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<b>let </b><font color="Maroon">F</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">V</font> be  <a href="matrlin.html#V2">finite-dimensional</a> <a href="vectsp_1.html#NM6">VectSp</a> of <font color="Maroon">F</font>;<br/><b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>





<b>assume </b><a NAME="E1:23"/><i><font color="Green">E24</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="vectsp_9.html#K1">dim</a> <font color="Maroon">V</font> &amp; 3 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="vectsp_9.html#K1">dim</a> <font color="Maroon">V</font> )
 ;<br/>

<b>set </b><font color="Maroon">S</font> =  <a href="pencil_4.html#K5" target="_self">PencilSpace</a> <font color="Maroon">V</font>,<font color="Maroon">k</font>;<br/>
<b>thus </b><a NAME="E2:23"/>
not  <a href="pencil_4.html#K5" target="_self">PencilSpace</a> <font color="Maroon">V</font>,<font color="Maroon">k</font> is <a href="pencil_1.html#V4">degenerated</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">B</font> be   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><a href="pencil_4.html#K5" target="_self">PencilSpace</a> <font color="Maroon">V</font>,<font color="Maroon">k</font></span>)</span>;<br/><a NAME="E1:23_1_1"/>
not the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="pencil_4.html#K5" target="_self">PencilSpace</a> <font color="Maroon">V</font>,<font color="Maroon">k</font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i>, </i>;<br/><b>then consider </b><font color="Maroon">W1</font> being    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font>, <font color="Maroon">W2</font> being    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:23_1_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">W1</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">W2</font> &amp; <span class="p1">(<span class="default"><a href="vectsp_9.html#K1">dim</a> <font color="Maroon">W1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp;  <a href="vectsp_9.html#K1">dim</a> <font color="Maroon">W2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="pencil_4.html#K3" target="_self">pencil</a> <font color="Maroon">W1</font>,<font color="Maroon">W2</font>,<font color="Maroon">k</font> )
 <b>by </b><i/>;<br/><a NAME="E4:23_1_1"/><i><font color="Green">E31</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D2">VECTSP_4:def 2</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_1_1"/>
( <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="vectsp_9.html#K1">dim</a> <font color="Maroon">V</font> or ( 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="vectsp_9.html#K1">dim</a> <font color="Maroon">V</font> ) )
 </span><b>by </b><i>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1_1"><b>suppose </b></a><a NAME="E1:23_1_1_1_1"/>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="vectsp_9.html#K1">dim</a> <font color="Maroon">V</font>
 ;<br/><div class="add"><a NAME="E2:23_1_1_1_1"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">W2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="txt" href="pencil_4.html#E1:23"><i><font color="Green">E24</font></i></a>, <a class="ref" href="vectsp_9.html#T32">VECTSP_9:32</a></i>;<br/><a NAME="E3:23_1_1_1_1"/><i><font color="Green">E33</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D2">VECTSP_4:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_1_1_1_1"/><i><font color="Green">E34</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font>
 ;<br/><a NAME="E2:23_1_1_1_1_1"/><i><font color="Green">E36</font></i>: 
 <a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">W2</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E3:23_1_1_1_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="vectsp_4.html#T37">VECTSP_4:37</a>, <a class="txt" href="pencil_4.html#E3:23_1_1"><i><font color="Green">E29</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E5:23_1_1_1_1"/><b>then </b>
not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><b>then consider </b><font color="Maroon">v</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:23_1_1_1_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> &amp; not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">v</font> as   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font> <b>by </b><i/>;<br/><b>set </b><font color="Maroon">X</font> = <font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>;<br/><a NAME="E9:23_1_1_1_1"/>
not <font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font>
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E10:23_1_1_1_1"/><b>then </b>
not <font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T16">VECTSP_4:16</a>, <a class="txt" href="pencil_4.html#E1:23"><i><font color="Green">E24</font></i></a></i>;<br/><a NAME="E11:23_1_1_1_1"/><b>then </b>
 <a href="vectsp_9.html#K1">dim</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="txt" href="pencil_4.html#E1:23"><i><font color="Green">E24</font></i></a></i>;<br/><a NAME="E12:23_1_1_1_1"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_4.html#K5" target="_self">PencilSpace</a> <font color="Maroon">V</font>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_9.html#D3">VECTSP_9:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_1_1_1_2"/>
<font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><a NAME="E2:23_1_1_1_1_2"/><b>then </b>
<font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">W2</font>
 <b>by </b><i>, <a class="txt" href="pencil_4.html#E1:23"><i><font color="Green">E24</font></i></a></i>;<br/><a NAME="E3:23_1_1_1_1_2"/><b>then </b><i><font color="Green">E68</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D2">VECTSP_4:def 2</a></i>;<br/><a NAME="E4:23_1_1_1_1_2"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:23_1_1_1_1_2"/><b>then </b>
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_7.html#K1">Lin</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="vectsp_7.html#T13">VECTSP_7:13</a></i>;<br/><a NAME="E6:23_1_1_1_1_2"/><b>then </b>
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_5.html#T6">VECTSP_5:6</a></i>;<br/><a NAME="E7:23_1_1_1_1_2"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><b>hence </b><a NAME="E8:23_1_1_1_1_2"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E14:23_1_1_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_4.html#K5" target="_self">PencilSpace</a> <font color="Maroon">V</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="txt" href="pencil_4.html#E3:23_1_1"><i><font color="Green">E29</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1_2"><b>suppose </b></a><a NAME="E1:23_1_1_1_2"/><i><font color="Green">E81</font></i>: 
( 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="vectsp_9.html#K1">dim</a> <font color="Maroon">V</font> )
 ;<br/><div class="add"><a NAME="E2:23_1_1_1_2"/><b>then </b>
( <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="vectsp_9.html#K1">dim</a> <font color="Maroon">V</font> &amp;  <a href="vectsp_9.html#K1">dim</a> <font color="Maroon">V</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 )
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:23_1_1_1_2"/><b>then </b><i><font color="Green">E82</font></i>: 
 <a href="vectsp_9.html#K1">dim</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>consider </b><font color="Maroon">T</font> being    <a href="vectsp_5.html#M1">Linear_Compl</a> of <font color="Maroon">W1</font>;<br/><a NAME="E5:23_1_1_1_2"/><i><font color="Green">E83</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D2">VECTSP_4:def 2</a></i>;<br/><a NAME="E6:23_1_1_1_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">V</font> <a href="vectsp_5.html#R1">is_the_direct_sum_of</a> <font color="Maroon">W1</font>,<font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#T48">VECTSP_5:48</a></i>;<br/><a NAME="E7:23_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_9.html#K1">dim</a> <font color="Maroon">W1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_9.html#K1">dim</a> <font color="Maroon">W1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="vectsp_9.html#K1">dim</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_9.html#K1">dim</a> <font color="Maroon">W1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="vectsp_9.html#T38">VECTSP_9:38</a></i>;<br/><a NAME="E8:23_1_1_1_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_9.html#K1">dim</a> <font color="Maroon">T</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="txt" href="pencil_4.html#E1:23"><i><font color="Green">E24</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">u</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">T</font>, <font color="Maroon">v</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E10:23_1_1_1_2"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font> &amp; <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">v</font></span><a href="domain_1.html#K7">}</a></span> is <a href="vectsp_7.html#V1">linearly-independent</a> &amp;  <a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_7.html#K1">Lin</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">v</font></span><a href="domain_1.html#K7">}</a></span> )
 <b>by </b><i><a class="ref" href="vectsp_9.html#T35">VECTSP_9:35</a></i>;<br/><a NAME="E11:23_1_1_1_2"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</font> )
 <b>by </b><i>, <a class="ref" href="vectsp_7.html#T6">VECTSP_7:6</a></i>;<br/><a NAME="E12:23_1_1_1_2"/>
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> )
 ;<br/><b>then reconsider </b><font color="Maroon">u1</font> = <font color="Maroon">u</font>, <font color="Maroon">v1</font> = <font color="Maroon">v</font> as   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font> <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">I</font> being    <a href="vectsp_7.html#M1">Basis</a> of <font color="Maroon">W1</font>;<br/><b>reconsider </b><font color="Maroon">I1</font> = <font color="Maroon">I</font> as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">W1</font> <b>by </b><i><a class="ref" href="vectsp_9.html#T24">VECTSP_9:24</a></i>;<br/><a NAME="E16:23_1_1_1_2"/>
2 <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_9.html#K1">dim</a> <font color="Maroon">W1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, <a class="txt" href="pencil_4.html#E1:23"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E17:23_1_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">I1</font>
 <b>by </b><i><a class="ref" href="vectsp_9.html#D2">VECTSP_9:def 2</a></i>;<br/><a NAME="E18:23_1_1_1_2"/><b>then </b>
not <font color="Maroon">I1</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="card_4.html#T17">CARD_4:17</a></i>;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E20:23_1_1_1_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">W1</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">J</font> = <font color="Maroon">I1</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i</font></span><a href="domain_1.html#K6">}</a></span> as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="pencil_4.html#T2" target="_self">Th2</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E23:23_1_1_1_2"/><i><font color="Green">E90</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_9.html#K1">dim</a> <font color="Maroon">W1</font>
 <b>by </b><i><a class="ref" href="vectsp_9.html#D2">VECTSP_9:def 2</a></i>;<br/><i><font color="Green">E91</font></i>:  <a href="card_1.html#K4">card</a> <font color="Maroon">J</font> = 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="euler_1.html#T5">EULER_1:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="vectsp_9.html#K1">dim</a> <font color="Maroon">W1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
<b>by </b><i><a class="ref" href="card_1.html#T79">CARD_1:79</a>, </i>
;<br/><a NAME="E25:23_1_1_1_2"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">J</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">J</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:23_1_1_1_2_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font>
 ;<br/>

<a NAME="E2:23_1_1_1_2_2"/><b>then </b>
<font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">J</font>
 
<b>by </b><i><a class="ref" href="vectsp_7.html#T13">VECTSP_7:13</a></i>;<br/>
<b>hence </b><a NAME="E3:23_1_1_1_2_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">J</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">J1</font> = <font color="Maroon">J</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">J</font></span>)</span> ;<br/><a NAME="E27:23_1_1_1_2"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">I</font> is <a href="vectsp_7.html#V1">linearly-independent</a>
 <b>by </b><i><a class="ref" href="vectsp_7.html#D3">VECTSP_7:def 3</a></i>;<br/><a NAME="E28:23_1_1_1_2"/>
<font color="Maroon">I</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/><a NAME="E29:23_1_1_1_2"/><b>then </b>
<font color="Maroon">I</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i</font></span><a href="domain_1.html#K6">}</a></span> is <a href="vectsp_7.html#V1">linearly-independent</a>
 <b>by </b><i>, <a class="ref" href="vectsp_7.html#T2">VECTSP_7:2</a></i>;<br/><b>then reconsider </b><font color="Maroon">JJ</font> = <font color="Maroon">I</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i</font></span><a href="domain_1.html#K6">}</a></span> as  <a href="vectsp_7.html#V1">linearly-independent</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="vectsp_9.html#T15">VECTSP_9:15</a></i>;<br/><b>reconsider </b><font color="Maroon">JJJ</font> = <font color="Maroon">JJ</font> as  <a href="vectsp_7.html#V1">linearly-independent</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">J</font></span>)</span> <b>by </b><i>, <a class="ref" href="vectsp_9.html#T16">VECTSP_9:16</a></i>;<br/><a NAME="E32:23_1_1_1_2"/>
 <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">JJJ</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">J</font>
 <b>by </b><i><a class="ref" href="vectsp_9.html#T21">VECTSP_9:21</a></i>;<br/><a NAME="E33:23_1_1_1_2"/><b>then </b>
<font color="Maroon">J</font> is    <a href="vectsp_7.html#M1">Basis</a> of  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">J</font>
 <b>by </b><i><a class="ref" href="vectsp_7.html#D3">VECTSP_7:def 3</a></i>;<br/><a NAME="E34:23_1_1_1_2"/><b>then </b><i><font color="Green">E106</font></i>: 
 <a href="vectsp_9.html#K1">dim</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">J</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1">dim</a> <font color="Maroon">W1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, <a class="ref" href="vectsp_9.html#D2">VECTSP_9:def 2</a></i>;<br/><a NAME="E35:23_1_1_1_2"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">W1</font> <a href="vectsp_5.html#K2">/\</a> <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="txt" href="pencil_4.html#E3:23_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="vectsp_5.html#D4">VECTSP_5:def 4</a></i>;<br/><div><i><font color="Green">E108</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_1_1_2_3"/>
<font color="Maroon">u1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font>
 ;<br/><a NAME="E2:23_1_1_1_2_3"/><b>then </b>
( <font color="Maroon">u1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">u1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">T</font> )
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E3:23_1_1_1_2_3"/><b>then </b>
<font color="Maroon">u1</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="pencil_4.html#T3" target="_self">Th3</a>, <a class="ref" href="vectsp_5.html#T7">VECTSP_5:7</a></i>;<br/><a NAME="E4:23_1_1_1_2_3"/><b>then </b>
<font color="Maroon">u1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E5:23_1_1_1_2_3"/><b>then </b>
<font color="Maroon">u1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D3">VECTSP_4:def 3</a></i>;<br/><a NAME="E6:23_1_1_1_2_3"/><b>then </b>
<font color="Maroon">u1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E7:23_1_1_1_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="pencil_4.html#E4:23_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="vectsp_4.html#T19">VECTSP_4:19</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E109</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_2_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_1_1_2_4"/>
<font color="Maroon">v1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font>
 ;<br/><a NAME="E2:23_1_1_1_2_4"/><b>then </b>
( <font color="Maroon">v1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">v1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">T</font> )
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E3:23_1_1_1_2_4"/><b>then </b>
<font color="Maroon">v1</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="pencil_4.html#T3" target="_self">Th3</a>, <a class="ref" href="vectsp_5.html#T7">VECTSP_5:7</a></i>;<br/><a NAME="E4:23_1_1_1_2_4"/><b>then </b>
<font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E5:23_1_1_1_2_4"/><b>then </b>
<font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D3">VECTSP_4:def 3</a></i>;<br/><a NAME="E6:23_1_1_1_2_4"/><b>then </b>
<font color="Maroon">v1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E7:23_1_1_1_2_4"/>
contradiction
 <b>by </b><i><a class="txt" href="pencil_4.html#E4:23_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="vectsp_4.html#T19">VECTSP_4:19</a></i>;<br/></div><b>end;</b></div><a NAME="E38:23_1_1_1_2"/><i><font color="Green">E110</font></i>: 
 <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">W1</font>
 <b>by </b><i><a class="ref" href="vectsp_7.html#D3">VECTSP_7:def 3</a></i>;<br/><a NAME="E39:23_1_1_1_2"/>
<font color="Maroon">I</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/><a NAME="E40:23_1_1_1_2"/><b>then </b>
 <a href="vectsp_7.html#K1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> is    <a href="vectsp_4.html#M1">Subspace</a> of  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="vectsp_7.html#T18">VECTSP_7:18</a></i>;<br/><a NAME="E41:23_1_1_1_2"/><b>then </b><i><font color="Green">E111</font></i>: 
 <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">J</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">W1</font>
 <b>by </b><i><a class="ref" href="vectsp_9.html#T21">VECTSP_9:21</a></i>;<br/><a NAME="E42:23_1_1_1_2"/><i><font color="Green">E112</font></i>: 
not <font color="Maroon">u1</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">J</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T16">VECTSP_4:16</a>, , <a class="ref" href="pencil_4.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E43:23_1_1_1_2"/><i><font color="Green">E113</font></i>: 
not <font color="Maroon">v1</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">J</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T16">VECTSP_4:16</a>, , <a class="ref" href="pencil_4.html#T4" target="_self">Th4</a></i>;<br/><b>reconsider </b><font color="Maroon">Y</font> = <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">v</font></span><a href="domain_1.html#K7">}</a></span> as  <a href="vectsp_7.html#V1">linearly-independent</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="vectsp_9.html#T15">VECTSP_9:15</a>, </i>;<br/><a NAME="E45:23_1_1_1_2"/>
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">v</font></span><a href="domain_1.html#K7">}</a></span>
 ;<br/><a NAME="E46:23_1_1_1_2"/><b>then </b><i><font color="Green">E114</font></i>: 
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u1</font>,<font color="Maroon">v1</font></span><a href="domain_1.html#K7">}</a></span> is <a href="vectsp_7.html#V1">linearly-independent</a>
 ;<br/><a NAME="E47:23_1_1_1_2"/><i><font color="Green">E115</font></i>: 
 <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">J</font></span>)</span> <a href="vectsp_5.html#K2">/\</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u1</font>,<font color="Maroon">v1</font></span><a href="domain_1.html#K7">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T50">VECTSP_4:50</a></i>;<br/><a NAME="E48:23_1_1_1_2"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">J</font></span>)</span> <a href="vectsp_5.html#K2">/\</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u1</font>,<font color="Maroon">v1</font></span><a href="domain_1.html#K7">}</a></span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:23_1_1_1_2_5"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">J</font></span>)</span> <a href="vectsp_5.html#K2">/\</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u1</font>,<font color="Maroon">v1</font></span><a href="domain_1.html#K7">}</a></span></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:23_1_1_1_2_5"/><b>then </b>
<font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">J</font></span>)</span> <a href="vectsp_5.html#K2">/\</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u1</font>,<font color="Maroon">v1</font></span><a href="domain_1.html#K7">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E3:23_1_1_1_2_5"/><b>then </b>
( <font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">J</font> &amp; <font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_7.html#K1">Lin</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u1</font>,<font color="Maroon">v1</font></span><a href="domain_1.html#K7">}</a></span> )
 
<b>by </b><i><a class="ref" href="vectsp_5.html#T7">VECTSP_5:7</a></i>;<br/>
<a NAME="E4:23_1_1_1_2_5"/><b>then </b>
( <font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_7.html#K1">Lin</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">v</font></span><a href="domain_1.html#K7">}</a></span> )
 
<b>by </b><i><a class="ref" href="pencil_4.html#T4" target="_self">Th4</a>, <a class="ref" href="vectsp_4.html#T16">VECTSP_4:16</a>, <a class="ref" href="vectsp_9.html#T21">VECTSP_9:21</a></i>;<br/>
<a NAME="E5:23_1_1_1_2_5"/><b>then </b>
( <font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">T</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">T</font>,the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">T</font> #) )
 
<b>by </b><i/>;<br/>
<a NAME="E6:23_1_1_1_2_5"/><b>then </b>
( <font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">T</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">T</font>,the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">T</font> #) )
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E7:23_1_1_1_2_5"/><b>then </b>
( <font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">T</font> )
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E8:23_1_1_1_2_5"/><b>then </b>
<font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> <a href="vectsp_5.html#K2">/\</a> <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="vectsp_5.html#T7">VECTSP_5:7</a></i>;<br/>
<b>hence </b><a NAME="E9:23_1_1_1_2_5"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font></span>)</span>
 <b>by </b><i><a class="ref" href="pencil_4.html#T3" target="_self">Th3</a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E49:23_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">J</font></span>)</span> <a href="vectsp_5.html#K2">/\</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u1</font>,<font color="Maroon">v1</font></span><a href="domain_1.html#K7">}</a></span></span>)</span> is    <a href="vectsp_4.html#M1">Subspace</a> of  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T35">VECTSP_4:35</a></i>;<br/><a NAME="E50:23_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">J</font></span>)</span> <a href="vectsp_5.html#K2">/\</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u1</font>,<font color="Maroon">v1</font></span><a href="domain_1.html#K7">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="vectsp_4.html#T33">VECTSP_4:33</a></i>;<br/><a NAME="E51:23_1_1_1_2"/><b>then </b><i><font color="Green">E116</font></i>: 
 <a href="vectsp_9.html#K1">dim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">J</font></span>)</span> <a href="vectsp_5.html#K1">+</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u1</font>,<font color="Maroon">v1</font></span><a href="domain_1.html#K7">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_9.html#K1">dim</a> <font color="Maroon">W1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i>, , , , <a class="ref" href="pencil_4.html#T5" target="_self">Th5</a>, <a class="ref" href="pencil_4.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E52:23_1_1_1_2"/><b>then </b><i><font color="Green">E117</font></i>: 
<span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">J</font></span>)</span> <a href="vectsp_5.html#K1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u1</font>,<font color="Maroon">v1</font></span><a href="domain_1.html#K7">}</a></span></span>)</span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_4.html#K5" target="_self">PencilSpace</a> <font color="Maroon">V</font>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_9.html#D3">VECTSP_9:def 3</a>, <a class="txt" href="pencil_4.html#E1:23"><i><font color="Green">E24</font></i></a></i>;<br/><a NAME="E53:23_1_1_1_2"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">i</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_2_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_1_1_2_6"/>
<span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">J</font></span>)</span> <a href="vectsp_5.html#K1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u1</font>,<font color="Maroon">v1</font></span><a href="domain_1.html#K7">}</a></span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><a NAME="E2:23_1_1_1_2_6"/><b>then </b><i><font color="Green">E119</font></i>: 
<font color="Maroon">W1</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">J</font></span>)</span> <a href="vectsp_5.html#K1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u1</font>,<font color="Maroon">v1</font></span><a href="domain_1.html#K7">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_4.html#E1:23"><i><font color="Green">E24</font></i></a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_2_6_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_1_1_2_6_1"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">i</font> <a href="rlvect_1.html#R1">in</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">J</font></span>)</span> <a href="vectsp_5.html#K1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u1</font>,<font color="Maroon">v1</font></span><a href="domain_1.html#K7">}</a></span></span>)</span>
 ;<br/><b>reconsider </b><font color="Maroon">IV</font> = <font color="Maroon">I</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="pencil_4.html#T2" target="_self">Th2</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E3:23_1_1_1_2_6_1"/>
<font color="Maroon">IV</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">J</font></span>)</span> <a href="vectsp_5.html#K1">+</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u1</font>,<font color="Maroon">v1</font></span><a href="domain_1.html#K7">}</a></span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_2_6_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:23_1_1_1_2_6_1_1"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IV</font>
 ;<br/>

<a NAME="E2:23_1_1_1_2_6_1_1"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E3:23_1_1_1_2_6_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/>
<a NAME="E4:23_1_1_1_2_6_1_1"/><b>then </b><i><font color="Green">E123</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1_2_6_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_1_1_2_6_1_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i</font></span><a href="domain_1.html#K6">}</a></span> )
 </span><b>by </b><i><a class="ref" href="pencil_4.html#T7" target="_self">Th7</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1_2_6_1_1_1_1"><b>suppose </b></a><a NAME="E1:23_1_1_1_2_6_1_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font>
 ;<br/><div class="add"><a NAME="E2:23_1_1_1_2_6_1_1_1_1"/><b>then </b><i><font color="Green">E124</font></i>: 
<font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">J</font>
 <b>by </b><i><a class="ref" href="vectsp_7.html#T13">VECTSP_7:13</a></i>;<br/><a NAME="E3:23_1_1_1_2_6_1_1_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T17">VECTSP_4:17</a></i>;<br/><b>then reconsider </b><font color="Maroon">o</font> = <font color="Maroon">a</font> as   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E5:23_1_1_1_2_6_1_1_1_1"/>
<font color="Maroon">o</font> <a href="rlvect_1.html#R1">in</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">J</font></span>)</span> <a href="vectsp_5.html#K1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u1</font>,<font color="Maroon">v1</font></span><a href="domain_1.html#K7">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_5.html#T6">VECTSP_5:6</a>, </i>;<br/><b>hence </b><a NAME="E6:23_1_1_1_2_6_1_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">J</font></span>)</span> <a href="vectsp_5.html#K1">+</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u1</font>,<font color="Maroon">v1</font></span><a href="domain_1.html#K7">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1_2_6_1_1_1_2"><b>suppose </b></a><a NAME="E1:23_1_1_1_2_6_1_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:23_1_1_1_2_6_1_1_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:23_1_1_1_2_6_1_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">J</font></span>)</span> <a href="vectsp_5.html#K1">+</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u1</font>,<font color="Maroon">v1</font></span><a href="domain_1.html#K7">}</a></span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E4:23_1_1_1_2_6_1"/><b>then </b>
 <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">IV</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">J</font></span>)</span> <a href="vectsp_5.html#K1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u1</font>,<font color="Maroon">v1</font></span><a href="domain_1.html#K7">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_9.html#T20">VECTSP_9:20</a></i>;<br/><a NAME="E5:23_1_1_1_2_6_1"/><b>then </b>
 <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">I</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">J</font></span>)</span> <a href="vectsp_5.html#K1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u1</font>,<font color="Maroon">v1</font></span><a href="domain_1.html#K7">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_9.html#T21">VECTSP_9:21</a></i>;<br/><a NAME="E6:23_1_1_1_2_6_1"/><b>then </b><i><font color="Green">E126</font></i>: 
<font color="Maroon">W1</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">J</font></span>)</span> <a href="vectsp_5.html#K1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u1</font>,<font color="Maroon">v1</font></span><a href="domain_1.html#K7">}</a></span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E7:23_1_1_1_2_6_1"/>
 <a href="vectsp_7.html#K1">Lin</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u1</font>,<font color="Maroon">v1</font></span><a href="domain_1.html#K7">}</a></span> is    <a href="vectsp_4.html#M1">Subspace</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">J</font></span>)</span> <a href="vectsp_5.html#K1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u1</font>,<font color="Maroon">v1</font></span><a href="domain_1.html#K7">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_5.html#T11">VECTSP_5:11</a></i>;<br/><a NAME="E8:23_1_1_1_2_6_1"/><b>then </b>
<font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u1</font>,<font color="Maroon">v1</font></span><a href="domain_1.html#K7">}</a></span></span>)</span> is    <a href="vectsp_4.html#M1">Subspace</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">J</font></span>)</span> <a href="vectsp_5.html#K1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u1</font>,<font color="Maroon">v1</font></span><a href="domain_1.html#K7">}</a></span></span>)</span>
 <b>by </b><i>, <a class="ref" href="vectsp_5.html#T40">VECTSP_5:40</a></i>;<br/><a NAME="E9:23_1_1_1_2_6_1"/><b>then </b><i><font color="Green">E127</font></i>: 
 <a href="vectsp_9.html#K1">dim</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u1</font>,<font color="Maroon">v1</font></span><a href="domain_1.html#K7">}</a></span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="vectsp_9.html#K1">dim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">J</font></span>)</span> <a href="vectsp_5.html#K1">+</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u1</font>,<font color="Maroon">v1</font></span><a href="domain_1.html#K7">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_9.html#T29">VECTSP_9:29</a></i>;<br/><a NAME="E10:23_1_1_1_2_6_1"/><i><font color="Green">E128</font></i>: 
 <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">W1</font> <a href="vectsp_5.html#K2">/\</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u1</font>,<font color="Maroon">v1</font></span><a href="domain_1.html#K7">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T50">VECTSP_4:50</a></i>;<br/><a NAME="E11:23_1_1_1_2_6_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K2">/\</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u1</font>,<font color="Maroon">v1</font></span><a href="domain_1.html#K7">}</a></span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_2_6_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:23_1_1_1_2_6_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K2">/\</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u1</font>,<font color="Maroon">v1</font></span><a href="domain_1.html#K7">}</a></span></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:23_1_1_1_2_6_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> <a href="vectsp_5.html#K2">/\</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u1</font>,<font color="Maroon">v1</font></span><a href="domain_1.html#K7">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E3:23_1_1_1_2_6_1_2"/><b>then </b>
( <font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_7.html#K1">Lin</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u1</font>,<font color="Maroon">v1</font></span><a href="domain_1.html#K7">}</a></span> )
 
<b>by </b><i><a class="ref" href="vectsp_5.html#T7">VECTSP_5:7</a></i>;<br/>
<a NAME="E4:23_1_1_1_2_6_1_2"/><b>then </b>
( <font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_7.html#K1">Lin</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">v</font></span><a href="domain_1.html#K7">}</a></span> )
 
<b>by </b><i><a class="ref" href="vectsp_9.html#T21">VECTSP_9:21</a></i>;<br/>
<a NAME="E5:23_1_1_1_2_6_1_2"/><b>then </b>
( <font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">T</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">T</font>,the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">T</font> #) )
 
<b>by </b><i/>;<br/>
<a NAME="E6:23_1_1_1_2_6_1_2"/><b>then </b>
( <font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">T</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">T</font>,the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">T</font> #) )
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E7:23_1_1_1_2_6_1_2"/><b>then </b>
( <font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">T</font> )
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E8:23_1_1_1_2_6_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> <a href="vectsp_5.html#K2">/\</a> <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="vectsp_5.html#T7">VECTSP_5:7</a></i>;<br/>
<b>hence </b><a NAME="E9:23_1_1_1_2_6_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font></span>)</span>
 <b>by </b><i><a class="ref" href="pencil_4.html#T3" target="_self">Th3</a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E12:23_1_1_1_2_6_1"/><b>then </b>
<font color="Maroon">W1</font> <a href="vectsp_5.html#K2">/\</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u1</font>,<font color="Maroon">v1</font></span><a href="domain_1.html#K7">}</a></span></span>)</span> is    <a href="vectsp_4.html#M1">Subspace</a> of  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T35">VECTSP_4:35</a></i>;<br/><a NAME="E13:23_1_1_1_2_6_1"/><b>then </b>
<font color="Maroon">W1</font> <a href="vectsp_5.html#K2">/\</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u1</font>,<font color="Maroon">v1</font></span><a href="domain_1.html#K7">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="vectsp_4.html#T33">VECTSP_4:33</a></i>;<br/><a NAME="E14:23_1_1_1_2_6_1"/><b>then </b>
 <a href="vectsp_9.html#K1">dim</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u1</font>,<font color="Maroon">v1</font></span><a href="domain_1.html#K7">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1">dim</a> <font color="Maroon">W1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i>, , , , </i>;<br/><a NAME="E15:23_1_1_1_2_6_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_9.html#K1">dim</a> <font color="Maroon">W1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1">dim</a> <font color="Maroon">W1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="pencil_4.html#D1" target="_self">Def1</a>, <a class="ref" href="pencil_4.html#T7" target="_self">Th7</a></i>;<br/><b>hence </b><a NAME="E16:23_1_1_1_2_6_1"/>
contradiction
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:23_1_1_1_2_6"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="vectsp_4.html#T16">VECTSP_4:16</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E55:23_1_1_1_2"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_4.html#K5" target="_self">PencilSpace</a> <font color="Maroon">V</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">B</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E2:23_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_4.html#K5" target="_self">PencilSpace</a> <font color="Maroon">V</font>,<font color="Maroon">k</font></span>)</span> is  not   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><a href="pencil_4.html#K5" target="_self">PencilSpace</a> <font color="Maroon">V</font>,<font color="Maroon">k</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E3:23_1"/>
not  <a href="pencil_4.html#K5" target="_self">PencilSpace</a> <font color="Maroon">V</font>,<font color="Maroon">k</font> is <a href="pencil_1.html#V4">degenerated</a>
 <b>by </b><i><a class="ref" href="pencil_1.html#D5">PENCIL_1:def 5</a></i>;<br/>


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


</div>
