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<b>let </b><font color="Maroon">V</font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon">W1</font> be    <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">V</font>, <font color="Maroon">W2</font> be    <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">V</font>;<br/>



<b>set </b><font color="Maroon">VW1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font>;<br/>
<b>set </b><font color="Maroon">VW2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font>;<br/>
<b>thus </b><a NAME="E1:41"/>
( for <font color="Olive">W</font> being   <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">V</font>  holds not the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">W</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font> or <font color="Maroon">W1</font> is    <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">W2</font> or <font color="Maroon">W2</font> is    <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">W1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">W</font> being    <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">V</font><b> such that </b><a NAME="E1:41_1"/><i><font color="Green">E20</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font>
 ;<br/>

<b>set </b><font color="Maroon">VW</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font>;<br/>
<b>assume </b><a NAME="E2:41_1"/>
( <font color="Maroon">W1</font> is  not    <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">W2</font> &amp; <font color="Maroon">W2</font> is  not    <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">W1</font> )
 ;<br/>

<a NAME="E3:41_1"/><b>then </b><i><font color="Green">E23</font></i>: 
( not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font> &amp; not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> )
 
<b>by </b><i><a class="ref" href="rusub_1.html#T22">RUSUB_1:22</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:41_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font>
 <b>and </b><br/><a NAME="E6:41_1"/><i><font color="Green">E25</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:41_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font>
 <b>and </b><br/><a NAME="E9:41_1"/><i><font color="Green">E27</font></i>: 
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font>
 <b>by </b><i><a class="ref" href="rusub_2.html#T2" target="_self">Th2</a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> <b>by </b><i><a class="ref" href="rusub_2.html#T3" target="_self">Th3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="rusub_1.html#T3">RUSUB_1:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="rusub_1.html#T3">RUSUB_1:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="rusub_1.html#D1">RUSUB_1:def 1</a></i>;<br/>
<a NAME="E15:41_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font> &amp; <font color="Maroon">A1</font> is <a href="rlsub_1.html#V1">lineary-closed</a> )
 
<b>by </b><i><a class="ref" href="rusub_2.html#T1" target="_self">Th1</a>, <a class="ref" href="rusub_1.html#T28">RUSUB_1:28</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E16:41_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="rlsub_1.html#D1">RLSUB_1:def 1</a></i>;<br/>
<div><i><font color="Green">E49</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font>
 ;<br/><b>reconsider </b><font color="Maroon">A2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="rusub_1.html#D1">RUSUB_1:def 1</a></i>;<br/><a NAME="E3:41_1_1"/>
<font color="Maroon">A2</font> is <a href="rlsub_1.html#V1">lineary-closed</a>
 <b>by </b><i><a class="ref" href="rusub_1.html#T28">RUSUB_1:28</a></i>;<br/><a NAME="E4:41_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font>
 <b>by </b><i>, <a class="ref" href="rlsub_1.html#T6">RLSUB_1:6</a></i>;<br/><a NAME="E5:41_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>;<br/><a NAME="E6:41_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>;<br/><b>hence </b><a NAME="E7:41_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font>
 ;<br/><b>reconsider </b><font color="Maroon">A2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="rusub_1.html#D1">RUSUB_1:def 1</a></i>;<br/><a NAME="E3:41_1_2"/>
<font color="Maroon">A2</font> is <a href="rlsub_1.html#V1">lineary-closed</a>
 <b>by </b><i><a class="ref" href="rusub_1.html#T28">RUSUB_1:28</a></i>;<br/><a NAME="E4:41_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font>
 <b>by </b><i>, <a class="ref" href="rlsub_1.html#T6">RLSUB_1:6</a></i>;<br/><a NAME="E5:41_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>;<br/><a NAME="E6:41_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>;<br/><b>hence </b><a NAME="E7:41_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="rusub_2.html#E6"><i><font color="Green">Lemma36</font></i></a>, <a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E19:41_1"/>
contradiction
 <b>by </b><i><a class="ref" href="rusub_2.html#T1" target="_self">Th1</a>, <a class="txt" href="rusub_2.html#E7"><i><font color="Green">Lemma39</font></i></a>, <a class="txt" href="rusub_2.html#E8"><i><font color="Green">Lemma40</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


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

<b>assume </b><a NAME="E2:41"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">W1</font> is    <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">W2</font> or <font color="Maroon">W2</font> is    <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">W1</font> )
 ;<br/>

<div><i><font color="Green">E53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_2"/>
<font color="Maroon">W1</font> is    <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">W2</font>
 ;<br/><a NAME="E2:41_2"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font>
 <b>by </b><i><a class="ref" href="rusub_1.html#D1">RUSUB_1:def 1</a></i>;<br/><a NAME="E3:41_2"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/><b>hence </b><a NAME="E4:41_2"/>
 ex <font color="Olive">W</font> being   <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">V</font> st the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">W</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font>
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_3"/>
<font color="Maroon">W2</font> is    <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">W1</font>
 ;<br/><a NAME="E2:41_3"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font>
 <b>by </b><i><a class="ref" href="rusub_1.html#D1">RUSUB_1:def 1</a></i>;<br/><a NAME="E3:41_3"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/><b>hence </b><a NAME="E4:41_3"/>
 ex <font color="Olive">W</font> being   <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">V</font> st the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">W</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:41"/>
 ex <font color="Olive">W</font> being   <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">V</font> st the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">W</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font>
 <b>by </b><i><a class="ref" href="rusub_2.html#T1" target="_self">Th1</a>, <a class="ref" href="rusub_2.html#T2" target="_self">Th2</a></i>;<br/>


</div>
