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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> holds   <a href="rusub_4.html#K4" title="RUSUB_4:func.4">Up</a> <font color="Olive" title="b1">W</font> is <a href="rusub_4.html#V3" title="RUSUB_4:attr.3">Subspace-like</a></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">  <a href="rusub_4.html#K4" title="RUSUB_4:func.4">Up</a> <font color="Maroon" title="c2">W</font> is <a href="rusub_4.html#V3" title="RUSUB_4:attr.3">Subspace-like</a></span><br/>



<a NAME="E1:43"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">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#T11" title="RUSUB_1:th.11">RUSUB_1:11</a></i>;<br/>
<b>hence </b><a NAME="E2:43"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_4.html#K4" title="RUSUB_4:func.4">Up</a> <font color="Maroon" title="c2">W</font>
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="rusub_4.html#D8" target="_self" title="RUSUB_4:def.8">RUSUB_4:def 8</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">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font><br/> for <font color="Olive" title="b3">a</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_4.html#K4" title="RUSUB_4:func.4">Up</a> <font color="Maroon" title="c2">W</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_4.html#K4" title="RUSUB_4:func.4">Up</a> <font color="Maroon" title="c2">W</font> holds <br/>( <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_4.html#K4" title="RUSUB_4:func.4">Up</a> <font color="Maroon" title="c2">W</font> &amp; <font color="Olive" title="b3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_4.html#K4" title="RUSUB_4:func.4">Up</a> <font color="Maroon" title="c2">W</font> )</span><br/>

<b>thus </b><a NAME="E3:43"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font><br/> for <font color="Olive" title="b3">a</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_4.html#K4" title="RUSUB_4:func.4">Up</a> <font color="Maroon" title="c2">W</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_4.html#K4" title="RUSUB_4:func.4">Up</a> <font color="Maroon" title="c2">W</font> holds <br/>( <font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_4.html#K4" title="RUSUB_4:func.4">Up</a> <font color="Maroon" title="c2">W</font> &amp; <font color="Olive" title="b3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_4.html#K4" title="RUSUB_4:func.4">Up</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"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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">a</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_4.html#K4" title="RUSUB_4:func.4">Up</a> <font color="Maroon" title="c2">W</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_4.html#K4" title="RUSUB_4:func.4">Up</a> <font color="Maroon" title="c2">W</font> holds <br/>( <font color="Maroon" title="c3">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_4.html#K4" title="RUSUB_4:func.4">Up</a> <font color="Maroon" title="c2">W</font> &amp; <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_4.html#K4" title="RUSUB_4:func.4">Up</a> <font color="Maroon" title="c2">W</font> )</span><br/><b>let </b><font color="Maroon" title="c5">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_4.html#K4" title="RUSUB_4:func.4">Up</a> <font color="Maroon" title="c2">W</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_4.html#K4" title="RUSUB_4:func.4">Up</a> <font color="Maroon" title="c2">W</font> implies ( <font color="Maroon" title="c3">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_4.html#K4" title="RUSUB_4:func.4">Up</a> <font color="Maroon" title="c2">W</font> &amp; <font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_4.html#K4" title="RUSUB_4:func.4">Up</a> <font color="Maroon" title="c2">W</font> ) )</span><br/>



<b>assume </b><a NAME="E1:43_1"/><i><font color="Green" title="E23">A1</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_4.html#K4" title="RUSUB_4:func.4">Up</a> <font color="Maroon" title="c2">W</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_4.html#K4" title="RUSUB_4:func.4">Up</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">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_4.html#K4" title="RUSUB_4:func.4">Up</a> <font color="Maroon" title="c2">W</font> &amp; <font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_4.html#K4" title="RUSUB_4:func.4">Up</a> <font color="Maroon" title="c2">W</font> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c7">y</font> = <font color="Maroon" title="c4">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> ;<br/>
<a NAME="E3:43_1"/>
( <font color="Maroon" title="c6">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W</font> &amp; <font color="Maroon" title="c7">y</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="#E1:43_1"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E4:43_1"/><b>then </b>
( <font color="Maroon" title="c6">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">y</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W</font> &amp; <font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">x</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#T14" title="RUSUB_1:th.14">RUSUB_1:14</a>, <a class="ref" href="rusub_1.html#T15" title="RUSUB_1:th.15">RUSUB_1:15</a></i>;<br/>
<b>hence </b><a NAME="E5:43_1"/>
( <font color="Maroon" title="c3">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_4.html#K4" title="RUSUB_4:func.4">Up</a> <font color="Maroon" title="c2">W</font> &amp; <font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_4.html#K4" title="RUSUB_4:func.4">Up</a> <font color="Maroon" title="c2">W</font> )
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</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>


</div>
