<?xml version="1.0"?>
<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="rlsub_2.html#K3" title="RLSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="rlsub_2.html#K3" title="RLSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="rlsub_2.html#K3" title="RLSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font>] <b>means </b>for <font color="Olive" title="b1">W1</font>, <font color="Olive" title="b2">W2</font> being   <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W1</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">W2</font> holds <br/>$3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W1</font> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <font color="Olive" title="b2">W2</font>;<br/>
<a NAME="E1:65_1_1"/><i><font color="Green" title="E49">A1</font></i>: 
for <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="rlsub_2.html#K3" title="RLSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font>  ex <font color="Olive" title="b3">B</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="rlsub_2.html#K3" title="RLSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A1</font>,<font color="Olive" title="b2">A2</font>,<font color="Olive" title="b3">B</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">A1</font>, <font color="Maroon" title="c3">A2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="rlsub_2.html#K3" title="RLSUB_2:func.3">Subspaces</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">  ex <font color="Olive" title="b1">B</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="rlsub_2.html#K3" title="RLSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">A1</font>,<font color="Maroon" title="c3">A2</font>,<font color="Olive" title="b1">B</font>]</span><br/>



<b>reconsider </b><font color="Maroon" title="c4">W1</font> = <font color="Maroon" title="c2">A1</font>, <font color="Maroon" title="c5">W2</font> = <font color="Maroon" title="c3">A2</font> as    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="rlsub_2.html#D3" target="_self" title="RLSUB_2:def.3">Def3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">C</font> = <font color="Maroon" title="c4">W1</font> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <font color="Maroon" title="c5">W2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="rlsub_2.html#K3" title="RLSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="rlsub_2.html#D3" target="_self" title="RLSUB_2:def.3">Def3</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">C</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">A1</font>,<font color="Maroon" title="c3">A2</font>,<font color="Maroon" title="c6">C</font>]</span><br/>

<b>thus </b><a NAME="E3:65_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">A1</font>,<font color="Maroon" title="c3">A2</font>,<font color="Maroon" title="c6">C</font>]
 ; <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="E2:65_1_1"/>
 ex <font color="Olive" title="b1">o</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of  <a href="rlsub_2.html#K3" title="RLSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font> st <br/>for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="rlsub_2.html#K3" title="RLSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font>,<font color="Olive" title="b1">o</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font>]
 
<b>from </b><i><a class="ref" href="binop_1.html#S3" title="BINOP_1:sch.3">BINOP_1:sch 3</a>(<a class="txt" href="#E1:65_1_1"><i><font color="Green" title="E49">A1</font></i></a>);<br/></i>
<b>hence </b><a NAME="E3:65_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of  <a href="rlsub_2.html#K3" title="RLSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font> st <br/>for <font color="Olive" title="b2">A1</font>, <font color="Olive" title="b3">A2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="rlsub_2.html#K3" title="RLSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font><br/> for <font color="Olive" title="b4">W1</font>, <font color="Olive" title="b5">W2</font> being   <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b2">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">W1</font> &amp; <font color="Olive" title="b3">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">W2</font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b2">A1</font>,<font color="Olive" title="b3">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">W1</font> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <font color="Olive" title="b5">W2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
