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<span class="kw">let </span><font color="Maroon" title="c1">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">R</font> being   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a><br/>  for <font color="Olive" title="b2">V</font> being   <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Olive" title="b1">R</font><br/>  for <font color="Olive" title="b3">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b2">V</font> holds <br/> ( <font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Olive" title="b3">A</font> iff  ex <font color="Olive" title="b4">l</font> being    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Olive" title="b3">A</font> st <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Olive" title="b4">l</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">R</font> be   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c2">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> holds <br/> ( <font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Olive" title="b2">A</font> iff  ex <font color="Olive" title="b3">l</font> being    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Olive" title="b2">A</font> st <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Olive" title="b3">l</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">V</font> be   <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c2">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c3">V</font> holds <br/> ( <font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Olive" title="b1">A</font> iff  ex <font color="Olive" title="b2">l</font> being    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Olive" title="b1">A</font> st <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Olive" title="b2">l</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Maroon" title="c4">A</font> iff  ex <font color="Olive" title="b1">l</font> being    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c4">A</font> st <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Olive" title="b1">l</font> )</span><br/>







<span class="kw">thus </span><a NAME="E1:6"/>
( <font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Maroon" title="c4">A</font> implies  ex <font color="Olive" title="b1">l</font> being    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c4">A</font> st <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Olive" title="b1">l</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">l</font> being    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c4">A</font> st <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Olive" title="b1">l</font> implies <font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Maroon" title="c4">A</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:6_1"/>
<font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Maroon" title="c4">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">l</font> being    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c4">A</font> st <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Olive" title="b1">l</font></span><br/>

<a NAME="E2:6_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Maroon" title="c4">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E3:6_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Olive" title="b1">l</font></span>)</span> where <font color="Olive" title="b1">l</font> is    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c4">A</font> : verum </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mod_3.html#D1" target="_self" title="MOD_3:def.1">Def1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:6_1"/>
 ex <font color="Olive" title="b1">l</font> being    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c4">A</font> st <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Olive" title="b1">l</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">given </span><font color="Maroon" title="c5">k</font> being    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c4">A</font><span class="kw"> such that </span><a NAME="E3:6"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c5">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Maroon" title="c4">A</font></span><br/>

<a NAME="E4:6"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Olive" title="b1">l</font></span>)</span> where <font color="Olive" title="b1">l</font> is    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c4">A</font> : verum </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:6"><span class="lab"><font color="Green" title="E7">A1</font></span></a></span>;<br/>
<a NAME="E5:6"/><span class="kw">then </span>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Maroon" title="c4">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mod_3.html#D1" target="_self" title="MOD_3:def.1">Def1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:6"/>
<font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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