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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="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font><br/> for <font color="Olive" title="b2">u</font>, <font color="Olive" title="b3">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">v</font> <a href="rusub_1.html#K3" title="RUSUB_1:func.3">+</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b3">v</font></span>)</span> <a href="rusub_1.html#K3" title="RUSUB_1:func.3">+</a> <font color="Olive" title="b1">W</font> holds <br/><font color="Olive" title="b3">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b1">W</font></span><br/><b>let </b><font color="Maroon" title="c2">W</font> be    <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">u</font>, <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>  st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">v</font> <a href="rusub_1.html#K3" title="RUSUB_1:func.3">+</a> <font color="Maroon" title="c2">W</font> &amp; <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b2">v</font></span>)</span> <a href="rusub_1.html#K3" title="RUSUB_1:func.3">+</a> <font color="Maroon" title="c2">W</font> holds <br/><font color="Olive" title="b2">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>let </b><font color="Maroon" title="c3">u</font>, <font color="Maroon" title="c4">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">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v</font> <a href="rusub_1.html#K3" title="RUSUB_1:func.3">+</a> <font color="Maroon" title="c2">W</font> &amp; <font color="Maroon" title="c3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">v</font></span>)</span> <a href="rusub_1.html#K3" title="RUSUB_1:func.3">+</a> <font color="Maroon" title="c2">W</font> implies <font color="Maroon" title="c4">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>assume </b><a NAME="E1:53"/>
( <font color="Maroon" title="c3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v</font> <a href="rusub_1.html#K3" title="RUSUB_1:func.3">+</a> <font color="Maroon" title="c2">W</font> &amp; <font color="Maroon" title="c3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">v</font></span>)</span> <a href="rusub_1.html#K3" title="RUSUB_1:func.3">+</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="c4">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W</font></span><br/>

<a NAME="E2:53"/><b>then </b>
<font color="Maroon" title="c4">v</font> <a href="rusub_1.html#K3" title="RUSUB_1:func.3">+</a> <font color="Maroon" title="c2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">v</font></span>)</span> <a href="rusub_1.html#K3" title="RUSUB_1:func.3">+</a> <font color="Maroon" title="c2">W</font>
 
<b>by </b><i><a class="ref" href="rusub_1.html#T50" target="_self" title="RUSUB_1:th.50">Th50</a></i>;<br/>
<b>hence </b><a NAME="E3:53"/>
<font color="Maroon" title="c4">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="ref" href="rusub_1.html#T49" target="_self" title="RUSUB_1:th.49">Th49</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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