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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">W1</font>, <font color="Olive" title="b2">W2</font>, <font color="Olive" title="b3">W3</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 <font color="Olive" title="b1">W1</font> is    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Olive" title="b2">W2</font> holds <br/><font color="Olive" title="b1">W1</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Olive" title="b3">W3</font> is    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Olive" title="b2">W2</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Olive" title="b3">W3</font></span><br/><b>let </b><font color="Maroon" title="c2">W1</font>, <font color="Maroon" title="c3">W2</font>, <font color="Maroon" title="c4">W3</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="c2">W1</font> is    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c3">W2</font> implies <font color="Maroon" title="c2">W1</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c4">W3</font> is    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c3">W2</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c4">W3</font> )</span><br/>



<b>set </b><font color="Maroon" title="c5">S</font> =  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="rusub_2.html#K3" title="RUSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> #);<br/>
<b>reconsider </b><font color="Maroon" title="c6">A</font> = <font color="Maroon" title="c2">W1</font>, <font color="Maroon" title="c7">B</font> = <font color="Maroon" title="c3">W2</font>, <font color="Maroon" title="c8">C</font> = <font color="Maroon" title="c4">W3</font>, <font color="Maroon" title="c9">AC</font> = <font color="Maroon" title="c2">W1</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c4">W3</font>, <font color="Maroon" title="c10">BC</font> = <font color="Maroon" title="c3">W2</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c4">W3</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="rusub_2.html#K3" title="RUSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> #) <b>by </b><i><a class="ref" href="rusub_2.html#D3" target="_self" title="RUSUB_2:def.3">Def3</a></i>;<br/>
<b>assume </b><a NAME="E2:76"/><i><font color="Green" title="E57">A1</font></i>: 
<font color="Maroon" title="c2">W1</font> is    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c3">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="c2">W1</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c4">W3</font> is    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c3">W2</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c4">W3</font></span><br/>

<font color="Maroon" title="c6">A</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c7">B</font> = 
<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">B</font>
<b>by </b><i><a class="ref" href="lattices.html#D1" title="LATTICES:def.1">LATTICES:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c2">W1</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c3">W2</font>
<b>by </b><i><a class="ref" href="rusub_2.html#D7" target="_self" title="RUSUB_2:def.7">Def7</a></i>
<br/>.= 
<font color="Maroon" title="c7">B</font>
<b>by </b><i><a class="txt" href="#E2:76"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="rusub_2.html#T8" target="_self" title="RUSUB_2:th.8">Th8</a></i>
;<br/>
<a NAME="E4:76"/><b>then </b>
<font color="Maroon" title="c6">A</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">B</font>
 
<b>by </b><i><a class="ref" href="lattices.html#D3" title="LATTICES:def.3">LATTICES:def 3</a></i>;<br/>
<a NAME="E5:76"/><b>then </b>
<font color="Maroon" title="c6">A</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">C</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">B</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">C</font>
 
<b>by </b><i><a class="ref" href="lattices.html#T27" title="LATTICES:th.27">LATTICES:27</a></i>;<br/>
<a NAME="E6:76"/><b>then </b><i><font color="Green" title="E58">A2</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">C</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">B</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">B</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">C</font>
 
<b>by </b><i><a class="ref" href="lattices.html#D3" title="LATTICES:def.3">LATTICES:def 3</a></i>;<br/>
<i><font color="Green" title="E59">A3</font></i>: <font color="Maroon" title="c7">B</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">C</font> = 
<span class="p1">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c8">C</font>
<b>by </b><i><a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c3">W2</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c4">W3</font>
<b>by </b><i><a class="ref" href="rusub_2.html#D8" target="_self" title="RUSUB_2:def.8">Def8</a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">C</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">B</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">C</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">C</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">B</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">C</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D1" title="LATTICES:def.1">LATTICES:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c8">C</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">B</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">C</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c8">C</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c8">C</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c8">C</font></span>)</span>,<font color="Maroon" title="c10">BC</font>
<b>by </b><i><a class="ref" href="rusub_2.html#D8" target="_self" title="RUSUB_2:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">AC</font>,<font color="Maroon" title="c10">BC</font>
<b>by </b><i><a class="ref" href="rusub_2.html#D8" target="_self" title="RUSUB_2:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c4">W3</font></span>)</span> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W2</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c4">W3</font></span>)</span>
<b>by </b><i><a class="ref" href="rusub_2.html#D7" target="_self" title="RUSUB_2:def.7">Def7</a></i>
;<br/>
<b>hence </b><a NAME="E9:76"/>
<font color="Maroon" title="c2">W1</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c4">W3</font> is    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c3">W2</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c4">W3</font>
 <b>by </b><i><a class="txt" href="#E6:76"><i><font color="Green" title="E58">A2</font></i></a>, <a class="txt" href="#E7:76"><i><font color="Green" title="E59">A3</font></i></a>, <a class="ref" href="rusub_2.html#T8" target="_self" title="RUSUB_2:th.8">Th8</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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