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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">W</font> be    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>







<b>assume </b><a NAME="E1:29"/><i><font color="Green">E24</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="vectsp_9.html#K1">dim</a> <font color="Maroon">V</font> &amp; <font color="Maroon">m</font> <a href="vectsp_9.html#K2">Subspaces_of</a> <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">m</font> <a href="vectsp_9.html#K2">Subspaces_of</a> <font color="Maroon">W</font> )
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:29_1"/><i><font color="Green">E29</font></i>: 
 <a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">V</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">W</font>
 ;<br/><a NAME="E2:29_1"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="vectsp_9.html#K1">dim</a> <font color="Maroon">W</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="vectsp_9.html#K1">dim</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="vectsp_9.html#T32">VECTSP_9:32</a></i>;<br/><a NAME="E3:29_1"/>
 <a href="vectsp_9.html#K1">dim</a> <font color="Maroon">W</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="vectsp_9.html#K1">dim</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="vectsp_9.html#T29">VECTSP_9:29</a></i>;<br/><a NAME="E4:29_1"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="vectsp_9.html#K1">dim</a> <font color="Maroon">W</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="vectsp_9.html#K1">dim</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="txt" href="pencil_4.html#E2:29_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1"/>
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_9.html#K1">dim</a> <font color="Maroon">V</font> or <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="vectsp_9.html#K1">dim</a> <font color="Maroon">V</font> )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a>, <a class="txt" href="pencil_4.html#E1:29_1"><i><font color="Green">E29</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1"><b>suppose </b></a><a NAME="E1:29_1_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_9.html#K1">dim</a> <font color="Maroon">V</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_1"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">m</font> <a href="vectsp_9.html#K2">Subspaces_of</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="vectsp_9.html#T41">VECTSP_9:41</a>, <a class="txt" href="pencil_4.html#E4:29_1"><i><font color="Green">E32</font></i></a></i>;<br/><a NAME="E3:29_1_1_1"/>
not <font color="Maroon">m</font> <a href="vectsp_9.html#K2">Subspaces_of</a> <font color="Maroon">V</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="vectsp_9.html#T40">VECTSP_9:40</a>, <a class="txt" href="pencil_4.html#E1:29_1_1_1"><i><font color="Green">E33</font></i></a></i>;<br/><b>hence </b><a NAME="E4:29_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pencil_4.html#E1:29_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="pencil_4.html#E2:29_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">m</font> <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="E2:29_1_1_2"/><i><font color="Green">E52</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</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="29_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_1_1_2_1"/><i><font color="Green">E67</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">W</font>
 ;<br/><a NAME="E2:29_1_1_2_1"/><i><font color="Green">E68</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">V</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">W</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:29_1_1_2_1"/>
 <a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">W</font> is  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:29_1_1_2_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#E1:29_1_1_2"><i><font color="Green">E36</font></i></a>, </i>;<br/></div><b>end;</b></div><a NAME="E4:29_1_1_2"/><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">W</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">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:29_1_1_2"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">x</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">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font> <b>by </b><i/>;<br/><a NAME="E8:29_1_1_2"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T25">VECTSP_4:25</a></i>;<br/><a NAME="E9:29_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <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="E10:29_1_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</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#T5">VECTSP_7:5</a></i>;<br/><b>then consider </b><font color="Maroon">I</font> being    <a href="vectsp_7.html#M1">Basis</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E12:29_1_1_2"/><i><font color="Green">E82</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</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="vectsp_7.html#T27">VECTSP_7:27</a></i>;<br/><a NAME="E13:29_1_1_2"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">I</font> is <a href="vectsp_7.html#V1">linearly-independent</a> &amp;  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">V</font>,the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">V</font> #) )
 <b>by </b><i><a class="ref" href="vectsp_7.html#D3">VECTSP_7:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">J</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">V</font> <b>by </b><i><a class="ref" href="vectsp_9.html#T24">VECTSP_9:24</a></i>;<br/><a NAME="E15:29_1_1_2"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">J</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_9.html#K1">dim</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="vectsp_9.html#D2">VECTSP_9:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">K</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">J</font><b> such that </b><br/><a NAME="E17:29_1_1_2"/><i><font color="Green">E84</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_2_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1_2_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> or  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_2_2_1"><b>suppose </b></a><a NAME="E1:29_1_1_2_2_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">L</font> = <font color="Maroon">K</font> 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="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E3:29_1_1_2_2_1"/><i><font color="Green">E86</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="pencil_4.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E4:29_1_1_2_2_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">L</font> 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/><a NAME="E5:29_1_1_2_2_1"/>
<font color="Maroon">L</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">L</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_2_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="E1:29_1_1_2_2_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font>
 ;<br/>

<a NAME="E2:29_1_1_2_2_1_1"/><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">L</font>
 
<b>by </b><i><a class="ref" href="vectsp_7.html#T13">VECTSP_7:13</a></i>;<br/>
<b>hence </b><a NAME="E3:29_1_1_2_2_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"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">L</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">L'</font> = <font color="Maroon">L</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">L</font></span>)</span> ;<br/><b>reconsider </b><font color="Maroon">LL1</font> = <font color="Maroon">L'</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">L</font></span>)</span> <b>by </b><i><a class="ref" href="vectsp_9.html#T16">VECTSP_9:16</a>, </i>;<br/><a NAME="E8:29_1_1_2_2_1"/>
 <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">LL1</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_1.html#G4">VectSpStr</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">L</font></span>)</span>,the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">L</font></span>)</span>,the <a href="struct_0.html#U2">Zero</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">L</font></span>)</span>,the <a href="vectsp_1.html#U2">lmult</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">L</font></span>)</span> #)
 <b>by </b><i><a class="ref" href="vectsp_9.html#T21">VECTSP_9:21</a></i>;<br/><a NAME="E9:29_1_1_2_2_1"/><b>then </b>
<font color="Maroon">L</font> is    <a href="vectsp_7.html#M1">Basis</a> of  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="vectsp_7.html#D3">VECTSP_7:def 3</a></i>;<br/><a NAME="E10:29_1_1_2_2_1"/><b>then </b>
 <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">L</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="vectsp_9.html#D2">VECTSP_9:def 2</a></i>;<br/><a NAME="E11:29_1_1_2_2_1"/><b>then </b>
 <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">m</font> <a href="vectsp_9.html#K2">Subspaces_of</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="vectsp_9.html#D3">VECTSP_9:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">M</font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">W</font><b> such that </b><br/><a NAME="E13:29_1_1_2_2_1"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">L</font> &amp;  <a href="vectsp_9.html#K1">dim</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> )
 <b>by </b><i><a class="txt" href="pencil_4.html#E1:29_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="vectsp_9.html#D3">VECTSP_9:def 3</a></i>;<br/><a NAME="E14:29_1_1_2_2_1"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="vectsp_7.html#T13">VECTSP_7:13</a>, </i>;<br/><a NAME="E15:29_1_1_2_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="pencil_4.html#T4" target="_self">Th4</a>, <a class="ref" href="vectsp_4.html#T17">VECTSP_4:17</a></i>;<br/><b>hence </b><a NAME="E16:29_1_1_2_2_1"/>
contradiction
 <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="29_1_1_2_2_2"><b>suppose </b></a><a NAME="E1:29_1_1_2_2_2"/><i><font color="Green">E91</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:29_1_1_2_2_2"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a>, <a class="ref" href="pencil_4.html#T3" target="_self">Th3</a>, <a class="txt" href="pencil_4.html#E1:29_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="card_1.html#T78">CARD_1:78</a></i>;<br/><a NAME="E4:29_1_1_2_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_2_2_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="E1:29_1_1_2_2_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E2:29_1_1_2_2_2_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> 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">x</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/>
<a NAME="E3:29_1_1_2_2_2_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</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">x</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E4:29_1_1_2_2_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">L</font> = <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</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> ;<br/><a NAME="E6:29_1_1_2_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</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="E7:29_1_1_2_2_2"/><b>then </b><i><font color="Green">E94</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E8:29_1_1_2_2_2"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a>, </i>;<br/><b>then </b><i><font color="Green">E106</font></i>:  <a href="card_1.html#K4">card</a> <font color="Maroon">L</font> = 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="card_2.html#T54">CARD_2:54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">K</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
<b>by </b><i><a class="ref" href="euler_1.html#T5">EULER_1:5</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">K</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="ref" href="card_1.html#T79">CARD_1:79</a></i>
<br/>.= 
<font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="pencil_4.html#T3" target="_self">Th3</a></i>
;<br/><a NAME="E10:29_1_1_2_2_2"/>
<font color="Maroon">L</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_2_2_3"><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="E1:29_1_1_2_2_2_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font>
 ;<br/>

<a NAME="E2:29_1_1_2_2_2_3"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> 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">x</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/>
<a NAME="E3:29_1_1_2_2_2_3"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</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">x</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E4:29_1_1_2_2_2_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="txt" href="pencil_4.html#E1:29_1_1_2"><i><font color="Green">E36</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E11:29_1_1_2_2_2"/><b>then </b><i><font color="Green">E107</font></i>: 
<font color="Maroon">L</font> 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/><a NAME="E12:29_1_1_2_2_2"/>
<font color="Maroon">L</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">L</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_2_2_4"><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="E1:29_1_1_2_2_2_4"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font>
 ;<br/>

<a NAME="E2:29_1_1_2_2_2_4"/><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">L</font>
 
<b>by </b><i><a class="ref" href="vectsp_7.html#T13">VECTSP_7:13</a></i>;<br/>
<b>hence </b><a NAME="E3:29_1_1_2_2_2_4"/>
<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">L</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">L'</font> = <font color="Maroon">L</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">L</font></span>)</span> ;<br/><b>reconsider </b><font color="Maroon">LL1</font> = <font color="Maroon">L'</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">L</font></span>)</span> <b>by </b><i><a class="ref" href="vectsp_9.html#T16">VECTSP_9:16</a>, <a class="ref" href="pencil_4.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E15:29_1_1_2_2_2"/>
 <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">LL1</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_1.html#G4">VectSpStr</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">L</font></span>)</span>,the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">L</font></span>)</span>,the <a href="struct_0.html#U2">Zero</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">L</font></span>)</span>,the <a href="vectsp_1.html#U2">lmult</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">L</font></span>)</span> #)
 <b>by </b><i><a class="ref" href="vectsp_9.html#T21">VECTSP_9:21</a></i>;<br/><a NAME="E16:29_1_1_2_2_2"/><b>then </b>
<font color="Maroon">L</font> is    <a href="vectsp_7.html#M1">Basis</a> of  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="vectsp_7.html#D3">VECTSP_7:def 3</a></i>;<br/><a NAME="E17:29_1_1_2_2_2"/><b>then </b>
 <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">L</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="pencil_4.html#T4" target="_self">Th4</a>, <a class="ref" href="vectsp_9.html#D2">VECTSP_9:def 2</a></i>;<br/><a NAME="E18:29_1_1_2_2_2"/><b>then </b>
 <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">m</font> <a href="vectsp_9.html#K2">Subspaces_of</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="vectsp_9.html#D3">VECTSP_9:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">M</font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">W</font><b> such that </b><br/><a NAME="E20:29_1_1_2_2_2"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">L</font> &amp;  <a href="vectsp_9.html#K1">dim</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> )
 <b>by </b><i><a class="txt" href="pencil_4.html#E1:29_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="vectsp_9.html#D3">VECTSP_9:def 3</a></i>;<br/><a NAME="E21:29_1_1_2_2_2"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="vectsp_7.html#T13">VECTSP_7:13</a>, </i>;<br/><a NAME="E22:29_1_1_2_2_2"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="pencil_4.html#T6" target="_self">Th6</a>, <a class="ref" href="vectsp_4.html#T17">VECTSP_4:17</a></i>;<br/><b>hence </b><a NAME="E23:29_1_1_2_2_2"/>
contradiction
 <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></div><b>end;</b></div></div>
<b>end;</b></div>


</div>
