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<span class="kw">let </span><font color="Maroon" title="c1">V</font> be   <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">W1</font>, <font color="Olive" title="b2">W2</font> being    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>  st <font color="Maroon" title="c1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W1</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Olive" title="b2">W2</font> &amp;  ex <font color="Olive" title="b3">v</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</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="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</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="c1">V</font> <a href="rusub_2.html#R1" title="RUSUB_2:pred.1">is_the_direct_sum_of</a> <font color="Olive" title="b1">W1</font>,<font color="Olive" title="b2">W2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">W1</font>, <font color="Maroon" title="c3">W2</font> be    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W1</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c3">W2</font> &amp;  ex <font color="Olive" title="b1">v</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</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="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</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="c2">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="c2">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="c3">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="c3">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="c1">V</font> <a href="rusub_2.html#R1" title="RUSUB_2:pred.1">is_the_direct_sum_of</a> <font color="Maroon" title="c2">W1</font>,<font color="Maroon" title="c3">W2</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:57"/><span class="lab"><font color="Green" title="E46">A1</font></span>: 
<font color="Maroon" title="c1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W1</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c3">W2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  for <font color="Olive" title="b1">v</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</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="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</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="c2">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="c2">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="c3">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="c3">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="c1">V</font> <a href="rusub_2.html#R1" title="RUSUB_2:pred.1">is_the_direct_sum_of</a> <font color="Maroon" title="c2">W1</font>,<font color="Maroon" title="c3">W2</font> )</span><br/>

<a NAME="E2:57"/>
(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rusub_1.html#K1" title="RUSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</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="c1">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp;  <a href="rusub_1.html#K1" title="RUSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font> is    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c2">W1</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c3">W2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rusub_1.html#T33" title="RUSUB_1:th.33">RUSUB_1:33</a>, <a class="ref" href="rusub_1.html#D2" title="RUSUB_1:def.2">RUSUB_1:def 2</a></span>;<br/>
<a NAME="E3:57"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A2</font></span>: 
<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="c1">V</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="c2">W1</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c3">W2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rusub_1.html#D1" title="RUSUB_1:def.1">RUSUB_1:def 1</a></span>;<br/>
<span class="kw">given </span><font color="Maroon" title="c4">v</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><a NAME="E5:57"/><span class="lab"><font color="Green" title="E48">A3</font></span>: 
 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="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st <font color="Maroon" title="c4">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="c4">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="c2">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="c2">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="c3">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="c3">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">V</font> <a href="rusub_2.html#R1" title="RUSUB_2:pred.1">is_the_direct_sum_of</a> <font color="Maroon" title="c2">W1</font>,<font color="Maroon" title="c3">W2</font></span><br/>

<span class="kw">assume </span><a NAME="E6:57"/>
 not <font color="Maroon" title="c1">V</font> <a href="rusub_2.html#R1" title="RUSUB_2:pred.1">is_the_direct_sum_of</a> <font color="Maroon" title="c2">W1</font>,<font color="Maroon" title="c3">W2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E7:57"/><span class="kw">then </span>
<font color="Maroon" title="c2">W1</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c3">W2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rusub_1.html#K1" title="RUSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="ref" href="rusub_2.html#D4" target="_self" title="RUSUB_2:def.4">Def4</a></span>;<br/>
<a NAME="E8:57"/><span class="kw">then </span>
 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="c2">W1</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c3">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="c1">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rusub_1.html#D2" title="RUSUB_1:def.2">RUSUB_1:def 2</a></span>;<br/>
<a NAME="E9:57"/><span class="kw">then </span>
<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="c1">V</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="c2">W1</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c3">W2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:57"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E11:57"/><span class="lab"><font color="Green" title="E49">A4</font></span>: 
<font color="Maroon" title="c5">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="c2">W1</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c3">W2</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E12:57"/><span class="lab"><font color="Green" title="E50">A5</font></span>: 
 not <font color="Maroon" title="c5">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="c1">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T6" title="XBOOLE_0:th.6">XBOOLE_0:6</a></span>;<br/>
<a NAME="E13:57"/><span class="lab"><font color="Green" title="E51">A6</font></span>: 
<font color="Maroon" title="c5">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="c1">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:57"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E14:57"/><span class="lab"><font color="Green" title="E52">A7</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W1</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c3">W2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:57"><span class="lab"><font color="Green" title="E49">A4</font></span></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E15:57"/><span class="kw">then </span>
<font color="Maroon" title="c5">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c1">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rusub_1.html#T2" title="RUSUB_1:th.2">RUSUB_1:2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">u</font> = <font color="Maroon" title="c5">x</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">v1</font>, <font color="Maroon" title="c8">v2</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E18:57"/><span class="lab"><font color="Green" title="E53">A8</font></span>: 
<font color="Maroon" title="c7">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W1</font>
 <span class="kw">and </span><br/><a NAME="E19:57"/><span class="lab"><font color="Green" title="E54">A9</font></span>: 
<font color="Maroon" title="c8">v2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W2</font>
 <span class="kw">and </span><br/><a NAME="E20:57"/><span class="lab"><font color="Green" title="E55">A10</font></span>: 
<font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="rusub_2.html#E50"><span class="lab"><font color="Green" title="E33">Lm13</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E56">A11</font></span>: <font color="Maroon" title="c4">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">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="c1">V</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:57"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="ref" href="rlvect_1.html#T4" title="RLVECT_1:th.4">RLVECT_1:4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">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="c6">u</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c6">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T15" title="RLVECT_1:th.15">RLVECT_1:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">v2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">u</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c6">u</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D3" title="RLVECT_1:def.3">RLVECT_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">v2</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c6">u</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D3" title="RLVECT_1:def.3">RLVECT_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">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="c8">v2</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c6">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D3" title="RLVECT_1:def.3">RLVECT_1:def 3</a></span>
;<br/>
<a NAME="E22:57"/>
<font color="Maroon" title="c5">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:57"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="ref" href="rusub_2.html#T3" target="_self" title="RUSUB_2:th.3">Th3</a></span>;<br/>
<a NAME="E23:57"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A12</font></span>: 
<font color="Maroon" title="c8">v2</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c6">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:57"><span class="lab"><font color="Green" title="E54">A9</font></span></a>, <a class="ref" href="rusub_1.html#T17" title="RUSUB_1:th.17">RUSUB_1:17</a></span>;<br/>
<a NAME="E24:57"/>
<font color="Maroon" title="c5">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:57"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="ref" href="rusub_2.html#T3" target="_self" title="RUSUB_2:th.3">Th3</a></span>;<br/>
<a NAME="E25:57"/><span class="kw">then </span>
<font color="Maroon" title="c7">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:57"><span class="lab"><font color="Green" title="E53">A8</font></span></a>, <a class="ref" href="rusub_1.html#T14" title="RUSUB_1:th.14">RUSUB_1:14</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c8">v2</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c6">u</font> = 
<font color="Maroon" title="c8">v2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:57"><span class="lab"><font color="Green" title="E48">A3</font></span></a>, <a class="txt" href="#E18:57"><span class="lab"><font color="Green" title="E53">A8</font></span></a>, <a class="txt" href="#E19:57"><span class="lab"><font color="Green" title="E54">A9</font></span></a>, <a class="txt" href="#E20:57"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="txt" href="#E21:57"><span class="lab"><font color="Green" title="E56">A11</font></span></a>, <a class="txt" href="#E23:57"><span class="lab"><font color="Green" title="E57">A12</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c8">v2</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T13" title="RLVECT_1:th.13">RLVECT_1:13</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E27:57"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:57"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="ref" href="rlvect_1.html#T23" title="RLVECT_1:th.23">RLVECT_1:23</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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