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<b>let </b><font color="Maroon" title="c1">F</font> be   <a href="vectsp_1.html#NM2" title="VECTSP_1:NM.2">Field</a>;<br/><b>let </b><font color="Maroon" title="c2">V</font> be  <a href="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a> <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">VectSp</a> of <font color="Maroon" title="c1">F</font>;<br/><b>let </b><font color="Maroon" title="c3">W1</font>, <font color="Maroon" title="c4">W2</font> be    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font>;<br/>





<b>assume </b><a NAME="E1:19"/><i><font color="Green" title="E20">A1</font></i>: 
<font color="Maroon" title="c3">W1</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c4">W2</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c5">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E2:19"/><i><font color="Green" title="E21">A2</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> &amp; <font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c2">V</font> &amp; <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c3">W1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> &amp;  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c4">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c6">U</font> = <font color="Maroon" title="c3">W1</font> as    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c4">W2</font> <b>by </b><i><a class="txt" href="pencil_4.html#E1:19"><i><font color="Green" title="E20">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">W</font> being    <a href="vectsp_5.html#M1" title="VECTSP_5:mode.1">Linear_Compl</a> of <font color="Maroon" title="c6">U</font>;<br/>
<a NAME="E5:19"/><i><font color="Green" title="E22">A3</font></i>: 
<font color="Maroon" title="c4">W2</font> <a href="vectsp_5.html#R1" title="VECTSP_5:pred.1">is_the_direct_sum_of</a> <font color="Maroon" title="c7">W</font>,<font color="Maroon" title="c6">U</font>
 
<b>by </b><i><a class="ref" href="vectsp_5.html#D5" title="VECTSP_5:def.5">VECTSP_5:def 5</a></i>;<br/>
<a NAME="E6:19"/><b>then </b>
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c4">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c6">U</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c7">W</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_9.html#T38" title="VECTSP_9:th.38">VECTSP_9:38</a></i>;<br/>
<a NAME="E7:19"/><b>then </b>
<span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c4">W2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c7">W</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c6">U</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
 
;<br/>
<a NAME="E8:19"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c7">W</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">k</font>
 
<b>by </b><i><a class="txt" href="pencil_4.html#E2:19"><i><font color="Green" title="E21">A2</font></i></a></i>;<br/>
<a NAME="E9:19"/><b>then </b>
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c7">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
;<br/>
<b>then consider </b><font color="Maroon" title="c8">u</font>, <font color="Maroon" title="c9">v</font> being   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c7">W</font><b> such that </b><br/><a NAME="E11:19"/><i><font color="Green" title="E23">A4</font></i>: 
( <font color="Maroon" title="c8">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">v</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c8">u</font>,<font color="Maroon" title="c9">v</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> &amp;  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c7">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c8">u</font>,<font color="Maroon" title="c9">v</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )
 <b>by </b><i><a class="ref" href="vectsp_9.html#T35" title="VECTSP_9:th.35">VECTSP_9:35</a></i>;<br/>
<div><i><font color="Green" title="E24">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_1"/>
<font color="Maroon" title="c9">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c8">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/><b>then consider </b><font color="Maroon" title="c10">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="E3:19_1"/><i><font color="Green" title="E25">A6</font></i>: 
<font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c8">u</font>
 <b>by </b><i><a class="ref" href="vectsp10.html#T4" title="VECTSP10:th.4">VECTSP10:4</a></i>;<br/><b>thus </b><a NAME="E4:19_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pencil_4.html#E3:19_1"><i><font color="Green" title="E25">A6</font></i></a>, <a class="ref" href="vectsp_7.html#T8" title="VECTSP_7:th.8">VECTSP_7:8</a>, <a class="txt" href="pencil_4.html#E11:19"><i><font color="Green" title="E23">A4</font></i></a></i>;<br/></div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c10">L</font> =  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c8">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<a NAME="E13:19"/><i><font color="Green" title="E25">A7</font></i>: 
(  <a href="vectsp_1.html#G4" title="VECTSP_1:aggr.4">VectSpStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">W2</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c4">W2</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">Zero</a> of <font color="Maroon" title="c4">W2</font>,the <a href="vectsp_1.html#U2" title="VECTSP_1:sel.2">lmult</a> of <font color="Maroon" title="c4">W2</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">W</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c6">U</font> &amp; <font color="Maroon" title="c7">W</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c6">U</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="c4">W2</font> )
 
<b>by </b><i><a class="txt" href="pencil_4.html#E5:19"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="vectsp_5.html#D4" title="VECTSP_5:def.4">VECTSP_5:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">u</font> = <font color="Maroon" title="c8">u</font>, <font color="Maroon" title="c12">v</font> = <font color="Maroon" title="c9">v</font> as   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c4">W2</font> <b>by </b><i><a class="ref" href="vectsp_4.html#T18" title="VECTSP_4:th.18">VECTSP_4:18</a></i>;<br/>
<a NAME="E15:19"/><i><font color="Green" title="E26">A8</font></i>: 
<font color="Maroon" title="c11">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c7">W</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E16:19"/><i><font color="Green" title="E27">A9</font></i>: 
<font color="Maroon" title="c11">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c4">W2</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/>
<div><i><font color="Green" title="E28">A10</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_2"/>
<font color="Maroon" title="c11">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">W1</font>
 ;<br/><a NAME="E2:19_2"/><b>then </b>
<font color="Maroon" title="c11">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c4">W2</font>
 <b>by </b><i><a class="txt" href="pencil_4.html#E15:19"><i><font color="Green" title="E26">A8</font></i></a>, <a class="ref" href="vectsp_5.html#T7" title="VECTSP_5:th.7">VECTSP_5:7</a>, <a class="txt" href="pencil_4.html#E13:19"><i><font color="Green" title="E25">A7</font></i></a></i>;<br/><a NAME="E3:19_2"/><b>then </b>
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c4">W2</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T46" title="VECTSP_4:th.46">VECTSP_4:46</a></i>;<br/><a NAME="E4:19_2"/><b>then </b>
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c7">W</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T19" title="VECTSP_4:th.19">VECTSP_4:19</a></i>;<br/><b>hence </b><a NAME="E5:19_2"/>
contradiction
 <b>by </b><i><a class="ref" href="vectsp_7.html#T6" title="VECTSP_7:th.6">VECTSP_7:6</a>, <a class="txt" href="pencil_4.html#E11:19"><i><font color="Green" title="E23">A4</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E18:19"/><i><font color="Green" title="E29">A11</font></i>: 
<font color="Maroon" title="c12">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c7">W</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E19:19"/><i><font color="Green" title="E30">A12</font></i>: 
<font color="Maroon" title="c12">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c4">W2</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/>
<div><i><font color="Green" title="E31">A13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_3"/>
<font color="Maroon" title="c12">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">W1</font>
 ;<br/><a NAME="E2:19_3"/><b>then </b>
<font color="Maroon" title="c12">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c4">W2</font>
 <b>by </b><i><a class="txt" href="pencil_4.html#E18:19"><i><font color="Green" title="E29">A11</font></i></a>, <a class="ref" href="vectsp_5.html#T7" title="VECTSP_5:th.7">VECTSP_5:7</a>, <a class="txt" href="pencil_4.html#E13:19"><i><font color="Green" title="E25">A7</font></i></a></i>;<br/><a NAME="E3:19_3"/><b>then </b>
<font color="Maroon" title="c12">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c4">W2</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T46" title="VECTSP_4:th.46">VECTSP_4:46</a></i>;<br/><a NAME="E4:19_3"/><b>then </b>
<font color="Maroon" title="c12">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c7">W</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T19" title="VECTSP_4:th.19">VECTSP_4:19</a></i>;<br/><b>hence </b><a NAME="E5:19_3"/>
contradiction
 <b>by </b><i><a class="ref" href="vectsp_7.html#T6" title="VECTSP_7:th.6">VECTSP_7:6</a>, <a class="txt" href="pencil_4.html#E11:19"><i><font color="Green" title="E23">A4</font></i></a></i>;<br/></div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c13">u1</font> = <font color="Maroon" title="c11">u</font>, <font color="Maroon" title="c14">v1</font> = <font color="Maroon" title="c12">v</font> as   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="ref" href="vectsp_4.html#T18" title="VECTSP_4:th.18">VECTSP_4:18</a></i>;<br/>
<a NAME="E22:19"/>
<font color="Maroon" title="c12">v</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"><font color="Maroon" title="c14">v1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E23:19"/><b>then </b>
<font color="Maroon" title="c12">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">v1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="vectsp_7.html#T13" title="VECTSP_7:th.13">VECTSP_7:13</a></i>;<br/>
<a NAME="E24:19"/><b>then </b><i><font color="Green" title="E32">A14</font></i>: 
<font color="Maroon" title="c12">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">v1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_5.html#T6" title="VECTSP_5:th.6">VECTSP_5:6</a></i>;<br/>
<a NAME="E25:19"/><i><font color="Green" title="E33">A15</font></i>: 
not <font color="Maroon" title="c12">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c11">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="pencil_4.html#E12:19"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="vectsp10.html#T14" title="VECTSP10:th.14">VECTSP10:14</a></i>;<br/>
<div><i><font color="Green" title="E34">A16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_4"/>
<font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">v1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c13">u1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c15">h</font>, <font color="Maroon" title="c16">j</font> being   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E3:19_4"/><i><font color="Green" title="E35">A17</font></i>: 
( <font color="Maroon" title="c15">h</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">W1</font> &amp; <font color="Maroon" title="c16">j</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c13">u1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c14">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">h</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c16">j</font> )
 <b>by </b><i><a class="ref" href="vectsp_5.html#T5" title="VECTSP_5:th.5">VECTSP_5:5</a>, <a class="txt" href="pencil_4.html#E24:19"><i><font color="Green" title="E32">A14</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c17">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="E5:19_4"/><i><font color="Green" title="E36">A18</font></i>: 
<font color="Maroon" title="c16">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">a</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c13">u1</font>
 <b>by </b><i><a class="ref" href="vectsp10.html#T4" title="VECTSP10:th.4">VECTSP10:4</a>, <a class="txt" href="pencil_4.html#E3:19_4"><i><font color="Green" title="E35">A17</font></i></a></i>;<br/><a NAME="E6:19_4"/>
<font color="Maroon" title="c14">v1</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">a</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c13">u1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">h</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">a</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c13">u1</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">a</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c13">u1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_4.html#E3:19_4"><i><font color="Green" title="E35">A17</font></i></a>, <a class="txt" href="pencil_4.html#E5:19_4"><i><font color="Green" title="E36">A18</font></i></a>, <a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>;<br/><a NAME="E7:19_4"/><b>then </b>
<font color="Maroon" title="c14">v1</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">a</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c13">u1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">h</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">V</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></i>;<br/><a NAME="E8:19_4"/><b>then </b><i><font color="Green" title="E37">A19</font></i>: 
<font color="Maroon" title="c14">v1</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">a</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c13">u1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">h</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">Wx</font> = <font color="Maroon" title="c7">W</font> as    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="ref" href="vectsp_4.html#T34" title="VECTSP_4:th.34">VECTSP_4:34</a></i>;<br/><a NAME="E10:19_4"/>
<font color="Maroon" title="c17">a</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c11">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c18">Wx</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T29" title="VECTSP_4:th.29">VECTSP_4:29</a>, <a class="txt" href="pencil_4.html#E15:19"><i><font color="Green" title="E26">A8</font></i></a></i>;<br/><a NAME="E11:19_4"/><b>then </b><i><font color="Green" title="E38">A20</font></i>: 
<font color="Maroon" title="c12">v</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">a</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c18">Wx</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T31" title="VECTSP_4:th.31">VECTSP_4:31</a>, <a class="txt" href="pencil_4.html#E18:19"><i><font color="Green" title="E29">A11</font></i></a></i>;<br/><a NAME="E12:19_4"/>
<font color="Maroon" title="c17">a</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">a</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c13">u1</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T22" title="VECTSP_4:th.22">VECTSP_4:22</a></i>;<br/><a NAME="E13:19_4"/><b>then </b><i><font color="Green" title="E39">A21</font></i>: 
<font color="Maroon" title="c14">v1</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">a</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c13">u1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">v</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">a</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c11">u</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T24" title="VECTSP_4:th.24">VECTSP_4:24</a></i>;<br/><a NAME="E14:19_4"/><b>then </b>
<font color="Maroon" title="c12">v</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">a</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c7">W</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c6">U</font>
 <b>by </b><i><a class="txt" href="pencil_4.html#E8:19_4"><i><font color="Green" title="E37">A19</font></i></a>, <a class="txt" href="pencil_4.html#E3:19_4"><i><font color="Green" title="E35">A17</font></i></a>, <a class="txt" href="pencil_4.html#E11:19_4"><i><font color="Green" title="E38">A20</font></i></a>, <a class="ref" href="vectsp_5.html#T7" title="VECTSP_5:th.7">VECTSP_5:7</a></i>;<br/><a NAME="E15:19_4"/><b>then </b>
<font color="Maroon" title="c12">v</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">a</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c4">W2</font>
 <b>by </b><i><a class="txt" href="pencil_4.html#E13:19"><i><font color="Green" title="E25">A7</font></i></a>, <a class="ref" href="vectsp_4.html#T46" title="VECTSP_4:th.46">VECTSP_4:46</a></i>;<br/><a NAME="E16:19_4"/><b>then </b>
<font color="Maroon" title="c15">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="txt" href="pencil_4.html#E13:19_4"><i><font color="Green" title="E39">A21</font></i></a>, <a class="txt" href="pencil_4.html#E8:19_4"><i><font color="Green" title="E37">A19</font></i></a>, <a class="ref" href="vectsp_4.html#T19" title="VECTSP_4:th.19">VECTSP_4:19</a></i>;<br/><a NAME="E17:19_4"/><b>then </b>
<font color="Maroon" title="c14">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j</font>
 <b>by </b><i><a class="txt" href="pencil_4.html#E3:19_4"><i><font color="Green" title="E35">A17</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="E18:19_4"/>
contradiction
 <b>by </b><i><a class="ref" href="vectsp10.html#T14" title="VECTSP10:th.14">VECTSP10:14</a>, <a class="txt" href="pencil_4.html#E3:19_4"><i><font color="Green" title="E35">A17</font></i></a>, <a class="txt" href="pencil_4.html#E25:19"><i><font color="Green" title="E33">A15</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E27:19"/>
( <font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c13">u1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pencil_4.html#K3" title="PENCIL_4:func.3">pencil</a> <font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font>,<font color="Maroon" title="c5">k</font> &amp; <font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">v1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pencil_4.html#K3" title="PENCIL_4:func.3">pencil</a> <font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font>,<font color="Maroon" title="c5">k</font> )
 
<b>by </b><i><a class="txt" href="pencil_4.html#E16:19"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="pencil_4.html#E17:19"><i><font color="Green" title="E28">A10</font></i></a>, <a class="txt" href="pencil_4.html#E2:19"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="pencil_4.html#T16" target="_self" title="PENCIL_4:th.16">Th16</a>, <a class="txt" href="pencil_4.html#E19:19"><i><font color="Green" title="E30">A12</font></i></a>, <a class="txt" href="pencil_4.html#E20:19"><i><font color="Green" title="E31">A13</font></i></a></i>;<br/>
<a NAME="E28:19"/><b>then </b>
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="pencil_4.html#K3" title="PENCIL_4:func.3">pencil</a> <font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font>,<font color="Maroon" title="c5">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_4.html#E26:19"><i><font color="Green" title="E34">A16</font></i></a>, <a class="ref" href="pencil_1.html#T2" title="PENCIL_1:th.2">PENCIL_1:2</a></i>;<br/>
<b>hence </b><a NAME="E29:19"/>
not  <a href="pencil_4.html#K3" title="PENCIL_4:func.3">pencil</a> <font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font>,<font color="Maroon" title="c5">k</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>
 <b>by </b><i><a class="ref" href="pencil_1.html#T4" title="PENCIL_1:th.4">PENCIL_1:4</a></i>;<br/>


</div>
