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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">W</font> be  <a href="bhsp_1.html#V1">strict</a>  <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">V</font>;<br/>



<b>assume </b><a NAME="E1:46"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> holds  <font color="Olive">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>thus </b><a NAME="E1:46_1"/>
( <font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> implies <font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a>  <a href="rusub_1.html#K2">(Omega).</a> <font color="Maroon">V</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:46_1_1"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font>
 ;<br/>

<a NAME="E2:46_1_1"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a>  <a href="bhsp_1.html#G1">UNITSTR</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</font>,the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">V</font> #)
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T3">RLVECT_1:3</a></i>;<br/>
<b>hence </b><a NAME="E3:46_1_1"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a>  <a href="rusub_1.html#K2">(Omega).</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="rusub_1.html#D3">RUSUB_1:def 3</a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:46_1"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a>  <a href="rusub_1.html#K2">(Omega).</a> <font color="Maroon">V</font>
 ;<br/><b>thus </b><a NAME="E3:46_1"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="txt" href="rusub_2.html#E6"><i><font color="Green">Lemma36</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:46"/><b>then </b>
<font color="Maroon">W</font> <a href="hidden.html#R1">=</a>  <a href="rusub_1.html#K2">(Omega).</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="rusub_1.html#T25">RUSUB_1:25</a></i>;<br/>
<b>hence </b><a NAME="E4:46"/>
<font color="Maroon">W</font> <a href="hidden.html#R1">=</a>  <a href="bhsp_1.html#G1">UNITSTR</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</font>,the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">V</font> #)
 <b>by </b><i><a class="ref" href="rusub_1.html#D3">RUSUB_1:def 3</a></i>;<br/>


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