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<b>let </b><font color="Maroon" title="c1">GF</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</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">M</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">VectSp-like</a> <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  <a href="vectsp_1.html#L1" title="VECTSP_1:struct.1">VectSpStr</a> of <font color="Maroon" title="c1">GF</font> holds <br/> ( <font color="Olive" title="b1">M</font> <a href="vectsp_5.html#R1" title="VECTSP_5:pred.1">is_the_direct_sum_of</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Olive" title="b1">M</font>, <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Olive" title="b1">M</font> &amp; <font color="Olive" title="b1">M</font> <a href="vectsp_5.html#R1" title="VECTSP_5:pred.1">is_the_direct_sum_of</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Olive" title="b1">M</font>, <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Olive" title="b1">M</font> )</span><br/><b>let </b><font color="Maroon" title="c2">M</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">VectSp-like</a> <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  <a href="vectsp_1.html#L1" title="VECTSP_1:struct.1">VectSpStr</a> of <font color="Maroon" title="c1">GF</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">M</font> <a href="vectsp_5.html#R1" title="VECTSP_5:pred.1">is_the_direct_sum_of</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">M</font>, <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c2">M</font> <a href="vectsp_5.html#R1" title="VECTSP_5:pred.1">is_the_direct_sum_of</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c2">M</font>, <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">M</font> )</span><br/>



<a NAME="E1:52"/>
( <span class="p1">(<span class="default"><a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p1">(<span class="default"><a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c2">M</font></span>)</span> <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">M</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">U7</a> of <font color="Maroon" title="c2">M</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c2">M</font>,the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c2">M</font> #) &amp;  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">M</font></span>)</span> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p1">(<span class="default"><a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c2">M</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="vectsp_5.html#T13" target="_self" title="VECTSP_5:th.13">Th13</a>, <a class="ref" href="vectsp_5.html#T25" target="_self" title="VECTSP_5:th.25">Th25</a></i>;<br/>
<b>hence </b><a NAME="E2:52"/>
<font color="Maroon" title="c2">M</font> <a href="vectsp_5.html#R1" title="VECTSP_5:pred.1">is_the_direct_sum_of</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">M</font>, <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#D4" target="_self" title="VECTSP_5:def.4">Def4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">M</font> <a href="vectsp_5.html#R1" title="VECTSP_5:pred.1">is_the_direct_sum_of</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c2">M</font>, <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">M</font></span><br/>

<b>hence </b><a NAME="E3:52"/>
<font color="Maroon" title="c2">M</font> <a href="vectsp_5.html#R1" title="VECTSP_5:pred.1">is_the_direct_sum_of</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c2">M</font>, <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="txt" href="vectsp_5.html#E66"><i><font color="Green" title="E36">Lm17</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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