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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">W2</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">W1</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> holds <br/> ( <font color="Olive" title="b2">W1</font> is    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Olive" title="b1">W2</font> iff <font color="Olive" title="b2">W1</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Olive" title="b1">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">W1</font> )</span><br/><b>let </b><font color="Maroon" title="c2">W2</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">W1</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> holds <br/> ( <font color="Olive" title="b1">W1</font> is    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c2">W2</font> iff <font color="Olive" title="b1">W1</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c2">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W1</font> )</span><br/><b>let </b><font color="Maroon" title="c3">W1</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"> ( <font color="Maroon" title="c3">W1</font> is    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c2">W2</font> iff <font color="Maroon" title="c3">W1</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c2">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W1</font> )</span><br/>





<b>thus </b><a NAME="E1:20"/>
( <font color="Maroon" title="c3">W1</font> is    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c2">W2</font> implies <font color="Maroon" title="c3">W1</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c2">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W1</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">W1</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c2">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W1</font> implies <font color="Maroon" title="c3">W1</font> is    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c2">W2</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_1"/>
<font color="Maroon" title="c3">W1</font> is    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c2">W2</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">W1</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c2">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W1</font></span><br/>

<a NAME="E2:20_1"/><b>then </b><i><font color="Green" title="E20">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">W1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">W2</font>
 
<b>by </b><i><a class="ref" href="rusub_1.html#D1" title="RUSUB_1:def.1">RUSUB_1:def 1</a></i>;<br/>
<a NAME="E3:20_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c2">W2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">W1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">W2</font>
 
<b>by </b><i><a class="ref" href="rusub_2.html#D2" target="_self" title="RUSUB_2:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E4:20_1"/>
<font color="Maroon" title="c3">W1</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c2">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W1</font>
 <b>by </b><i><a class="txt" href="rusub_2.html#E2:20_1"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="rusub_1.html#T24" title="RUSUB_1:th.24">RUSUB_1:24</a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</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>thus </b><a NAME="E2:20"/>
( <font color="Maroon" title="c3">W1</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c2">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W1</font> implies <font color="Maroon" title="c3">W1</font> is    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c2">W2</font> )
 <b>by </b><i><a class="ref" href="rusub_2.html#T16" target="_self" title="RUSUB_2:th.16">Th16</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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