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<b>let </b><font color="Maroon" title="c1">R</font> be   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>;<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>;<br/>



<b>set </b><font color="Maroon" title="c3">W</font> =  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">B'</font> =  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font></span>)</span> <b>by </b><i><a class="ref" href="subset_1.html#T4" title="SUBSET_1:th.4">SUBSET_1:4</a></i>;<br/>
<a NAME="E2:18"/><i><font color="Green" title="E14">A1</font></i>: 
<font color="Maroon" title="c4">B'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font></span>)</span>
 
;<br/>
<a NAME="E3:18"/><b>then </b><i><font color="Green" title="E15">A2</font></i>: 
<font color="Maroon" title="c4">B'</font> is <a href="lmod_5.html#V1" title="LMOD_5:attr.1">linearly-independent</a>
 
<b>by </b><i><a class="ref" href="lmod_5.html#T4" title="LMOD_5:th.4">LMOD_5:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">V'</font> = <font color="Maroon" title="c2">V</font> as    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="ref" href="vectsp_4.html#T32" title="VECTSP_4:th.32">VECTSP_4:32</a></i>;<br/>
<a NAME="E5:18"/>
 <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c5">V'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <span class="p1">(<span class="default"><a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T48" title="VECTSP_4:th.48">VECTSP_4:48</a></i>;<br/>
<a NAME="E6:18"/><b>then </b>
 <a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Maroon" title="c4">B'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font>
 
<b>by </b><i><a class="txt" href="mod_3.html#E2:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="mod_3.html#T13" target="_self" title="MOD_3:th.13">Th13</a></i>;<br/>
<a NAME="E7:18"/><b>then </b>
<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="txt" href="mod_3.html#E3:18"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="mod_3.html#D2" target="_self" title="MOD_3:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E8:18"/>
 <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font> is <a href="mod_3.html#V2" title="MOD_3:attr.2">free</a>
 <b>by </b><i><a class="ref" href="mod_3.html#D3" target="_self" title="MOD_3:def.3">Def3</a></i>;<br/>


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