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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">W1</font>, <font color="Maroon" title="c3">W2</font>, <font color="Maroon" title="c4">W3</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>assume </b><a NAME="E1:32"/><i><font color="Green" title="E29">A1</font></i>: 
<font color="Maroon" title="c2">W1</font> is    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c3">W2</font>
 ;<br/>

<b>thus </b><a NAME="E2:32"/>
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">W2</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c4">W3</font></span>)</span></span>)</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"><span class="p2">(<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="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">W2</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c4">W3</font></span>)</span></span>)</span>
  <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="32_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">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="E1:32_1"/>
<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="c3">W2</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c4">W3</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:32_1"/><b>then </b><i><font color="Green" title="E30">A2</font></i>: 
<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 <font color="Maroon" title="c3">W2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</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#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c4">W3</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rusub_2.html#D2" target="_self" title="RUSUB_2:def.2">Def2</a></i>;<br/>
<a NAME="E3:32_1"/><b>then </b>
<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#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c4">W3</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:32_1"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">u</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">v</font></span>)</span> where <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</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">u</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">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W3</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rusub_2.html#D1" target="_self" title="RUSUB_2:def.1">Def1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">u1</font>, <font color="Maroon" title="c7">v1</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="E6:32_1"/><i><font color="Green" title="E31">A3</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">u1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">v1</font>
 <b>and </b><br/><a NAME="E7:32_1"/><i><font color="Green" title="E32">A4</font></i>: 
<font color="Maroon" title="c6">u1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W1</font>
 <b>and </b><br/><a NAME="E8:32_1"/><i><font color="Green" title="E33">A5</font></i>: 
<font color="Maroon" title="c7">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W3</font>
 ;<br/>
<a NAME="E9:32_1"/><i><font color="Green" title="E34">A6</font></i>: 
<font color="Maroon" title="c6">u1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W2</font>
 
<b>by </b><i><a class="txt" href="rusub_2.html#E1:32"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="rusub_2.html#E7:32_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="rusub_1.html#T1" title="RUSUB_1:th.1">RUSUB_1:1</a></i>;<br/>
<a NAME="E10:32_1"/><b>then </b><i><font color="Green" title="E35">A7</font></i>: 
<font color="Maroon" title="c6">u1</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>
 
<b>by </b><i><a class="txt" href="rusub_2.html#E7:32_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="rusub_2.html#T3" target="_self" title="RUSUB_2:th.3">Th3</a></i>;<br/>
<a NAME="E11:32_1"/>
<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 <font color="Maroon" title="c3">W2</font>
 
<b>by </b><i><a class="txt" href="rusub_2.html#E2:32_1"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E12:32_1"/><b>then </b>
<font color="Maroon" title="c6">u1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W2</font>
 
<b>by </b><i><a class="txt" href="rusub_2.html#E6:32_1"><i><font color="Green" title="E31">A3</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="E13:32_1"/><b>then </b>
<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">u1</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c6">u1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W2</font>
 
<b>by </b><i><a class="txt" href="rusub_2.html#E9:32_1"><i><font color="Green" title="E34">A6</font></i></a>, <a class="ref" href="rusub_1.html#T17" title="RUSUB_1:th.17">RUSUB_1:17</a></i>;<br/>
<a NAME="E14:32_1"/><b>then </b>
<font color="Maroon" title="c7">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">u1</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c6">u1</font></span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W2</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/>
<a NAME="E15:32_1"/><b>then </b>
<font color="Maroon" title="c7">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> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W2</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></i>;<br/>
<a NAME="E16:32_1"/><b>then </b>
<font color="Maroon" title="c7">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W2</font>
 
<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="E17:32_1"/><b>then </b>
<font color="Maroon" title="c7">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W2</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c4">W3</font>
 
<b>by </b><i><a class="txt" href="rusub_2.html#E8:32_1"><i><font color="Green" title="E33">A5</font></i></a>, <a class="ref" href="rusub_2.html#T3" target="_self" title="RUSUB_2:th.3">Th3</a></i>;<br/>
<a NAME="E18:32_1"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <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="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W2</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c4">W3</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rusub_2.html#E6:32_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="rusub_2.html#E10:32_1"><i><font color="Green" title="E35">A7</font></i></a>, <a class="ref" href="rusub_2.html#T1" target="_self" title="RUSUB_2:th.1">Th1</a></i>;<br/>
<b>hence </b><a NAME="E19:32_1"/>
<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"><span class="p2">(<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="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">W2</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c4">W3</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>


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

<b>thus </b><a NAME="E3:32"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<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="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">W2</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c4">W3</font></span>)</span></span>)</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">W2</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c4">W3</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rusub_2.html#E34"><i><font color="Green" title="E27">Lm8</font></i></a></i>;<br/>


</div>
