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

<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>set </b><font color="Maroon" title="c4">VW1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">W1</font>;<br/>
<b>set </b><font color="Maroon" title="c5">VW2</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">W2</font>;<br/>
<a NAME="E1:60"/><i><font color="Green" title="E47">A1</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">W1</font> is    <a href="rlsub_1.html#M2" title="RLSUB_1:mode.2">Coset</a> of <font color="Maroon" title="c2">W1</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">W2</font> is    <a href="rlsub_1.html#M2" title="RLSUB_1:mode.2">Coset</a> of <font color="Maroon" title="c3">W2</font> )
 
<b>by </b><i><a class="ref" href="rlsub_1.html#T90" title="RLSUB_1:th.90">RLSUB_1:90</a></i>;<br/>
<b>thus </b><a NAME="E2:60"/>
( <font color="Maroon" title="c1">V</font> <a href="rlsub_2.html#R1" title="RLSUB_2:pred.1">is_the_direct_sum_of</a> <font color="Maroon" title="c2">W1</font>,<font color="Maroon" title="c3">W2</font> implies for <font color="Olive" title="b1">C1</font> being   <a href="rlsub_1.html#M2" title="RLSUB_1:mode.2">Coset</a> of <font color="Maroon" title="c2">W1</font><br/> for <font color="Olive" title="b2">C2</font> being   <a href="rlsub_1.html#M2" title="RLSUB_1:mode.2">Coset</a> of <font color="Maroon" title="c3">W2</font>  ex <font color="Olive" title="b3">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> st <font color="Olive" title="b1">C1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b3">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:60_1"/><i><font color="Green" title="E48">A2</font></i>: 
<font color="Maroon" title="c1">V</font> <a href="rlsub_2.html#R1" title="RLSUB_2:pred.1">is_the_direct_sum_of</a> <font color="Maroon" title="c2">W1</font>,<font color="Maroon" title="c3">W2</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c6">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="c7">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>consider </b><font color="Maroon" title="c8">v1</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="E3:60_1"/><i><font color="Green" title="E49">A3</font></i>: 
<font color="Maroon" title="c6">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">v1</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#D6" title="RLSUB_1:def.6">RLSUB_1:def 6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">v2</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="E5:60_1"/><i><font color="Green" title="E50">A4</font></i>: 
<font color="Maroon" title="c7">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v2</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#D6" title="RLSUB_1:def.6">RLSUB_1:def 6</a></i>;<br/>
<a NAME="E6:60_1"/><i><font color="Green" title="E51">A5</font></i>: 
 <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c1">V</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">V</font>,the <a href="rlvect_1.html#U2" title="RLVECT_1:sel.2">Mult</a> of <font color="Maroon" title="c1">V</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W1</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c3">W2</font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E1:60_1"><i><font color="Green" title="E48">A2</font></i></a>, <a class="ref" href="rlsub_2.html#D4" target="_self" title="RLSUB_2:def.4">Def4</a></i>;<br/>
<a NAME="E7:60_1"/>
<font color="Maroon" title="c8">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c1">V</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">V</font>,the <a href="rlvect_1.html#U2" title="RLVECT_1:sel.2">Mult</a> of <font color="Maroon" title="c1">V</font> #)
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T3" title="RLVECT_1:th.3">RLVECT_1:3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">v11</font>, <font color="Maroon" title="c11">v12</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="E9:60_1"/><i><font color="Green" title="E52">A6</font></i>: 
<font color="Maroon" title="c10">v11</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="E10:60_1"/><i><font color="Green" title="E53">A7</font></i>: 
<font color="Maroon" title="c11">v12</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="E11:60_1"/><i><font color="Green" title="E54">A8</font></i>: 
<font color="Maroon" title="c8">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">v11</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">v12</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E6:60_1"><i><font color="Green" title="E51">A5</font></i></a>, <a class="ref" href="rlsub_2.html#T5" target="_self" title="RLSUB_2:th.5">Th5</a></i>;<br/>
<a NAME="E12:60_1"/>
<font color="Maroon" title="c9">v2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c1">V</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">V</font>,the <a href="rlvect_1.html#U2" title="RLVECT_1:sel.2">Mult</a> of <font color="Maroon" title="c1">V</font> #)
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T3" title="RLVECT_1:th.3">RLVECT_1:3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">v21</font>, <font color="Maroon" title="c13">v22</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="E14:60_1"/><i><font color="Green" title="E55">A9</font></i>: 
<font color="Maroon" title="c12">v21</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="E15:60_1"/><i><font color="Green" title="E56">A10</font></i>: 
<font color="Maroon" title="c13">v22</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="E16:60_1"/><i><font color="Green" title="E57">A11</font></i>: 
<font color="Maroon" title="c9">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">v21</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c13">v22</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E6:60_1"><i><font color="Green" title="E51">A5</font></i></a>, <a class="ref" href="rlsub_2.html#T5" target="_self" title="RLSUB_2:th.5">Th5</a></i>;<br/>
<b>take </b><font color="Maroon" title="c14">v</font> = <font color="Maroon" title="c11">v12</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c12">v21</font>;<br/>

<a NAME="E17:60_1"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">C1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">C2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:60_1_1"/><i><font color="Green" title="E58">A12</font></i>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">C1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">C2</font>
  <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="60_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">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="E1:60_1_1_1"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/>

<a NAME="E2:60_1_1_1"/><b>then </b><i><font color="Green" title="E59">A13</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">v</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:60_1_1_1"/>
<font color="Maroon" title="c11">v12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">v1</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c10">v11</font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E11:60_1"><i><font color="Green" title="E54">A8</font></i></a>, <a class="txt" href="rlsub_2.html#E57"><i><font color="Green" title="E36">Lm14</font></i></a></i>;<br/>
<a NAME="E4:60_1_1_1"/><b>then </b>
<font color="Maroon" title="c11">v12</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C1</font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E3:60_1"><i><font color="Green" title="E49">A3</font></i></a>, <a class="txt" href="rlsub_2.html#E9:60_1"><i><font color="Green" title="E52">A6</font></i></a>, <a class="ref" href="rlsub_1.html#T80" title="RLSUB_1:th.80">RLSUB_1:80</a></i>;<br/>
<a NAME="E5:60_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">v12</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="E6:60_1_1_1"/><b>then </b><i><font color="Green" title="E60">A14</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C1</font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E14:60_1"><i><font color="Green" title="E55">A9</font></i></a>, <a class="txt" href="rlsub_2.html#E2:60_1_1_1"><i><font color="Green" title="E59">A13</font></i></a></i>;<br/>
<a NAME="E7:60_1_1_1"/>
<font color="Maroon" title="c12">v21</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v2</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c13">v22</font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E16:60_1"><i><font color="Green" title="E57">A11</font></i></a>, <a class="txt" href="rlsub_2.html#E57"><i><font color="Green" title="E36">Lm14</font></i></a></i>;<br/>
<a NAME="E8:60_1_1_1"/><b>then </b>
<font color="Maroon" title="c12">v21</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C2</font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E5:60_1"><i><font color="Green" title="E50">A4</font></i></a>, <a class="txt" href="rlsub_2.html#E15:60_1"><i><font color="Green" title="E56">A10</font></i></a>, <a class="ref" href="rlsub_1.html#T80" title="RLSUB_1:th.80">RLSUB_1:80</a></i>;<br/>
<a NAME="E9:60_1_1_1"/><b>then </b>
( <font color="Maroon" title="c7">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">v21</font> <a href="rlsub_1.html#K3" title="RLSUB_1:func.3">+</a> <font color="Maroon" title="c3">W2</font> &amp; <font color="Maroon" title="c14">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">v21</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">v12</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="E10:60_1_1_1"/><b>then </b>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C2</font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E10:60_1"><i><font color="Green" title="E53">A7</font></i></a>, <a class="txt" href="rlsub_2.html#E2:60_1_1_1"><i><font color="Green" title="E59">A13</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:60_1_1_1"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">C2</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E6:60_1_1_1"><i><font color="Green" title="E60">A14</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>let </b><font color="Maroon" title="c15">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="E2:60_1_1"/><i><font color="Green" title="E59">A15</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">C2</font>
 ;<br/>

<a NAME="E3:60_1_1"/><b>then </b>
<font color="Maroon" title="c6">C1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c7">C2</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#T4" title="XBOOLE_0:th.4">XBOOLE_0:4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c16">C</font> = <font color="Maroon" title="c6">C1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">C2</font> as    <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> <b>by </b><i><a class="ref" href="rlsub_2.html#T50" target="_self" title="RLSUB_2:th.50">Th50</a></i>;<br/>
<a NAME="E5:60_1_1"/>
<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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlsub_1.html#K1" title="RLSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E1:60_1"><i><font color="Green" title="E48">A2</font></i></a>, <a class="ref" href="rlsub_2.html#D4" target="_self" title="RLSUB_2:def.4">Def4</a></i>;<br/>
<a NAME="E6:60_1_1"/><b>then </b><i><font color="Green" title="E60">A16</font></i>: 
 ex <font color="Olive" title="b1">u</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> st <font color="Maroon" title="c16">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="rlsub_1.html#T89" title="RLSUB_1:th.89">RLSUB_1:89</a></i>;<br/>
<a NAME="E7:60_1_1"/>
<font color="Maroon" title="c14">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E8:60_1_1"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E1:60_1_1"><i><font color="Green" title="E58">A12</font></i></a>, <a class="txt" href="rlsub_2.html#E2:60_1_1"><i><font color="Green" title="E59">A15</font></i></a>, <a class="txt" href="rlsub_2.html#E6:60_1_1"><i><font color="Green" title="E60">A16</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>

<b>thus </b><a NAME="E9:60_1_1"/>
verum
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E18:60_1"/>
<font color="Maroon" title="c6">C1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/>


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

<b>assume </b><a NAME="E3:60"/><i><font color="Green" title="E48">A17</font></i>: 
for <font color="Olive" title="b1">C1</font> being   <a href="rlsub_1.html#M2" title="RLSUB_1:mode.2">Coset</a> of <font color="Maroon" title="c2">W1</font><br/> for <font color="Olive" title="b2">C2</font> being   <a href="rlsub_1.html#M2" title="RLSUB_1:mode.2">Coset</a> of <font color="Maroon" title="c3">W2</font>  ex <font color="Olive" title="b3">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> st <font color="Olive" title="b1">C1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b3">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">u</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>;<br/><b>consider </b><font color="Maroon" title="c7">C1</font> being    <a href="rlsub_1.html#M2" title="RLSUB_1:mode.2">Coset</a> of <font color="Maroon" title="c2">W1</font><b> such that </b><br/><a NAME="E2:60_2"/><i><font color="Green" title="E49">A18</font></i>: 
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C1</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E72"><i><font color="Green" title="E45">Lm17</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c8">v</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="E4:60_2"/><i><font color="Green" title="E50">A19</font></i>: 
<font color="Maroon" title="c7">C1</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E1:60"><i><font color="Green" title="E47">A1</font></i></a>, <a class="txt" href="rlsub_2.html#E3:60"><i><font color="Green" title="E48">A17</font></i></a></i>;<br/><a NAME="E5:60_2"/><i><font color="Green" title="E51">A20</font></i>: 
<font color="Maroon" title="c8">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E6:60_2"/><b>then </b>
<font color="Maroon" title="c8">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">W2</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E4:60_2"><i><font color="Green" title="E50">A19</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="E7:60_2"/><b>then </b><i><font color="Green" title="E52">A21</font></i>: 
<font color="Maroon" title="c8">v</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="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><a NAME="E8:60_2"/>
<font color="Maroon" title="c8">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C1</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E4:60_2"><i><font color="Green" title="E50">A19</font></i></a>, <a class="txt" href="rlsub_2.html#E5:60_2"><i><font color="Green" title="E51">A20</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="c9">v1</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="E10:60_2"/><i><font color="Green" title="E53">A22</font></i>: 
<font color="Maroon" title="c9">v1</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="E11:60_2"/><i><font color="Green" title="E54">A23</font></i>: 
<font color="Maroon" title="c6">u</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c9">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">v</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E2:60_2"><i><font color="Green" title="E49">A18</font></i></a>, <a class="ref" href="rlsub_1.html#T96" title="RLSUB_1:th.96">RLSUB_1:96</a></i>;<br/><a NAME="E12:60_2"/>
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">v</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E11:60_2"><i><font color="Green" title="E54">A23</font></i></a>, <a class="txt" href="rlsub_2.html#E57"><i><font color="Green" title="E36">Lm14</font></i></a></i>;<br/><b>hence </b><a NAME="E13:60_2"/>
<font color="Maroon" title="c6">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#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c3">W2</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E7:60_2"><i><font color="Green" title="E52">A21</font></i></a>, <a class="txt" href="rlsub_2.html#E10:60_2"><i><font color="Green" title="E53">A22</font></i></a>, <a class="ref" href="rlsub_2.html#T5" target="_self" title="RLSUB_2:th.5">Th5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:60"/>
 <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c1">V</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">V</font>,the <a href="rlvect_1.html#U2" title="RLVECT_1:sel.2">Mult</a> of <font color="Maroon" title="c1">V</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W1</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c3">W2</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E55"><i><font color="Green" title="E34">Lm12</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="rlsub_2.html#D4" target="_self" title="RLSUB_2:def.4">RLSUB_2:def 4</a><br/>

<b>consider </b><font color="Maroon" title="c6">v</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="E7:60"/><i><font color="Green" title="E49">A24</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">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="c3">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E1:60"><i><font color="Green" title="E47">A1</font></i></a>, <a class="txt" href="rlsub_2.html#E3:60"><i><font color="Green" title="E48">A17</font></i></a></i>;<br/>
<a NAME="E8:60"/>
(  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W1</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</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="ref" href="rlsub_1.html#T25" title="RLSUB_1:th.25">RLSUB_1:25</a></i>;<br/>
<a NAME="E9:60"/><b>then </b>
(  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">W1</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">W2</font> )
 
<b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E10:60"/><b>then </b><i><font color="Green" title="E50">A25</font></i>: 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E7:60"><i><font color="Green" title="E49">A24</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlsub_1.html#K1" title="RLSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font></span>)</span> = 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<b>by </b><i><a class="ref" href="rlsub_1.html#D3" title="RLSUB_1:def.3">RLSUB_1:def 3</a></i>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">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="c3">W2</font>
<b>by </b><i><a class="txt" href="rlsub_2.html#E7:60"><i><font color="Green" title="E49">A24</font></i></a>, <a class="txt" href="rlsub_2.html#E10:60"><i><font color="Green" title="E50">A25</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
<br/>.= 
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="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="ref" href="rlsub_2.html#D2" target="_self" title="RLSUB_2:def.2">Def2</a></i>
;<br/>
<b>hence </b><a NAME="E12:60"/>
<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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlsub_1.html#K1" title="RLSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="ref" href="rlsub_1.html#T38" title="RLSUB_1:th.38">RLSUB_1:38</a></i>;<br/>


</div>
