" title="E58">A40</font></i></a>, <a class="txt" href="#E13:46_1_1_4_1_1_2_3_1"><i><font color="Green" title="E60">A42</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_4_1_1_2_3_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_4_1_1_2_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_1_1_4_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="c22">y1</font>,<font color="Maroon" title="c23">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="c23">y2</font>,<font color="Maroon" title="c22">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><b>by </b><i><a class="txt" href="#E2:46_1_1_4"><i><font color="Green" title="E42">A16</font></i></a>, <a class="txt" href="#E4:46_1_1_4_1_1_2"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="#E10:46_1_1_4_1_1_2_3_1"><i><font color="Green" title="E58">A40</font></i></a>, <a class="txt" href="#E13:46_1_1_4_1_1_2_3_1"><i><font color="Green" title="E60">A42</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_4_1_1_2_3_1_1_1_1"><b>suppose </b></a><a NAME="E1:46_1_1_4_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="c22">y1</font>,<font color="Maroon" title="c23">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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> <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_4_1_1_2_3_1_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">W3</font>, <font color="Olive" title="b2">W4</font> being <a href="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="c22">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W3</font> &amp; <font color="Maroon" title="c23">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> )
 <b>by </b><i><a class="txt" href="#E6:46_1_1"><i><font color="Green" title="E36">A2</font></i></a></i>;<br/><a NAME="E3:46_1_1_4_1_1_2_3_1_1_1_1"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c20">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="c21">W2</font>
 <b>by </b><i><a class="txt" href="#E16:46_1_1_4_1_1_2_3_1"><i><font color="Green" title="E62">A44</font></i></a>, <a class="txt" href="#E19:46_1_1_4_1_1_2_3_1"><i><font color="Green" title="E64">A46</font></i></a>, <a class="ref" href="vectsp_4.html#D2" title="VECTSP_4:def.2">VECTSP_4:def 2</a></i>;<br/><a NAME="E4:46_1_1_4_1_1_2_3_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c14">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c21">W2</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="c21">W2</font> )
 <b>by </b><i><a class="txt" href="#E4:46_1_1_4_1_1_2_3_1"><i><font color="Green" title="E54">A36</font></i></a>, <a class="txt" href="#E7:46_1_1_4_1_1_2_3_1"><i><font color="Green" title="E56">A38</font></i></a>, <a class="txt" href="#E11:46_1_1_4_1_1_2_3_1"><i><font color="Green" title="E59">A41</font></i></a>, <a class="txt" href="#E14:46_1_1_4_1_1_2_3_1"><i><font color="Green" title="E61">A43</font></i></a>, <a class="txt" href="#E17:46_1_1_4_1_1_2_3_1"><i><font color="Green" title="E63">A45</font></i></a>, <a class="txt" href="#E20:46_1_1_4_1_1_2_3_1"><i><font color="Green" title="E65">A47</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><a NAME="E5:46_1_1_4_1_1_2_3_1_1_1_1"/><b>then </b>
<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> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c21">W2</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T28" title="VECTSP_4:th.28">VECTSP_4:28</a></i>;<br/><a NAME="E6:46_1_1_4_1_1_2_3_1_1_1_1"/><b>then </b><i><font color="Green" title="E66">A48</font></i>: 
<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> <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="c21">W2</font>
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><a NAME="E7:46_1_1_4_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="c23">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>
 <b>by </b><i><a class="txt" href="#E13:46_1_1_4_1_1_2_3_1"><i><font color="Green" title="E60">A42</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E8:46_1_1_4_1_1_2_3_1_1_1_1"/>
<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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font>
 <b>by </b><i><a class="txt" href="#E20:46_1_1_4_1_1_2_3_1"><i><font color="Green" title="E65">A47</font></i></a>, <a class="txt" href="#E6:46_1_1_4_1_1_2_3_1_1_1_1"><i><font color="Green" title="E66">A48</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_4_1_1_2_3_1_1_1_2"><b>suppose </b></a><a NAME="E1:46_1_1_4_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="c23">y2</font>,<font color="Maroon" title="c22">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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> <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_4_1_1_2_3_1_1_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">W3</font>, <font color="Olive" title="b2">W4</font> being <a href="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="c23">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W3</font> &amp; <font color="Maroon" title="c22">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> )
 <b>by </b><i><a class="txt" href="#E6:46_1_1"><i><font color="Green" title="E36">A2</font></i></a></i>;<br/><a NAME="E3:46_1_1_4_1_1_2_3_1_1_1_2"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c21">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="c20">W1</font>
 <b>by </b><i><a class="txt" href="#E16:46_1_1_4_1_1_2_3_1"><i><font color="Green" title="E62">A44</font></i></a>, <a class="txt" href="#E19:46_1_1_4_1_1_2_3_1"><i><font color="Green" title="E64">A46</font></i></a>, <a class="ref" href="vectsp_4.html#D2" title="VECTSP_4:def.2">VECTSP_4:def 2</a></i>;<br/><a NAME="E4:46_1_1_4_1_1_2_3_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c14">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c20">W1</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="c20">W1</font> )
 <b>by </b><i><a class="txt" href="#E4:46_1_1_4_1_1_2_3_1"><i><font color="Green" title="E54">A36</font></i></a>, <a class="txt" href="#E7:46_1_1_4_1_1_2_3_1"><i><font color="Green" title="E56">A38</font></i></a>, <a class="txt" href="#E11:46_1_1_4_1_1_2_3_1"><i><font color="Green" title="E59">A41</font></i></a>, <a class="txt" href="#E14:46_1_1_4_1_1_2_3_1"><i><font color="Green" title="E61">A43</font></i></a>, <a class="txt" href="#E17:46_1_1_4_1_1_2_3_1"><i><font color="Green" title="E63">A45</font></i></a>, <a class="txt" href="#E20:46_1_1_4_1_1_2_3_1"><i><font color="Green" title="E65">A47</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><a NAME="E5:46_1_1_4_1_1_2_3_1_1_1_2"/><b>then </b>
<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> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c20">W1</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T28" title="VECTSP_4:th.28">VECTSP_4:28</a></i>;<br/><a NAME="E6:46_1_1_4_1_1_2_3_1_1_1_2"/><b>then </b><i><font color="Green" title="E66">A49</font></i>: 
<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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c20">W1</font>
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><a NAME="E7:46_1_1_4_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="c22">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>
 <b>by </b><i><a class="txt" href="#E10:46_1_1_4_1_1_2_3_1"><i><font color="Green" title="E58">A40</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E8:46_1_1_4_1_1_2_3_1_1_1_2"/>
<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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font>
 <b>by </b><i><a class="txt" href="#E17:46_1_1_4_1_1_2_3_1"><i><font color="Green" title="E63">A45</font></i></a>, <a class="txt" href="#E6:46_1_1_4_1_1_2_3_1_1_1_2"><i><font color="Green" title="E66">A49</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E23:46_1_1_4_1_1_2_3_1"/>
<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> <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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>let </b><font color="Maroon" title="c14">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:()"><i><font color="Red">::  thesis: </font></i></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="c14">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/><b>let </b><font color="Maroon" title="c15">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c15">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="c14">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c15">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font> )</span><br/>



<b>assume </b><a NAME="E2:46_1_1_4_1_1_2_3"/>
<font color="Maroon" title="c15">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c15">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font></span><br/>

<b>then consider </b><font color="Maroon" title="c16">Y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:46_1_1_4_1_1_2_3"/><i><font color="Green" title="E52">A50</font></i>: 
<font color="Maroon" title="c15">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Y1</font>
 <b>and </b><br/><a NAME="E5:46_1_1_4_1_1_2_3"/><i><font color="Green" title="E53">A51</font></i>: 
<font color="Maroon" title="c16">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">f</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:46_1_1_4_1_1_2_3"/><i><font color="Green" title="E54">A52</font></i>: 
<font color="Maroon" title="c17">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">f</font>
 <b>and </b><br/><a NAME="E8:46_1_1_4_1_1_2_3"/><i><font color="Green" title="E55">A53</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">Y1</font>
 <b>by </b><i><a class="txt" href="#E5:46_1_1_4_1_1_2_3"><i><font color="Green" title="E53">A51</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">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><b> such that </b><br/><a NAME="E10:46_1_1_4_1_1_2_3"/>
<font color="Maroon" title="c17">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">W1</font>
 <b>and </b><br/><a NAME="E11:46_1_1_4_1_1_2_3"/><i><font color="Green" title="E56">A54</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c18">W1</font>
 <b>by </b><i><a class="txt" href="#E4:46_1_1_4_1_1_2"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="#E5:46_1_1_4_1_1_2"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="#E7:46_1_1_4_1_1_2_3"><i><font color="Green" title="E54">A52</font></i></a></i>;<br/>
<a NAME="E12:46_1_1_4_1_1_2_3"/>
<font color="Maroon" title="c15">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c18">W1</font>
 
<b>by </b><i><a class="txt" href="#E4:46_1_1_4_1_1_2_3"><i><font color="Green" title="E52">A50</font></i></a>, <a class="txt" href="#E8:46_1_1_4_1_1_2_3"><i><font color="Green" title="E55">A53</font></i></a>, <a class="txt" href="#E11:46_1_1_4_1_1_2_3"><i><font color="Green" title="E56">A54</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E13:46_1_1_4_1_1_2_3"/><b>then </b>
<font color="Maroon" title="c14">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c15">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c18">W1</font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T29" title="VECTSP_4:th.29">VECTSP_4:29</a></i>;<br/>
<a NAME="E14:46_1_1_4_1_1_2_3"/><b>then </b><i><font color="Green" title="E57">A55</font></i>: 
<font color="Maroon" title="c14">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c15">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="c18">W1</font>
 
<b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E15:46_1_1_4_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="c17">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>
 
<b>by </b><i><a class="txt" href="#E7:46_1_1_4_1_1_2_3"><i><font color="Green" title="E54">A52</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E16:46_1_1_4_1_1_2_3"/>
<font color="Maroon" title="c14">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c15">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font>
 <b>by </b><i><a class="txt" href="#E11:46_1_1_4_1_1_2_3"><i><font color="Green" title="E56">A54</font></i></a>, <a class="txt" href="#E14:46_1_1_4_1_1_2_3"><i><font color="Green" title="E57">A55</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c14">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><b> such that </b><br/><a NAME="E18:46_1_1_4_1_1_2"/><i><font color="Green" title="E52">A56</font></i>: 
<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="c14">E</font>
 <b>by </b><i><a class="txt" href="#E15:46_1_1_4_1_1_2"><i><font color="Green" title="E51">A33</font></i></a>, <a class="ref" href="vectsp_4.html#T42" title="VECTSP_4:th.42">VECTSP_4:42</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_4_1_1_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c15">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="c14">E</font> implies <font color="Maroon" title="c15">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="c15">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="c15">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="c14">E</font> ) )</span><br/><b>thus </b><a NAME="E1:46_1_1_4_1_1_2_4"/>
( <font color="Maroon" title="c15">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="c14">E</font> implies <font color="Maroon" title="c15">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c15">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="c15">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="c14">E</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_4_1_1_2_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:46_1_1_4_1_1_2_4_1"/><i><font color="Green" title="E53">A57</font></i>: 
<font color="Maroon" title="c15">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="c14">E</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c15">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_4_1_1_2_4_1"/><b>then </b>
<font color="Maroon" title="c15">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c14">E</font>
 
<b>by </b><i><a class="ref" href="vectsp_5.html#T7" target="_self" title="VECTSP_5:th.7">Th7</a></i>;<br/>
<a NAME="E3:46_1_1_4_1_1_2_4_1"/><b>then </b>
<font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font>
 
<b>by </b><i><a class="txt" href="#E18:46_1_1_4_1_1_2"><i><font color="Green" title="E52">A56</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c16">Y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:46_1_1_4_1_1_2_4_1"/><i><font color="Green" title="E54">A58</font></i>: 
<font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Y1</font>
 <b>and </b><br/><a NAME="E6:46_1_1_4_1_1_2_4_1"/><i><font color="Green" title="E55">A59</font></i>: 
<font color="Maroon" title="c16">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">f</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:46_1_1_4_1_1_2_4_1"/><i><font color="Green" title="E56">A60</font></i>: 
<font color="Maroon" title="c17">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">f</font>
 <b>and </b><br/><a NAME="E9:46_1_1_4_1_1_2_4_1"/><i><font color="Green" title="E57">A61</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">Y1</font>
 <b>by </b><i><a class="txt" href="#E6:46_1_1_4_1_1_2_4_1"><i><font color="Green" title="E55">A59</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">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><b> such that </b><br/><a NAME="E11:46_1_1_4_1_1_2_4_1"/><i><font color="Green" title="E58">A62</font></i>: 
<font color="Maroon" title="c17">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">W1</font>
 <b>and </b><br/><a NAME="E12:46_1_1_4_1_1_2_4_1"/><i><font color="Green" title="E59">A63</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c18">W1</font>
 <b>by </b><i><a class="txt" href="#E4:46_1_1_4_1_1_2"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="#E5:46_1_1_4_1_1_2"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="#E8:46_1_1_4_1_1_2_4_1"><i><font color="Green" title="E56">A60</font></i></a></i>;<br/>
<a NAME="E13:46_1_1_4_1_1_2_4_1"/><i><font color="Green" title="E60">A64</font></i>: 
( <font color="Maroon" title="c15">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c18">W1</font> &amp; <font color="Maroon" title="c15">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W</font> )
 
<b>by </b><i><a class="txt" href="#E1:46_1_1_4_1_1_2_4_1"><i><font color="Green" title="E53">A57</font></i></a>, <a class="txt" href="#E5:46_1_1_4_1_1_2_4_1"><i><font color="Green" title="E54">A58</font></i></a>, <a class="txt" href="#E9:46_1_1_4_1_1_2_4_1"><i><font color="Green" title="E57">A61</font></i></a>, <a class="txt" href="#E12:46_1_1_4_1_1_2_4_1"><i><font color="Green" title="E59">A63</font></i></a>, <a class="ref" href="vectsp_5.html#T7" target="_self" title="VECTSP_5:th.7">Th7</a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E14:46_1_1_4_1_1_2_4_1"/>
 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="c17">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> )
 
<b>by </b><i><a class="txt" href="#E2:46_1_1"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="#E1:46_1_1_4"><i><font color="Green" title="E41">A15</font></i></a>, <a class="txt" href="#E4:46_1_1_4_1_1_2"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="#E8:46_1_1_4_1_1_2_4_1"><i><font color="Green" title="E56">A60</font></i></a></i>;<br/>
<b>hence </b><a NAME="E15:46_1_1_4_1_1_2_4_1"/>
<font color="Maroon" title="c15">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>
 <b>by </b><i><a class="txt" href="#E11:46_1_1_4_1_1_2_4_1"><i><font color="Green" title="E58">A62</font></i></a>, <a class="txt" href="#E13:46_1_1_4_1_1_2_4_1"><i><font color="Green" title="E60">A64</font></i></a>, <a class="ref" href="vectsp_5.html#T7" target="_self" title="VECTSP_5:th.7">Th7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:46_1_1_4_1_1_2_4"/>
<font color="Maroon" title="c15">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c15">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="c14">E</font></span><br/><a NAME="E3:46_1_1_4_1_1_2_4"/><b>then </b>
<font color="Maroon" title="c15">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>
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><a NAME="E4:46_1_1_4_1_1_2_4"/><b>then </b>
<font color="Maroon" title="c15">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>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D3" title="VECTSP_4:def.3">VECTSP_4:def 3</a></i>;<br/><a NAME="E5:46_1_1_4_1_1_2_4"/><b>then </b>
<font color="Maroon" title="c15">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>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:46_1_1_4_1_1_2_4"/>
<font color="Maroon" title="c15">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="c14">E</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T25" title="VECTSP_4:th.25">VECTSP_4:25</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E20:46_1_1_4_1_1_2"/><b>then </b><i><font color="Green" title="E53">A65</font></i>: 
<font color="Maroon" title="c3">W</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c14">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>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T38" title="VECTSP_4:th.38">VECTSP_4:38</a></i>;<br/><a NAME="E21:46_1_1_4_1_1_2"/>
<font color="Maroon" title="c14">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>
 <b>by </b><i><a class="ref" href="vectsp_5.html#D3" target="_self" title="VECTSP_5:def.3">Def3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c15">y'</font> = <font color="Maroon" title="c14">E</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">X</font> <b>by </b><i><a class="txt" href="#E2:46_1_1"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="#E20:46_1_1_4_1_1_2"><i><font color="Green" title="E53">A65</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c16">y</font> = <font color="Maroon" title="c15">y'</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">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="c16">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/><b>let </b><font color="Maroon" title="c17">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c17">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="c17">x</font>,<font color="Maroon" title="c16">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/><b>assume </b><a NAME="E23:46_1_1_4_1_1_2"/><i><font color="Green" title="E54">A66</font></i>: 
<font color="Maroon" title="c17">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:()"><i><font color="Red">::  thesis: </font></i></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="c17">x</font>,<font color="Maroon" title="c16">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/><b>then consider </b><font color="Maroon" title="c18">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><b> such that </b><br/><a NAME="E25:46_1_1_4_1_1_2"/><i><font color="Green" title="E55">A67</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">W1</font>
 <b>and </b><br/><a NAME="E26:46_1_1_4_1_1_2"/><i><font color="Green" title="E56">A68</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">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="c18">W1</font>
 <b>by </b><i><a class="txt" href="#E5:46_1_1_4_1_1_2"><i><font color="Green" title="E46">A23</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_4_1_1_2_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c19">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c19">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c18">W1</font> implies <font color="Maroon" title="c19">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c14">E</font> )</span><br/><b>assume </b><a NAME="E1:46_1_1_4_1_1_2_5"/>
<font color="Maroon" title="c19">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c18">W1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c19">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c14">E</font></span><br/><a NAME="E2:46_1_1_4_1_1_2_5"/><b>then </b><i><font color="Green" title="E57">A69</font></i>: 
<font color="Maroon" title="c19">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="c18">W1</font>
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><a NAME="E3:46_1_1_4_1_1_2_5"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c18">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>
 <b>by </b><i><a class="txt" href="#E4:46_1_1_4_1_1_2"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="#E23:46_1_1_4_1_1_2"><i><font color="Green" title="E54">A66</font></i></a>, <a class="txt" href="#E26:46_1_1_4_1_1_2"><i><font color="Green" title="E56">A68</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:46_1_1_4_1_1_2_5"/><b>then </b>
<font color="Maroon" title="c19">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font>
 <b>by </b><i><a class="txt" href="#E2:46_1_1_4_1_1_2_5"><i><font color="Green" title="E57">A69</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E5:46_1_1_4_1_1_2_5"/>
<font color="Maroon" title="c19">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c14">E</font>
 <b>by </b><i><a class="txt" href="#E18:46_1_1_4_1_1_2"><i><font color="Green" title="E52">A56</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E28:46_1_1_4_1_1_2"/><b>then </b>
<font color="Maroon" title="c18">W1</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c14">E</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T36" title="VECTSP_4:th.36">VECTSP_4:36</a></i>;<br/><b>hence </b><a NAME="E29:46_1_1_4_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="c17">x</font>,<font color="Maroon" title="c16">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>
 <b>by </b><i><a class="txt" href="#E6:46_1_1"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="#E25:46_1_1_4_1_1_2"><i><font color="Green" title="E55">A67</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:46_1_1_4"/>
 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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>consider </b><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><b> such that </b><br/><a NAME="E12:46_1_1"/><i><font color="Green" title="E41">A70</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="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>]
 <b>from </b><i><a class="ref" href="orders_1.html#S1" title="ORDERS_1:sch.1">ORDERS_1:sch 1</a>(<a class="txt" href="#E7:46_1_1"><i><font color="Green" title="E37">A3</font></i></a>, <a class="txt" href="#E8:46_1_1"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="#E9:46_1_1"><i><font color="Green" title="E39">A9</font></i></a>, <a class="txt" href="#E10:46_1_1"><i><font color="Green" title="E40">A14</font></i></a>);<br/></i>
<b>consider </b><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><b> such that </b><br/><a NAME="E14:46_1_1"/><i><font color="Green" title="E42">A71</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L</font>
 <b>and </b><br/><a NAME="E15:46_1_1"/><i><font color="Green" title="E43">A72</font></i>: 
<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>
 <b>by </b><i><a class="txt" href="#E2:46_1_1"><i><font color="Green" title="E35">A1</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">L</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>thus </b><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">U7</a> of <font color="Maroon" title="c2">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</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>
  <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></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"><b>proof </b></a><div class="add">

<b>assume </b><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">U7</a> of <font color="Maroon" title="c2">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><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><b> such that </b><br/><a NAME="E3:46_1_1_5"/><i><font color="Green" title="E44">A73</font></i>: 
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> )
 <b>by </b><i><a class="txt" href="vectsp_5.html#E64"><i><font color="Green" title="E34">Lm16</font></i></a></i>;<br/>
<a NAME="E4:46_1_1_5"/><i><font color="Green" title="E45">A74</font></i>: 
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>
 
<b>by </b><i><a class="txt" href="#E3:46_1_1_5"><i><font color="Green" title="E44">A73</font></i></a>, <a class="ref" href="vectsp_5.html#T5" target="_self" title="VECTSP_5:th.5">Th5</a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">A</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="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="E5:46_1_1_5"/><i><font color="Green" title="E46">A75</font></i>: 
<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"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/><b>assume </b><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:()"><i><font color="Red">::  thesis: </font></i></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"/><b>then </b>
 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/><b>hence </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>then reconsider </b><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> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E8:46_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"><b>proof </b></a><div class="add">

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

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

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

<b>then consider </b><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><b> such that </b><br/><a NAME="E3:46_1_1_5_2_1"/><i><font color="Green" title="E47">A76</font></i>: 
<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/>
<b>assume </b><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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>then consider </b><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><b> such that </b><br/><a NAME="E6:46_1_1_5_2_1"/><i><font color="Green" title="E48">A77</font></i>: 
<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>
 
<b>by </b><i><a class="txt" href="#E3:46_1_1_5_2_1"><i><font color="Green" title="E47">A76</font></i></a>, <a class="txt" href="#E6:46_1_1_5_2_1"><i><font color="Green" title="E48">A77</font></i></a>, <a class="ref" href="vectsp_1.html#D26" title="VECTSP_1:def.26">VECTSP_1:def 26</a></i>;<br/>
<b>hence </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

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



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

<b>then consider </b><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><b> such that </b><br/><a NAME="E4:46_1_1_5_2"/><i><font color="Green" title="E47">A78</font></i>: 
<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>
 
<b>by </b><i><a class="txt" href="#E4:46_1_1_5_2"><i><font color="Green" title="E47">A78</font></i></a>, <a class="ref" href="vectsp_1.html#D26" title="VECTSP_1:def.26">VECTSP_1:def 26</a></i>;<br/>
<b>hence </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then consider </b><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><b> such that </b><br/><a NAME="E10:46_1_1_5"/><i><font color="Green" title="E47">A79</font></i>: 
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>
 <b>by </b><i><a class="txt" href="#E5:46_1_1_5"><i><font color="Green" title="E46">A75</font></i></a>, <a class="ref" href="vectsp_4.html#T42" title="VECTSP_4:th.42">VECTSP_4:42</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_5_3"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/><b>thus </b><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:()"><i><font color="Red">::  thesis: </font></i></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"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:46_1_1_5_3_1"/><i><font color="Green" title="E48">A80</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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"/><b>then </b>
<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>
 
<b>by </b><i><a class="ref" href="vectsp_5.html#T7" target="_self" title="VECTSP_5:th.7">Th7</a></i>;<br/>
<a NAME="E3:46_1_1_5_3_1"/><b>then </b>
<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>
 
<b>by </b><i><a class="txt" href="#E10:46_1_1_5"><i><font color="Green" title="E47">A79</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<b>then consider </b><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><b> such that </b><br/><a NAME="E5:46_1_1_5_3_1"/><i><font color="Green" title="E49">A81</font></i>: 
<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"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:46_1_1_5_3_1_1"/><i><font color="Green" title="E50">A82</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:46_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>
 <b>by </b><i><a class="txt" href="#E1:46_1_1_5_3_1"><i><font color="Green" title="E48">A80</font></i></a>, <a class="ref" href="vectsp_5.html#T7" target="_self" title="VECTSP_5:th.7">Th7</a></i>;<br/><a NAME="E3:46_1_1_5_3_1_1"/><b>then </b>
<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>
 <b>by </b><i><a class="txt" href="#E5:46_1_1_5_3_1"><i><font color="Green" title="E49">A81</font></i></a>, <a class="ref" href="vectsp_4.html#T29" title="VECTSP_4:th.29">VECTSP_4:29</a></i>;<br/><a NAME="E4:46_1_1_5_3_1_1"/><b>then </b>
<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>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D26" title="VECTSP_1:def.26">VECTSP_1:def 26</a></i>;<br/><a NAME="E5:46_1_1_5_3_1_1"/><b>then </b>
<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>
 <b>by </b><i><a class="txt" href="#E1:46_1_1_5_3_1_1"><i><font color="Green" title="E50">A82</font></i></a>, <a class="ref" href="vectsp_1.html#D22" title="VECTSP_1:def.22">VECTSP_1:def 22</a></i>;<br/><a NAME="E6:46_1_1_5_3_1_1"/><b>then </b>
<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>
 <b>by </b><i><a class="txt" href="vectsp_5.html#E2"><i><font color="Green" title="E2">Lm1</font></i></a></i>;<br/><b>hence </b><a NAME="E7:46_1_1_5_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E4:46_1_1_5"><i><font color="Green" title="E45">A74</font></i></a>, <a class="ref" href="vectsp_1.html#D26" title="VECTSP_1:def.26">VECTSP_1:def 26</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E7:46_1_1_5_3_1"/><b>then </b>
<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>
 
<b>by </b><i><a class="txt" href="#E5:46_1_1_5_3_1"><i><font color="Green" title="E49">A81</font></i></a>, <a class="ref" href="vectsp_1.html#T59" title="VECTSP_1:th.59">VECTSP_1:59</a></i>;<br/>
<b>hence </b><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>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T25" title="VECTSP_4:th.25">VECTSP_4:25</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>assume </b><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:()"><i><font color="Red">::  thesis: </font></i></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"/><b>then </b>
<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>
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><a NAME="E4:46_1_1_5_3"/><b>then </b>
<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>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D3" title="VECTSP_4:def.3">VECTSP_4:def 3</a></i>;<br/><a NAME="E5:46_1_1_5_3"/><b>then </b>
<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>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:46_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>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T25" title="VECTSP_4:th.25">VECTSP_4:25</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E12:46_1_1_5"/><b>then </b><i><font color="Green" title="E48">A83</font></i>: 
<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>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T38" title="VECTSP_4:th.38">VECTSP_4:38</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_5_4"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/><b>thus </b><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:()"><i><font color="Red">::  thesis: </font></i></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"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:46_1_1_5_4_1"/><i><font color="Green" title="E49">A84</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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"/><b>then </b>
<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>
 
<b>by </b><i><a class="ref" href="vectsp_5.html#T7" target="_self" title="VECTSP_5:th.7">Th7</a></i>;<br/>
<b>then consider </b><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><b> such that </b><br/><a NAME="E4:46_1_1_5_4_1"/><i><font color="Green" title="E50">A85</font></i>: 
<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>
 <b>and </b><br/><a NAME="E5:46_1_1_5_4_1"/><i><font color="Green" title="E51">A86</font></i>: 
<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>
 <b>and </b><br/><a NAME="E6:46_1_1_5_4_1"/><i><font color="Green" title="E52">A87</font></i>: 
<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>
 <b>by </b><i><a class="ref" href="vectsp_5.html#T5" target="_self" title="VECTSP_5:th.5">Th5</a></i>;<br/>
<a NAME="E7:46_1_1_5_4_1"/><i><font color="Green" title="E53">A88</font></i>: 
<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="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c15">v2</font>
 
<b>by </b><i><a class="txt" href="#E6:46_1_1_5_4_1"><i><font color="Green" title="E52">A87</font></i></a>, <a class="ref" href="vectsp_2.html#T22" title="VECTSP_2:th.22">VECTSP_2:22</a></i>;<br/>
<a NAME="E8:46_1_1_5_4_1"/><i><font color="Green" title="E54">A89</font></i>: 
<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>
 
<b>by </b><i><a class="txt" href="#E1:46_1_1_5_4_1"><i><font color="Green" title="E49">A84</font></i></a>, <a class="ref" href="vectsp_5.html#T7" target="_self" title="VECTSP_5:th.7">Th7</a></i>;<br/>
<a NAME="E9:46_1_1_5_4_1"/><b>then </b>
( <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> &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> )
 
<b>by </b><i><a class="txt" href="#E5:46_1_1_5_4_1"><i><font color="Green" title="E51">A86</font></i></a>, <a class="ref" href="vectsp_5.html#T6" target="_self" title="VECTSP_5:th.6">Th6</a></i>;<br/>
<a NAME="E10:46_1_1_5_4_1"/><b>then </b>
<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>
 
<b>by </b><i><a class="txt" href="#E7:46_1_1_5_4_1"><i><font color="Green" title="E53">A88</font></i></a>, <a class="ref" href="vectsp_4.html#T31" title="VECTSP_4:th.31">VECTSP_4:31</a></i>;<br/>
<a NAME="E11:46_1_1_5_4_1"/><b>then </b>
<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>
 
<b>by </b><i><a class="txt" href="#E4:46_1_1_5_4_1"><i><font color="Green" title="E50">A85</font></i></a>, <a class="ref" href="vectsp_5.html#T7" target="_self" title="VECTSP_5:th.7">Th7</a></i>;<br/>
<a NAME="E12:46_1_1_5_4_1"/><b>then </b>
<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>
 
<b>by </b><i><a class="txt" href="#E12:46_1_1_5"><i><font color="Green" title="E48">A83</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E13:46_1_1_5_4_1"/><b>then </b>
<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>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#D3" title="VECTSP_4:def.3">VECTSP_4:def 3</a></i>;<br/>
<a NAME="E14:46_1_1_5_4_1"/><b>then </b>
<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>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E15:46_1_1_5_4_1"/><b>then </b>
<font color="Maroon" title="c15">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">u</font>
 
<b>by </b><i><a class="txt" href="#E6:46_1_1_5_4_1"><i><font color="Green" title="E52">A87</font></i></a>, <a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>;<br/>
<b>hence </b><a NAME="E16:46_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>
 <b>by </b><i><a class="txt" href="#E15:46_1_1"><i><font color="Green" title="E43">A72</font></i></a>, <a class="txt" href="#E5:46_1_1_5_4_1"><i><font color="Green" title="E51">A86</font></i></a>, <a class="txt" href="#E8:46_1_1_5_4_1"><i><font color="Green" title="E54">A89</font></i></a>, <a class="ref" href="vectsp_5.html#T7" target="_self" title="VECTSP_5:th.7">Th7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>assume </b><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:()"><i><font color="Red">::  thesis: </font></i></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"/><b>then </b>
<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>
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><a NAME="E4:46_1_1_5_4"/><b>then </b>
<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>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D3" title="VECTSP_4:def.3">VECTSP_4:def 3</a></i>;<br/><a NAME="E5:46_1_1_5_4"/><b>then </b>
<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>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:46_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>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T25" title="VECTSP_4:th.25">VECTSP_4:25</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E14:46_1_1_5"/><b>then </b>
( <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> &amp; <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> )
 
<b>by </b><i><a class="ref" href="vectsp_5.html#D3" target="_self" title="VECTSP_5:def.3">Def3</a>, <a class="ref" href="vectsp_4.html#T38" title="VECTSP_4:th.38">VECTSP_4:38</a></i>;<br/>
<b>then reconsider </b><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> <b>by </b><i><a class="txt" href="#E2:46_1_1"><i><font color="Green" title="E35">A1</font></i></a></i>;<br/>
<a NAME="E16:46_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; <font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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> &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> &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="c8">L</font> )
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a>, <a class="ref" href="vectsp_4.html#T25" title="VECTSP_4:th.25">VECTSP_4:25</a></i>;<br/>
<a NAME="E17:46_1_1_5"/><b>then </b><i><font color="Green" title="E49">A90</font></i>: 
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>
 
<b>by </b><i><a class="txt" href="#E3:46_1_1_5"><i><font color="Green" title="E44">A73</font></i></a></i>;<br/>
<a NAME="E18:46_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>
 
<b>by </b><i><a class="txt" href="#E5:46_1_1_5"><i><font color="Green" title="E46">A75</font></i></a>, <a class="ref" href="vectsp_1.html#D26" title="VECTSP_1:def.26">VECTSP_1:def 26</a></i>;<br/>
<a NAME="E19:46_1_1_5"/><b>then </b>
<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>
 
<b>by </b><i><a class="txt" href="#E10:46_1_1_5"><i><font color="Green" title="E47">A79</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E20:46_1_1_5"/><b>then </b><i><font color="Green" title="E50">A91</font></i>: 
<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>
 
<b>by </b><i><a class="txt" href="#E17:46_1_1_5"><i><font color="Green" title="E49">A90</font></i></a>, <a class="ref" href="vectsp_5.html#T6" target="_self" title="VECTSP_5:th.6">Th6</a></i>;<br/>
<a NAME="E21:46_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>
 
<b>by </b><i><a class="ref" href="vectsp_5.html#T11" target="_self" title="VECTSP_5:th.11">Th11</a></i>;<br/>
<a NAME="E22:46_1_1_5"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="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>
 
<b>by </b><i><a class="txt" href="#E6:46_1_1"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="#E14:46_1_1"><i><font color="Green" title="E42">A71</font></i></a></i>;<br/>
<b>hence </b><a NAME="E23:46_1_1_5"/>
contradiction
 <b>by </b><i><a class="txt" href="#E12:46_1_1"><i><font color="Green" title="E41">A70</font></i></a>, <a class="txt" href="#E14:46_1_1"><i><font color="Green" title="E42">A71</font></i></a>, <a class="txt" href="#E20:46_1_1_5"><i><font color="Green" title="E50">A91</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><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>
 <b>by </b><i><a class="txt" href="#E15:46_1_1"><i><font color="Green" title="E43">A72</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
