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<b>let </b><font color="Maroon" title="c1">K</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="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="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="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">V</font> being  <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">K</font><br/> for <font color="Olive" title="b2">v</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Olive" title="b1">V</font><br/> for <font color="Olive" title="b3">W1</font>, <font color="Olive" title="b4">W2</font> being   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Olive" title="b1">V</font>  ex <font color="Olive" title="b5">v1</font>, <font color="Olive" title="b6">v2</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Olive" title="b1">V</font> st <font color="Olive" title="b2">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> <font color="Olive" title="b3">W1</font>,<font color="Olive" title="b4">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b5">v1</font>,<font color="Olive" title="b6">v2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><br/><b>let </b><font color="Maroon" title="c2">V</font> be   <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">K</font>; <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">v</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font><br/> for <font color="Olive" title="b2">W1</font>, <font color="Olive" title="b3">W2</font> being   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font>  ex <font color="Olive" title="b4">v1</font>, <font color="Olive" title="b5">v2</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font> st <font color="Olive" title="b1">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> <font color="Olive" title="b2">W1</font>,<font color="Olive" title="b3">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b4">v1</font>,<font color="Olive" title="b5">v2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><br/><b>let </b><font color="Maroon" title="c3">v</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font>; <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">W1</font>, <font color="Olive" title="b2">W2</font> being   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font>  ex <font color="Olive" title="b3">v1</font>, <font color="Olive" title="b4">v2</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font> st <font color="Maroon" title="c3">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> <font color="Olive" title="b1">W1</font>,<font color="Olive" title="b2">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b3">v1</font>,<font color="Olive" title="b4">v2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><br/><b>let </b><font color="Maroon" title="c4">W1</font>, <font color="Maroon" title="c5">W2</font> be    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font> st <font color="Maroon" title="c3">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> <font color="Maroon" title="c4">W1</font>,<font color="Maroon" title="c5">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">v2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><br/>







<b>take </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> <font color="Maroon" title="c4">W1</font>,<font color="Maroon" title="c5">W2</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> 
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">v2</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font> st <font color="Maroon" title="c3">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> <font color="Maroon" title="c4">W1</font>,<font color="Maroon" title="c5">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> <font color="Maroon" title="c4">W1</font>,<font color="Maroon" title="c5">W2</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<font color="Olive" title="b1">v2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><br/>

<b>take </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> <font color="Maroon" title="c4">W1</font>,<font color="Maroon" title="c5">W2</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> 
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> <font color="Maroon" title="c4">W1</font>,<font color="Maroon" title="c5">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> <font color="Maroon" title="c4">W1</font>,<font color="Maroon" title="c5">W2</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> <font color="Maroon" title="c4">W1</font>,<font color="Maroon" title="c5">W2</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><br/>

<b>thus </b><a NAME="E1:29"/>
<font color="Maroon" title="c3">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> <font color="Maroon" title="c4">W1</font>,<font color="Maroon" title="c5">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> <font color="Maroon" title="c4">W1</font>,<font color="Maroon" title="c5">W2</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> <font color="Maroon" title="c4">W1</font>,<font color="Maroon" title="c5">W2</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</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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