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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="rusub_1.html#M1" target="_self">Subspace</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<a NAME="E1:50"/><i><font color="Green">E38</font></i>: 
( (  <a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>4</sub></font> <a href="rusub_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="rusub_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font> ) &amp; ( <font color="Maroon">c<sub>4</sub></font> <a href="rusub_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="rusub_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font> implies  <a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> ) &amp; <font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="rusub_1.html#T46" target="_self">Th46</a></i>;<br/>
<a NAME="E2:50"/>
(  <a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:50_1"/>
 <a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:50_1"/><b>then </b>
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="rusub_1.html#T16" target="_self">Th16</a></i>;<br/>
<b>hence </b><a NAME="E3:50_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T30">RLVECT_1:30</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:50"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> iff <font color="Maroon">c<sub>4</sub></font> <a href="rusub_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rusub_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="rusub_1.html#E1:50"><i><font color="Green">E38</font></i></a>, <a class="ref" href="rusub_1.html#T16" target="_self">Th16</a></i>;<br/>


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