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<b>let </b><font color="Maroon" title="c1">K</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">K</font>;<br/><b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">A1</font>, <font color="Maroon" title="c5">A2</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font>;<br/>









<b>assume </b><a NAME="E1:17"/><i><font color="Green" title="E7">A1</font></i>: 
( <font color="Maroon" title="c3">A</font> is <a href="mod_3.html#V1" title="MOD_3:attr.1">base</a> &amp; <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">A2</font> &amp; <font color="Maroon" title="c4">A1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">A2</font> )
 ;<br/>

<b>set </b><font color="Maroon" title="c6">W</font> =  <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">VectSpStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c2">V</font>,the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c2">V</font> #);<br/>
<a NAME="E2:17"/><i><font color="Green" title="E8">A2</font></i>: 
( <font color="Maroon" title="c3">A</font> is <a href="lmod_5.html#V1" title="LMOD_5:attr.1">linearly-independent</a> &amp;  <a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">VectSpStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c2">V</font>,the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c2">V</font> #) )
 
<b>by </b><i><a class="txt" href="lmod_7.html#E1:17"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="mod_3.html#D2" title="MOD_3:def.2">MOD_3:def 2</a></i>;<br/>
<a NAME="E3:17"/><b>then </b><i><font color="Green" title="E9">A3</font></i>: 
 <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">VectSpStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c2">V</font>,the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c2">V</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Maroon" title="c4">A1</font></span>)</span> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p1">(<span class="default"><a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Maroon" title="c5">A2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="lmod_7.html#E1:17"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="mod_3.html#T19" title="MOD_3:th.19">MOD_3:19</a></i>;<br/>
<a NAME="E4:17"/>
<span class="p1">(<span class="default"><a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Maroon" title="c4">A1</font></span>)</span> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p1">(<span class="default"><a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Maroon" title="c5">A2</font></span>)</span> <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="lmod_7.html#E1:17"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="lmod_7.html#E2:17"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="lmod_7.html#T5" target="_self" title="LMOD_7:th.5">Th5</a></i>;<br/>
<b>hence </b><a NAME="E5:17"/>
<font color="Maroon" title="c2">V</font> <a href="vectsp_5.html#R1" title="VECTSP_5:pred.1">is_the_direct_sum_of</a>  <a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Maroon" title="c4">A1</font>, <a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Maroon" title="c5">A2</font>
 <b>by </b><i><a class="txt" href="lmod_7.html#E3:17"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="vectsp_5.html#D4" title="VECTSP_5:def.4">VECTSP_5:def 4</a></i>;<br/>


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