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<span class="kw">let </span><font color="Maroon" title="c1">F</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">V</font> being  <a href="matrlin.html#V1" title="MATRLIN:attr.1">finite-dimensional</a>  <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">F</font><br/> for <font color="Olive" title="b2">W1</font>, <font color="Olive" title="b3">W2</font> being   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Olive" title="b1">V</font>  st <font color="Olive" title="b2">W1</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Olive" title="b3">W2</font> holds <br/>for <font color="Olive" title="b4">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Olive" title="b2">W1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">k</font> &amp;  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Olive" title="b3">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 holds <br/> not  <a href="pencil_4.html#K3" title="PENCIL_4:func.3">pencil</a> <font color="Olive" title="b2">W1</font>,<font color="Olive" title="b3">W2</font>,<font color="Olive" title="b4">k</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">V</font> be   <a href="matrlin.html#V1" title="MATRLIN:attr.1">finite-dimensional</a>  <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">F</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">W1</font>, <font color="Olive" title="b2">W2</font> being   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font>  st <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> holds <br/>for <font color="Olive" title="b3">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Olive" title="b1">W1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">k</font> &amp;  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Olive" title="b2">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 holds <br/> not  <a href="pencil_4.html#K3" title="PENCIL_4:func.3">pencil</a> <font color="Olive" title="b1">W1</font>,<font color="Olive" title="b2">W2</font>,<font color="Olive" title="b3">k</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> </span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> implies for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <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="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 holds <br/> 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="Olive" title="b1">k</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  )</span><br/>





<span class="kw">assume </span><a NAME="E1:19"/>
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <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="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 holds <br/> 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="Olive" title="b1">k</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> </span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">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> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">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="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies  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="c6">k</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E3:19"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
( <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="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">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="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 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="c6">k</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> </span><br/>

<span class="kw">consider </span><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="c5">U</font>;<br/>
<a NAME="E5:19"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
<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="c5">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_5.html#D5" title="VECTSP_5:def.5">VECTSP_5:def 5</a></span>;<br/>
<a NAME="E6:19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A3</font></span>: 
<font color="Maroon" title="c7">W</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c5">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_5.html#D4" title="VECTSP_5:def.4">VECTSP_5:def 4</a></span>;<br/>
<a NAME="E7:19"/>
 <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="c5">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:19"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="vectsp_9.html#T38" title="VECTSP_9:th.38">VECTSP_9:38</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">u</font>, <font color="Maroon" title="c9">v</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c7">W</font><span class="kw"> such that </span><br/><a NAME="E9:19"/><span class="lab"><font color="Green" title="E21">A4</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E10:19"/><span class="lab"><font color="Green" title="E22">A5</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E11:19"/>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="vectsp_9.html#T35" title="VECTSP_9:th.35">VECTSP_9:35</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E23">A6</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:19_1"/>
<font color="Maroon" title="c9">v</font> <a href="struct_0.html#R1" title="STRUCT_0: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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:19_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font> st <font color="Maroon" title="c9">v</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="c8">u</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp10.html#T4" title="VECTSP10:th.4">VECTSP10:4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:19_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:19"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E10:19"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="ref" href="vectsp_7.html#T8" title="VECTSP_7:th.8">VECTSP_7:8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">reconsider </span><font color="Maroon" title="c10">u</font> = <font color="Maroon" title="c8">u</font>, <font color="Maroon" title="c11">v</font> = <font color="Maroon" title="c9">v</font> as   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c4">W2</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T18" title="VECTSP_4:th.18">VECTSP_4:18</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">u1</font> = <font color="Maroon" title="c10">u</font>, <font color="Maroon" title="c13">v1</font> = <font color="Maroon" title="c11">v</font> as   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T18" title="VECTSP_4:th.18">VECTSP_4:18</a></span>;<br/>
<a NAME="E15:19"/><span class="lab"><font color="Green" title="E24">A7</font></span>: 
<font color="Maroon" title="c11">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c7">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E25">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:19_2"/>
<font color="Maroon" title="c11">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:19_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">v</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="c4">W2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:19"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E15:19"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="ref" href="vectsp_5.html#T7" title="VECTSP_5:th.7">VECTSP_5:7</a></span>;<br/><a NAME="E3:19_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">v</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="c4">W2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T46" title="VECTSP_4:th.46">VECTSP_4:46</a></span>;<br/><a NAME="E4:19_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">v</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="c7">W</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T19" title="VECTSP_4:th.19">VECTSP_4:19</a></span>;<br/><span class="kw">hence </span><a NAME="E5:19_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:19"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="ref" href="vectsp_7.html#T6" title="VECTSP_7:th.6">VECTSP_7:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E17:19"/><span class="lab"><font color="Green" title="E26">A9</font></span>: 
<font color="Maroon" title="c10">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c7">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E27">A10</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:19_3"/>
<font color="Maroon" title="c10">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:19_3"/><span class="kw">then </span>
<font color="Maroon" title="c10">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="c4">W2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:19"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E17:19"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="ref" href="vectsp_5.html#T7" title="VECTSP_5:th.7">VECTSP_5:7</a></span>;<br/><a NAME="E3:19_3"/><span class="kw">then </span>
<font color="Maroon" title="c10">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="c4">W2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T46" title="VECTSP_4:th.46">VECTSP_4:46</a></span>;<br/><a NAME="E4:19_3"/><span class="kw">then </span>
<font color="Maroon" title="c10">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="c7">W</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T19" title="VECTSP_4:th.19">VECTSP_4:19</a></span>;<br/><span class="kw">hence </span><a NAME="E5:19_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:19"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="ref" href="vectsp_7.html#T6" title="VECTSP_7:th.6">VECTSP_7:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E19:19"/>
<font color="Maroon" title="c11">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="c13">v1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E20:19"/><span class="kw">then </span>
<font color="Maroon" title="c11">v</font> <a href="struct_0.html#R1" title="STRUCT_0: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">v1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_7.html#T13" title="VECTSP_7:th.13">VECTSP_7:13</a></span>;<br/>
<a NAME="E21:19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A11</font></span>: 
<font color="Maroon" title="c11">v</font> <a href="struct_0.html#R1" title="STRUCT_0: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="c13">v1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_5.html#T6" title="VECTSP_5:th.6">VECTSP_5:6</a></span>;<br/>
<a NAME="E22:19"/><span class="lab"><font color="Green" title="E29">A12</font></span>: 
not <font color="Maroon" title="c11">v</font> <a href="struct_0.html#R1" title="STRUCT_0: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="c10">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:19"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="ref" href="vectsp10.html#T14" title="VECTSP10:th.14">VECTSP10:14</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E30">A13</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4"><span class="kw">now </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c14">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T34" title="VECTSP_4:th.34">VECTSP_4:34</a></span>;<br/><span class="kw">assume </span><a NAME="E2:19_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="c13">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="c12">u1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">h</font>, <font color="Maroon" title="c16">j</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E4:19_4"/><span class="lab"><font color="Green" title="E31">A14</font></span>: 
<font color="Maroon" title="c15">h</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</font>
 <span class="kw">and </span><br/><a NAME="E5:19_4"/><span class="lab"><font color="Green" title="E32">A15</font></span>: 
<font color="Maroon" title="c16">j</font> <a href="struct_0.html#R1" title="STRUCT_0: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="c12">u1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">and </span><br/><a NAME="E6:19_4"/><span class="lab"><font color="Green" title="E33">A16</font></span>: 
<font color="Maroon" title="c13">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">h</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c16">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:19"><span class="lab"><font color="Green" title="E28">A11</font></span></a>, <a class="ref" href="vectsp_5.html#T5" title="VECTSP_5:th.5">VECTSP_5:5</a></span>;<br/><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E8:19_4"/><span class="lab"><font color="Green" title="E34">A17</font></span>: 
<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#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c12">u1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:19_4"><span class="lab"><font color="Green" title="E32">A15</font></span></a>, <a class="ref" href="vectsp10.html#T4" title="VECTSP10:th.4">VECTSP10:4</a></span>;<br/><a NAME="E9:19_4"/>
<font color="Maroon" title="c13">v1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c12">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#K3" title="RLVECT_1:func.3">+</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#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c12">u1</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c12">u1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:19_4"><span class="lab"><font color="Green" title="E33">A16</font></span></a>, <a class="txt" href="#E8:19_4"><span class="lab"><font color="Green" title="E34">A17</font></span></a>, <a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></span>;<br/><a NAME="E10:19_4"/><span class="kw">then </span>
<font color="Maroon" title="c13">v1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c12">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#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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></span>;<br/><a NAME="E11:19_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A18</font></span>: 
<font color="Maroon" title="c13">v1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c12">u1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">h</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></span>;<br/><a NAME="E12:19_4"/>
<font color="Maroon" title="c17">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c10">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c12">u1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T22" title="VECTSP_4:th.22">VECTSP_4:22</a></span>;<br/><a NAME="E13:19_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A19</font></span>: 
<font color="Maroon" title="c13">v1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c12">u1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c10">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T24" title="VECTSP_4:th.24">VECTSP_4:24</a></span>;<br/><a NAME="E14:19_4"/>
<font color="Maroon" title="c17">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c10">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c14">Wx</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:19"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="ref" href="vectsp_4.html#T29" title="VECTSP_4:th.29">VECTSP_4:29</a></span>;<br/><a NAME="E15:19_4"/><span class="kw">then </span>
<font color="Maroon" title="c11">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c10">u</font></span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c14">Wx</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:19"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="ref" href="vectsp_4.html#T31" title="VECTSP_4:th.31">VECTSP_4:31</a></span>;<br/><a NAME="E16:19_4"/><span class="kw">then </span>
<font color="Maroon" title="c11">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c10">u</font></span>)</span> <a href="struct_0.html#R1" title="STRUCT_0: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="c5">U</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:19_4"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="txt" href="#E11:19_4"><span class="lab"><font color="Green" title="E35">A18</font></span></a>, <a class="txt" href="#E13:19_4"><span class="lab"><font color="Green" title="E36">A19</font></span></a>, <a class="ref" href="vectsp_5.html#T7" title="VECTSP_5:th.7">VECTSP_5:7</a></span>;<br/><a NAME="E17:19_4"/><span class="kw">then </span>
<font color="Maroon" title="c11">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c10">u</font></span>)</span> <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="c4">W2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:19"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="vectsp_4.html#T46" title="VECTSP_4:th.46">VECTSP_4:46</a></span>;<br/><a NAME="E18:19_4"/><span class="kw">then </span>
<font color="Maroon" title="c15">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:19_4"><span class="lab"><font color="Green" title="E35">A18</font></span></a>, <a class="txt" href="#E13:19_4"><span class="lab"><font color="Green" title="E36">A19</font></span></a>, <a class="ref" href="vectsp_4.html#T19" title="VECTSP_4:th.19">VECTSP_4:19</a></span>;<br/><a NAME="E19:19_4"/><span class="kw">then </span>
<font color="Maroon" title="c13">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:19_4"><span class="lab"><font color="Green" title="E33">A16</font></span></a>, <a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></span>;<br/><span class="kw">hence </span><a NAME="E20:19_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:19"><span class="lab"><font color="Green" title="E29">A12</font></span></a>, <a class="txt" href="#E5:19_4"><span class="lab"><font color="Green" title="E32">A15</font></span></a>, <a class="ref" href="vectsp10.html#T14" title="VECTSP10:th.14">VECTSP10:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E24:19"/>
<font color="Maroon" title="c11">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E25:19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A20</font></span>: 
<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">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="c6">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E16:19"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="ref" href="pencil_4.html#T16" target="_self" title="PENCIL_4:th.16">Th16</a></span>;<br/>
<a NAME="E26:19"/>
<font color="Maroon" title="c10">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E27:19"/><span class="kw">then </span>
<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="c12">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="c6">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E18:19"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="ref" href="pencil_4.html#T16" target="_self" title="PENCIL_4:th.16">Th16</a></span>;<br/>
<a NAME="E28:19"/><span class="kw">then </span>
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="c6">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:19"><span class="lab"><font color="Green" title="E30">A13</font></span></a>, <a class="txt" href="#E25:19"><span class="lab"><font color="Green" title="E31">A20</font></span></a>, <a class="ref" href="pencil_1.html#T2" title="PENCIL_1:th.2">PENCIL_1:2</a></span>;<br/>
<span class="kw">hence </span><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="c6">k</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_1.html#T4" title="PENCIL_1:th.4">PENCIL_1:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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