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<b>let </b><font color="Maroon" title="c2">o1</font>, <font color="Maroon" title="c3">o2</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of  <a href="rusub_2.html#K3" title="RUSUB_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"> ( ( 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="rusub_2.html#K3" title="RUSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font><br/> for <font color="Olive" title="b3">W1</font>, <font color="Olive" title="b4">W2</font> being   <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">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">W1</font> &amp; <font color="Olive" title="b2">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">W2</font> holds <br/><font color="Maroon" title="c2">o1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">A1</font>,<font color="Olive" title="b2">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">W1</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Olive" title="b4">W2</font> ) &amp; ( 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="rusub_2.html#K3" title="RUSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font><br/> for <font color="Olive" title="b3">W1</font>, <font color="Olive" title="b4">W2</font> being   <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">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">W1</font> &amp; <font color="Olive" title="b2">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">W2</font> holds <br/><font color="Maroon" title="c3">o2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">A1</font>,<font color="Olive" title="b2">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">W1</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Olive" title="b4">W2</font> ) implies <font color="Maroon" title="c2">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">o2</font> )</span><br/>

<b>assume </b><a NAME="E1:66_1_2"/><i><font color="Green" title="E49">A3</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="rusub_2.html#K3" title="RUSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font><br/> for <font color="Olive" title="b3">W1</font>, <font color="Olive" title="b4">W2</font> being   <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">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">W1</font> &amp; <font color="Olive" title="b2">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">W2</font> holds <br/><font color="Maroon" title="c2">o1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">A1</font>,<font color="Olive" title="b2">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">W1</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Olive" title="b4">W2</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">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="rusub_2.html#K3" title="RUSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font> ex <font color="Olive" title="b3">W1</font>, <font color="Olive" title="b4">W2</font> being   <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Olive" title="b1">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">W1</font> &amp; <font color="Olive" title="b2">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">W2</font> &amp; not <font color="Maroon" title="c3">o2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">A1</font>,<font color="Olive" title="b2">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">W1</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Olive" title="b4">W2</font> ) or <font color="Maroon" title="c2">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">o2</font> )</span><br/>

<b>assume </b><a NAME="E2:66_1_2"/><i><font color="Green" title="E50">A4</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="rusub_2.html#K3" title="RUSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font><br/> for <font color="Olive" title="b3">W1</font>, <font color="Olive" title="b4">W2</font> being   <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">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">W1</font> &amp; <font color="Olive" title="b2">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">W2</font> holds <br/><font color="Maroon" title="c3">o2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">A1</font>,<font color="Olive" title="b2">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">W1</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Olive" title="b4">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">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">o2</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_2.html#K3" title="RUSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_2.html#K3" title="RUSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font> implies <font color="Maroon" title="c2">o1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">o2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> )</span><br/><b>assume </b><a NAME="E1:66_1_2_1"/><i><font color="Green" title="E51">A5</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_2.html#K3" title="RUSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_2.html#K3" title="RUSUB_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"> <font color="Maroon" title="c2">o1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">o2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><br/><b>then reconsider </b><font color="Maroon" title="c6">A</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c7">B</font> = <font color="Maroon" title="c5">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="rusub_2.html#K3" title="RUSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font> ;<br/><b>reconsider </b><font color="Maroon" title="c8">W1</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c9">W2</font> = <font color="Maroon" title="c5">y</font> as    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="#E1:66_1_2_1"><i><font color="Green" title="E51">A5</font></i></a>, <a class="ref" href="rusub_2.html#D3" target="_self" title="RUSUB_2:def.3">Def3</a></i>;<br/><a NAME="E4:66_1_2_1"/>
( <font color="Maroon" title="c2">o1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">W1</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c9">W2</font> &amp; <font color="Maroon" title="c3">o2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">W1</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c9">W2</font> )
 <b>by </b><i><a class="txt" href="#E1:66_1_2"><i><font color="Green" title="E49">A3</font></i></a>, <a class="txt" href="#E2:66_1_2"><i><font color="Green" title="E50">A4</font></i></a></i>;<br/><b>hence </b><a NAME="E5:66_1_2_1"/>
<font color="Maroon" title="c2">o1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">o2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</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>
<b>hence </b><a NAME="E4:66_1_2"/>
<font color="Maroon" title="c2">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">o2</font>
 <b>by </b><i><a class="ref" href="binop_1.html#T1" title="BINOP_1:th.1">BINOP_1:1</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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