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<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon" title="c2">W</font> be    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>;<br/><b>let </b><font color="Maroon" title="c3">u</font>, <font color="Maroon" title="c4">v1</font>, <font color="Maroon" title="c5">v2</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:53"/><i><font color="Green" title="E40">A1</font></i>: 
<font color="Maroon" title="c3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v1</font> <a href="rusub_1.html#K3" title="RUSUB_1:func.3">+</a> <font color="Maroon" title="c2">W</font>
 <b>and </b><br/><a NAME="E2:53"/><i><font color="Green" title="E41">A2</font></i>: 
<font color="Maroon" title="c3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">v2</font> <a href="rusub_1.html#K3" title="RUSUB_1:func.3">+</a> <font color="Maroon" title="c2">W</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c6">A</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b1">u1</font></span>)</span> where <font color="Olive" title="b1">u1</font> is   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">u1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c7">B</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c5">v2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b1">u2</font></span>)</span> where <font color="Olive" title="b1">u2</font> is   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">u2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W</font> </span>}</span> ;<br/>
<b>consider </b><font color="Maroon" title="c8">x1</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E4:53"/><i><font color="Green" title="E42">A3</font></i>: 
<font color="Maroon" title="c3">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">x1</font>
 <b>and </b><br/><a NAME="E5:53"/><i><font color="Green" title="E43">A4</font></i>: 
<font color="Maroon" title="c8">x1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W</font>
 <b>by </b><i><a class="txt" href="rusub_1.html#E1:53"><i><font color="Green" title="E40">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">x2</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E7:53"/><i><font color="Green" title="E44">A5</font></i>: 
<font color="Maroon" title="c3">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">x2</font>
 <b>and </b><br/><a NAME="E8:53"/><i><font color="Green" title="E45">A6</font></i>: 
<font color="Maroon" title="c9">x2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W</font>
 <b>by </b><i><a class="txt" href="rusub_1.html#E2:53"><i><font color="Green" title="E41">A2</font></i></a></i>;<br/>
<b>thus </b><a NAME="E9:53"/>
<font color="Maroon" title="c4">v1</font> <a href="rusub_1.html#K3" title="RUSUB_1:func.3">+</a> <font color="Maroon" title="c2">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">v2</font> <a href="rusub_1.html#K3" title="RUSUB_1:func.3">+</a> <font color="Maroon" title="c2">W</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:53_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v1</font> <a href="rusub_1.html#K3" title="RUSUB_1:func.3">+</a> <font color="Maroon" title="c2">W</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">u1</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E3:53_1"/><i><font color="Green" title="E46">A7</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">u1</font>
 <b>and </b><br/><a NAME="E4:53_1"/><i><font color="Green" title="E47">A8</font></i>: 
<font color="Maroon" title="c11">u1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W</font>
 ;<br/>
<font color="Maroon" title="c3">u</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c8">x1</font> = 
<font color="Maroon" title="c4">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x1</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c8">x1</font></span>)</span>
<b>by </b><i><a class="txt" href="rusub_1.html#E4:53"><i><font color="Green" title="E42">A3</font></i></a>, <a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon" title="c4">v1</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="c1">V</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></i>
<br/>.= 
<font color="Maroon" title="c4">v1</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
;<br/>
<b>then </b><i><font color="Green" title="E48">A9</font></i>: <font color="Maroon" title="c10">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">v2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">x2</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c8">x1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">u1</font>
<b>by </b><i><a class="txt" href="rusub_1.html#E7:53"><i><font color="Green" title="E44">A5</font></i></a>, <a class="txt" href="rusub_1.html#E3:53_1"><i><font color="Green" title="E46">A7</font></i></a>, <a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon" title="c5">v2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">x2</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c8">x1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">u1</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="E7:53_1"/>
<font color="Maroon" title="c9">x2</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c8">x1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W</font>
 
<b>by </b><i><a class="txt" href="rusub_1.html#E5:53"><i><font color="Green" title="E43">A4</font></i></a>, <a class="txt" href="rusub_1.html#E8:53"><i><font color="Green" title="E45">A6</font></i></a>, <a class="ref" href="rusub_1.html#T17" target="_self" title="RUSUB_1:th.17">Th17</a></i>;<br/>
<a NAME="E8:53_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">x2</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c8">x1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">u1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W</font>
 
<b>by </b><i><a class="txt" href="rusub_1.html#E4:53_1"><i><font color="Green" title="E47">A8</font></i></a>, <a class="ref" href="rusub_1.html#T14" target="_self" title="RUSUB_1:th.14">Th14</a></i>;<br/>
<b>hence </b><a NAME="E9:53_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">v2</font> <a href="rusub_1.html#K3" title="RUSUB_1:func.3">+</a> <font color="Maroon" title="c2">W</font>
 <b>by </b><i><a class="txt" href="rusub_1.html#E6:53_1"><i><font color="Green" title="E48">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E10:53"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">v2</font> <a href="rusub_1.html#K3" title="RUSUB_1:func.3">+</a> <font color="Maroon" title="c2">W</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">u1</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E12:53"/><i><font color="Green" title="E46">A10</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">u1</font>
 <b>and </b><br/><a NAME="E13:53"/><i><font color="Green" title="E47">A11</font></i>: 
<font color="Maroon" title="c11">u1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W</font>
 ;<br/>
<font color="Maroon" title="c3">u</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c9">x2</font> = 
<font color="Maroon" title="c5">v2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x2</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c9">x2</font></span>)</span>
<b>by </b><i><a class="txt" href="rusub_1.html#E7:53"><i><font color="Green" title="E44">A5</font></i></a>, <a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon" title="c5">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="c1">V</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></i>
<br/>.= 
<font color="Maroon" title="c5">v2</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
;<br/>
<b>then </b><i><font color="Green" title="E48">A12</font></i>: <font color="Maroon" title="c10">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">x1</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c9">x2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">u1</font>
<b>by </b><i><a class="txt" href="rusub_1.html#E4:53"><i><font color="Green" title="E42">A3</font></i></a>, <a class="txt" href="rusub_1.html#E12:53"><i><font color="Green" title="E46">A10</font></i></a>, <a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon" title="c4">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">x1</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c9">x2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">u1</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="E16:53"/>
<font color="Maroon" title="c8">x1</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c9">x2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W</font>
 
<b>by </b><i><a class="txt" href="rusub_1.html#E5:53"><i><font color="Green" title="E43">A4</font></i></a>, <a class="txt" href="rusub_1.html#E8:53"><i><font color="Green" title="E45">A6</font></i></a>, <a class="ref" href="rusub_1.html#T17" target="_self" title="RUSUB_1:th.17">Th17</a></i>;<br/>
<a NAME="E17:53"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">x1</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c9">x2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">u1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W</font>
 
<b>by </b><i><a class="txt" href="rusub_1.html#E13:53"><i><font color="Green" title="E47">A11</font></i></a>, <a class="ref" href="rusub_1.html#T14" target="_self" title="RUSUB_1:th.14">Th14</a></i>;<br/>
<b>hence </b><a NAME="E18:53"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v1</font> <a href="rusub_1.html#K3" title="RUSUB_1:func.3">+</a> <font color="Maroon" title="c2">W</font>
 <b>by </b><i><a class="txt" href="rusub_1.html#E15:53"><i><font color="Green" title="E48">A12</font></i></a></i>;<br/>


</div>
