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<b>let </b><font color="Maroon">K</font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="vectsp_2.html#NM6">LeftMod</a> of <font color="Maroon">K</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">A1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">A2</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>;<br/>









<b>assume </b><a NAME="E1:17"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">A</font> is <a href="mod_3.html#V1">base</a> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">A1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A2</font> )
 ;<br/>

<b>set </b><font color="Maroon">W</font> =  <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> #);<br/>
<a NAME="E2:17"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">A</font> is <a href="lmod_5.html#V1">linearly-independent</a> &amp;  <a href="mod_3.html#K1">Lin</a> <font color="Maroon">A</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="mod_3.html#D2">MOD_3:def 2</a></i>;<br/>
<a NAME="E3:17"/><b>then </b><i><font color="Green">E51</font></i>: 
 <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> #) <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="mod_3.html#K1">Lin</a> <font color="Maroon">A1</font></span>)</span> <a href="vectsp_5.html#K1">+</a> <span class="p1">(<span class="default"><a href="mod_3.html#K1">Lin</a> <font color="Maroon">A2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="mod_3.html#T19">MOD_3:19</a></i>;<br/>
<a NAME="E4:17"/>
<span class="p1">(<span class="default"><a href="mod_3.html#K1">Lin</a> <font color="Maroon">A1</font></span>)</span> <a href="vectsp_5.html#K2">/\</a> <span class="p1">(<span class="default"><a href="mod_3.html#K1">Lin</a> <font color="Maroon">A2</font></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>, , </i>;<br/>
<b>hence </b><a NAME="E5:17"/>
<font color="Maroon">V</font> <a href="vectsp_5.html#R1">is_the_direct_sum_of</a>  <a href="mod_3.html#K1">Lin</a> <font color="Maroon">A1</font>, <a href="mod_3.html#K1">Lin</a> <font color="Maroon">A2</font>
 <b>by </b><i>, <a class="ref" href="vectsp_5.html#D4">VECTSP_5:def 4</a></i>;<br/>


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