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<b>let </b><font color="Maroon" title="c1">R</font> be   <a href="vectsp_2.html#NM3" title="VECTSP_2:NM.3">Skew-Field</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">V</font> being  <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c1">R</font><br/> for <font color="Olive" title="b2">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">V</font>  st <font color="Olive" title="b2">A</font> is <a href="lmod_5.html#V1" title="LMOD_5:attr.1">linearly-independent</a> holds <br/> ex <font color="Olive" title="b3">I</font> being   <a href="mod_3.html#M1" title="MOD_3:mode.1">Basis</a> of <font color="Olive" title="b1">V</font> st <font color="Olive" title="b2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">I</font></span><br/><b>let </b><font color="Maroon" title="c2">V</font> be   <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font>  st <font color="Olive" title="b1">A</font> is <a href="lmod_5.html#V1" title="LMOD_5:attr.1">linearly-independent</a> holds <br/> ex <font color="Olive" title="b2">I</font> being   <a href="mod_3.html#M1" title="MOD_3:mode.1">Basis</a> of <font color="Maroon" title="c2">V</font> st <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">I</font></span><br/><b>let </b><font color="Maroon" title="c3">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">A</font> is <a href="lmod_5.html#V1" title="LMOD_5:attr.1">linearly-independent</a> implies  ex <font color="Olive" title="b1">I</font> being   <a href="mod_3.html#M1" title="MOD_3:mode.1">Basis</a> of <font color="Maroon" title="c2">V</font> st <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">I</font> )</span><br/>





<b>assume </b><a NAME="E1:29"/>
<font color="Maroon" title="c3">A</font> is <a href="lmod_5.html#V1" title="LMOD_5:attr.1">linearly-independent</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">I</font> being   <a href="mod_3.html#M1" title="MOD_3:mode.1">Basis</a> of <font color="Maroon" title="c2">V</font> st <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">I</font></span><br/>

<b>then consider </b><font color="Maroon" title="c4">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E3:29"/><i><font color="Green" title="E20">A1</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B</font>
 <b>and </b><br/><a NAME="E4:29"/><i><font color="Green" title="E21">A2</font></i>: 
<font color="Maroon" title="c4">B</font> is <a href="mod_3.html#V1" title="MOD_3:attr.1">base</a>
 <b>by </b><i><a class="ref" href="mod_3.html#T26" target="_self" title="MOD_3:th.26">Th26</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">B</font> = <font color="Maroon" title="c4">B</font> as    <a href="mod_3.html#M1" title="MOD_3:mode.1">Basis</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="txt" href="#E4:29"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="mod_3.html#D5" target="_self" title="MOD_3:def.5">Def5</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">B</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font></span><br/>

<b>thus </b><a NAME="E6:29"/>
<font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font>
 <b>by </b><i><a class="txt" href="#E3:29"><i><font color="Green" title="E20">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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