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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">W</font> being <a href="bhsp_1.html#V1" title="BHSP_1:attr.1">strict</a>  <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>  st ( for <font color="Olive" title="b2">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> holds  <font color="Olive" title="b2">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b1">W</font> ) holds <br/><font color="Olive" title="b1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c1">V</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">U6</a> of <font color="Maroon" title="c1">V</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>,the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #)</span><br/><b>let </b><font color="Maroon" title="c2">W</font> be  <a href="bhsp_1.html#V1" title="BHSP_1:attr.1">strict</a>  <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> holds  <font color="Olive" title="b1">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W</font> ) implies <font color="Maroon" title="c2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c1">V</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">U6</a> of <font color="Maroon" title="c1">V</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>,the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #) )</span><br/>



<b>assume </b><a NAME="E1:39"/><i><font color="Green" title="E33">A1</font></i>: 
for <font color="Olive" title="b1">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> holds  <font color="Olive" title="b1">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c1">V</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">U6</a> of <font color="Maroon" title="c1">V</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>,the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #)</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c3">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W</font> implies <font color="Maroon" title="c3">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="rusub_1.html#K2" title="RUSUB_1:func.2">(Omega).</a> <font color="Maroon" title="c1">V</font> ) &amp; ( <font color="Maroon" title="c3">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="rusub_1.html#K2" title="RUSUB_1:func.2">(Omega).</a> <font color="Maroon" title="c1">V</font> implies <font color="Maroon" title="c3">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W</font> ) )</span><br/><b>thus </b><a NAME="E1:39_1"/>
( <font color="Maroon" title="c3">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W</font> implies <font color="Maroon" title="c3">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="rusub_1.html#K2" title="RUSUB_1:func.2">(Omega).</a> <font color="Maroon" title="c1">V</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="rusub_1.html#K2" title="RUSUB_1:func.2">(Omega).</a> <font color="Maroon" title="c1">V</font> implies <font color="Maroon" title="c3">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_1"/>
<font color="Maroon" title="c3">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="rusub_1.html#K2" title="RUSUB_1:func.2">(Omega).</a> <font color="Maroon" title="c1">V</font></span><br/>

<a NAME="E2:39_1_1"/>
<font color="Maroon" title="c3">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c1">V</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">U6</a> of <font color="Maroon" title="c1">V</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>,the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #)
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T3" title="RLVECT_1:th.3">RLVECT_1:3</a></i>;<br/>
<b>hence </b><a NAME="E3:39_1_1"/>
<font color="Maroon" title="c3">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="rusub_1.html#K2" title="RUSUB_1:func.2">(Omega).</a> <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="ref" href="rusub_1.html#D3" title="RUSUB_1:def.3">RUSUB_1:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:39_1"/>
<font color="Maroon" title="c3">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="rusub_1.html#K2" title="RUSUB_1:func.2">(Omega).</a> <font color="Maroon" title="c1">V</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W</font></span><br/><b>thus </b><a NAME="E3:39_1"/>
<font color="Maroon" title="c3">v</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_2.html#E1:39"><i><font color="Green" title="E33">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E3:39"/><b>then </b>
<font color="Maroon" title="c2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rusub_1.html#K2" title="RUSUB_1:func.2">(Omega).</a> <font color="Maroon" title="c1">V</font>
 
<b>by </b><i><a class="ref" href="rusub_1.html#T25" title="RUSUB_1:th.25">RUSUB_1:25</a></i>;<br/>
<b>hence </b><a NAME="E4:39"/>
<font color="Maroon" title="c2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c1">V</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">U6</a> of <font color="Maroon" title="c1">V</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>,the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #)
 <b>by </b><i><a class="ref" href="rusub_1.html#D3" title="RUSUB_1:def.3">RUSUB_1:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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