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<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">W1</font>, <font color="Maroon" title="c3">W2</font> be    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>;<br/><b>let </b><font color="Maroon" title="c4">C1</font> be    <a href="rlsub_1.html#M2" title="RLSUB_1:mode.2">Coset</a> of <font color="Maroon" title="c2">W1</font>;<br/><b>let </b><font color="Maroon" title="c5">C2</font> be    <a href="rlsub_1.html#M2" title="RLSUB_1:mode.2">Coset</a> of <font color="Maroon" title="c3">W2</font>;<br/>









<b>assume </b><a NAME="E1:51"/><i><font color="Green" title="E42">A1</font></i>: 
<font color="Maroon" title="c4">C1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">C2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a><br/>

<b>consider </b><font color="Maroon" title="c6">v</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">C1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">C2</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">v</font> = <font color="Maroon" title="c6">v</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="rlsub_2.html#E1:51"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">C</font> = <font color="Maroon" title="c4">C1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">C2</font>;<br/>
<a NAME="E4:51"/>
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C1</font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E1:51"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:51"/><b>then </b><i><font color="Green" title="E43">A2</font></i>: 
<font color="Maroon" title="c4">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> <a href="rlsub_1.html#K3" title="RLSUB_1:func.3">+</a> <font color="Maroon" title="c2">W1</font>
 
<b>by </b><i><a class="ref" href="rlsub_1.html#T94" title="RLSUB_1:th.94">RLSUB_1:94</a></i>;<br/>
<a NAME="E6:51"/>
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C2</font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E1:51"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E7:51"/><b>then </b><i><font color="Green" title="E44">A3</font></i>: 
<font color="Maroon" title="c5">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> <a href="rlsub_1.html#K3" title="RLSUB_1:func.3">+</a> <font color="Maroon" title="c3">W2</font>
 
<b>by </b><i><a class="ref" href="rlsub_1.html#T94" title="RLSUB_1:th.94">RLSUB_1:94</a></i>;<br/>
<a NAME="E8:51"/>
<font color="Maroon" title="c4">C1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">C2</font> is    <a href="rlsub_1.html#M2" title="RLSUB_1:mode.2">Coset</a> of <font color="Maroon" title="c2">W1</font> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <font color="Maroon" title="c3">W2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c7">v</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="rlsub_1.html#D6" title="RLSUB_1:def.6">RLSUB_1:def 6</a><br/>

<b>thus </b><a NAME="E1:51_1"/>
<font color="Maroon" title="c4">C1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">C2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">v</font> <a href="rlsub_1.html#K3" title="RLSUB_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <font color="Maroon" title="c3">W2</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:51_1_1"/><i><font color="Green" title="E45">A4</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">C2</font>
 ;<br/>

<a NAME="E2:51_1_1"/><b>then </b>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C1</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">u1</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E4:51_1_1"/><i><font color="Green" title="E46">A5</font></i>: 
<font color="Maroon" title="c10">u1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W1</font>
 <b>and </b><br/><a NAME="E5:51_1_1"/><i><font color="Green" title="E47">A6</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">u1</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E5:51"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="rlsub_1.html#T79" title="RLSUB_1:th.79">RLSUB_1:79</a></i>;<br/>
<a NAME="E6:51_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C2</font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E1:51_1_1"><i><font color="Green" title="E45">A4</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">u2</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E8:51_1_1"/><i><font color="Green" title="E48">A7</font></i>: 
<font color="Maroon" title="c11">u2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W2</font>
 <b>and </b><br/><a NAME="E9:51_1_1"/><i><font color="Green" title="E49">A8</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">u2</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E7:51"><i><font color="Green" title="E44">A3</font></i></a>, <a class="ref" href="rlsub_1.html#T79" title="RLSUB_1:th.79">RLSUB_1:79</a></i>;<br/>
<a NAME="E10:51_1_1"/>
<font color="Maroon" title="c10">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">u2</font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E5:51_1_1"><i><font color="Green" title="E47">A6</font></i></a>, <a class="txt" href="rlsub_2.html#E9:51_1_1"><i><font color="Green" title="E49">A8</font></i></a>, <a class="ref" href="rlvect_1.html#T21" title="RLVECT_1:th.21">RLVECT_1:21</a></i>;<br/>
<a NAME="E11:51_1_1"/><b>then </b>
<font color="Maroon" title="c10">u1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W1</font> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <font color="Maroon" title="c3">W2</font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E4:51_1_1"><i><font color="Green" title="E46">A5</font></i></a>, <a class="txt" href="rlsub_2.html#E8:51_1_1"><i><font color="Green" title="E48">A7</font></i></a>, <a class="ref" href="rlsub_2.html#T7" target="_self" title="RLSUB_2:th.7">Th7</a></i>;<br/>
<b>hence </b><a NAME="E12:51_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">v</font> <a href="rlsub_1.html#K3" title="RLSUB_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <font color="Maroon" title="c3">W2</font></span>)</span>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E5:51_1_1"><i><font color="Green" title="E47">A6</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:51_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">v</font> <a href="rlsub_1.html#K3" title="RLSUB_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <font color="Maroon" title="c3">W2</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">u</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E4:51_1"/><i><font color="Green" title="E45">A9</font></i>: 
<font color="Maroon" title="c10">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W1</font> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <font color="Maroon" title="c3">W2</font>
 <b>and </b><br/><a NAME="E5:51_1"/><i><font color="Green" title="E46">A10</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">u</font>
 <b>by </b><i><a class="ref" href="rlsub_1.html#T79" title="RLSUB_1:th.79">RLSUB_1:79</a></i>;<br/>
<a NAME="E6:51_1"/>
( <font color="Maroon" title="c10">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W1</font> &amp; <font color="Maroon" title="c10">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W2</font> )
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E4:51_1"><i><font color="Green" title="E45">A9</font></i></a>, <a class="ref" href="rlsub_2.html#T7" target="_self" title="RLSUB_2:th.7">Th7</a></i>;<br/>
<a NAME="E7:51_1"/><b>then </b>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b1">u1</font></span>)</span> where <font color="Olive" title="b1">u1</font> is   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">u1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W1</font> </span>}</span>  &amp; <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b1">u2</font></span>)</span> where <font color="Olive" title="b1">u2</font> is   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">u2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W2</font> </span>}</span>  )
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E5:51_1"><i><font color="Green" title="E46">A10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:51_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">C2</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E5:51"><i><font color="Green" title="E43">A2</font></i></a>, <a class="txt" href="rlsub_2.html#E7:51"><i><font color="Green" title="E44">A3</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:51"/>
<font color="Maroon" title="c4">C1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">C2</font> is    <a href="rlsub_1.html#M2" title="RLSUB_1:mode.2">Coset</a> of <font color="Maroon" title="c2">W1</font> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <font color="Maroon" title="c3">W2</font>
 ;<br/>


</div>
