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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><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="Olive" title="b1">M</font>  st <font color="Olive" title="b1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Olive" title="b3">W2</font> &amp;  ex <font color="Olive" title="b4">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">M</font> st <br/>for <font color="Olive" title="b5">v1</font>, <font color="Olive" title="b6">v2</font>, <font color="Olive" title="b7">u1</font>, <font color="Olive" title="b8">u2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">M</font>  st <font color="Olive" title="b4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b6">v2</font> &amp; <font color="Olive" title="b4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">u1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b8">u2</font> &amp; <font color="Olive" title="b5">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b2">W1</font> &amp; <font color="Olive" title="b7">u1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b2">W1</font> &amp; <font color="Olive" title="b6">v2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b3">W2</font> &amp; <font color="Olive" title="b8">u2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b3">W2</font> holds <br/>( <font color="Olive" title="b5">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">u1</font> &amp; <font color="Olive" title="b6">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">u2</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> <font color="Olive" title="b2">W1</font>,<font color="Olive" title="b3">W2</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"> 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">M</font>  st <font color="Maroon" title="c2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Olive" title="b2">W2</font> &amp;  ex <font color="Olive" title="b3">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M</font> st <br/>for <font color="Olive" title="b4">v1</font>, <font color="Olive" title="b5">v2</font>, <font color="Olive" title="b6">u1</font>, <font color="Olive" title="b7">u2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M</font>  st <font color="Olive" title="b3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b5">v2</font> &amp; <font color="Olive" title="b3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">u1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b7">u2</font> &amp; <font color="Olive" title="b4">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b1">W1</font> &amp; <font color="Olive" title="b6">u1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b1">W1</font> &amp; <font color="Olive" title="b5">v2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b2">W2</font> &amp; <font color="Olive" title="b7">u2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b2">W2</font> holds <br/>( <font color="Olive" title="b4">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">u1</font> &amp; <font color="Olive" title="b5">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">u2</font> ) holds <br/><font color="Maroon" title="c2">M</font> <a href="vectsp_5.html#R1" title="VECTSP_5:pred.1">is_the_direct_sum_of</a> <font color="Olive" title="b1">W1</font>,<font color="Olive" title="b2">W2</font></span><br/><b>let </b><font color="Maroon" title="c3">W1</font>, <font color="Maroon" title="c4">W2</font> be    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">M</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c4">W2</font> &amp;  ex <font color="Olive" title="b1">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M</font> st <br/>for <font color="Olive" title="b2">v1</font>, <font color="Olive" title="b3">v2</font>, <font color="Olive" title="b4">u1</font>, <font color="Olive" title="b5">u2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M</font>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b3">v2</font> &amp; <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">u1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b5">u2</font> &amp; <font color="Olive" title="b2">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</font> &amp; <font color="Olive" title="b4">u1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</font> &amp; <font color="Olive" title="b3">v2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W2</font> &amp; <font color="Olive" title="b5">u2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W2</font> holds <br/>( <font color="Olive" title="b2">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">u1</font> &amp; <font color="Olive" title="b3">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">u2</font> ) implies <font color="Maroon" title="c2">M</font> <a href="vectsp_5.html#R1" title="VECTSP_5:pred.1">is_the_direct_sum_of</a> <font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font> )</span><br/>







<b>assume </b><a NAME="E1:58"/><i><font color="Green" title="E44">A1</font></i>: 
<font color="Maroon" title="c2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c4">W2</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="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M</font>  ex <font color="Olive" title="b2">v1</font>, <font color="Olive" title="b3">v2</font>, <font color="Olive" title="b4">u1</font>, <font color="Olive" title="b5">u2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M</font> st <br/>( <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b3">v2</font> &amp; <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">u1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b5">u2</font> &amp; <font color="Olive" title="b2">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</font> &amp; <font color="Olive" title="b4">u1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</font> &amp; <font color="Olive" title="b3">v2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W2</font> &amp; <font color="Olive" title="b5">u2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W2</font> &amp; not <!--HUMANRECFAILED-->( <font color="Olive" title="b2">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">u1</font> &amp; <font color="Olive" title="b3">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">u2</font> ) ) or <font color="Maroon" title="c2">M</font> <a href="vectsp_5.html#R1" title="VECTSP_5:pred.1">is_the_direct_sum_of</a> <font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font> )</span><br/>

<b>given </b><font color="Maroon" title="c5">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M</font><b> such that </b><a NAME="E3:58"/><i><font color="Green" title="E45">A2</font></i>: 
for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font>, <font color="Olive" title="b3">u1</font>, <font color="Olive" title="b4">u2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M</font>  st <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">v2</font> &amp; <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">u1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b4">u2</font> &amp; <font color="Olive" title="b1">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</font> &amp; <font color="Olive" title="b3">u1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</font> &amp; <font color="Olive" title="b2">v2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W2</font> &amp; <font color="Olive" title="b4">u2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W2</font> holds <br/>( <font color="Olive" title="b1">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">u1</font> &amp; <font color="Olive" title="b2">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">u2</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> <font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font></span><br/>

<a NAME="E4:58"/><i><font color="Green" title="E46">A3</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">M</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#D3" title="VECTSP_4:def.3">VECTSP_4:def 3</a></i>;<br/>
<b>assume </b><a NAME="E5:58"/>
not <font color="Maroon" title="c2">M</font> <a href="vectsp_5.html#R1" title="VECTSP_5:pred.1">is_the_direct_sum_of</a> <font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E6:58"/><b>then </b>
<font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c4">W2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <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#E1:58"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="vectsp_5.html#D4" target="_self" title="VECTSP_5:def.4">Def4</a></i>;<br/>
<a NAME="E7:58"/><b>then </b>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c4">W2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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> &amp;  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">M</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c4">W2</font> )
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T37" title="VECTSP_4:th.37">VECTSP_4:37</a>, <a class="ref" href="vectsp_4.html#T50" title="VECTSP_4:th.50">VECTSP_4:50</a></i>;<br/>
<a NAME="E8:58"/><b>then </b>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c4">W2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">M</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">M</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c4">W2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="vectsp_5.html#E4:58"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="vectsp_4.html#D2" title="VECTSP_4:def.2">VECTSP_4:def 2</a></i>;<br/>
<a NAME="E9:58"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">M</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c4">W2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:58"/><i><font color="Green" title="E47">A4</font></i>: 
<font color="Maroon" title="c6">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"><font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c4">W2</font></span>)</span>
 <b>and </b><br/><a NAME="E12:58"/><i><font color="Green" title="E48">A5</font></i>: 
not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">M</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#T6" title="XBOOLE_0:th.6">XBOOLE_0:6</a></i>;<br/>
<a NAME="E13:58"/><i><font color="Green" title="E49">A6</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c4">W2</font>
 
<b>by </b><i><a class="txt" href="vectsp_5.html#E11:58"><i><font color="Green" title="E47">A4</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E14:58"/><b>then </b><i><font color="Green" title="E50">A7</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</font> &amp; <font color="Maroon" title="c6">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W2</font> )
 
<b>by </b><i><a class="ref" href="vectsp_5.html#T7" target="_self" title="VECTSP_5:th.7">Th7</a></i>;<br/>
<a NAME="E15:58"/><i><font color="Green" title="E51">A8</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="txt" href="vectsp_5.html#E12:58"><i><font color="Green" title="E48">A5</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E16:58"/>
<font color="Maroon" title="c6">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="txt" href="vectsp_5.html#E13:58"><i><font color="Green" title="E49">A6</font></i></a>, <a class="ref" href="vectsp_4.html#T17" title="VECTSP_4:th.17">VECTSP_4:17</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">u</font> = <font color="Maroon" title="c6">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M</font> <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">v1</font>, <font color="Maroon" title="c9">v2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M</font><b> such that </b><br/><a NAME="E19:58"/><i><font color="Green" title="E52">A9</font></i>: 
( <font color="Maroon" title="c8">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</font> &amp; <font color="Maroon" title="c9">v2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W2</font> )
 <b>and </b><br/><a NAME="E20:58"/><i><font color="Green" title="E53">A10</font></i>: 
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">v2</font>
 <b>by </b><i><a class="txt" href="vectsp_5.html#E1:58"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="vectsp_5.html#E64"><i><font color="Green" title="E34">Lm16</font></i></a></i>;<br/>
<i><font color="Green" title="E54">A11</font></i>: <font color="Maroon" title="c5">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">v2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">M</font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp_5.html#E20:58"><i><font color="Green" title="E53">A10</font></i></a>, <a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">v2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c7">u</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T66" title="VECTSP_1:th.66">VECTSP_1:66</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">v2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">u</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c7">u</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">v2</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c7">u</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">v2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c7">u</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
;<br/>
<a NAME="E22:58"/>
( <font color="Maroon" title="c8">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</font> &amp; <font color="Maroon" title="c9">v2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c7">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W2</font> )
 
<b>by </b><i><a class="txt" href="vectsp_5.html#E14:58"><i><font color="Green" title="E50">A7</font></i></a>, <a class="txt" href="vectsp_5.html#E19:58"><i><font color="Green" title="E52">A9</font></i></a>, <a class="ref" href="vectsp_4.html#T28" title="VECTSP_4:th.28">VECTSP_4:28</a>, <a class="ref" href="vectsp_4.html#T31" title="VECTSP_4:th.31">VECTSP_4:31</a></i>;<br/>
<a NAME="E23:58"/><b>then </b>
<font color="Maroon" title="c9">v2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c7">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v2</font>
 
<b>by </b><i><a class="txt" href="vectsp_5.html#E3:58"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="vectsp_5.html#E19:58"><i><font color="Green" title="E52">A9</font></i></a>, <a class="txt" href="vectsp_5.html#E20:58"><i><font color="Green" title="E53">A10</font></i></a>, <a class="txt" href="vectsp_5.html#E21:58"><i><font color="Green" title="E54">A11</font></i></a></i>;<br/>
<a NAME="E24:58"/><b>then </b>
<font color="Maroon" title="c9">v2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">u</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">M</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>;<br/>
<a NAME="E25:58"/><b>then </b>
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T21" title="RLVECT_1:th.21">RLVECT_1:21</a></i>;<br/>
<a NAME="E26:58"/><b>then </b>
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">M</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T30" title="RLVECT_1:th.30">RLVECT_1:30</a></i>;<br/>
<b>hence </b><a NAME="E27:58"/>
contradiction
 <b>by </b><i><a class="txt" href="vectsp_5.html#E15:58"><i><font color="Green" title="E51">A8</font></i></a>, <a class="ref" href="rlvect_1.html#T25" title="RLVECT_1:th.25">RLVECT_1:25</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
