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

<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon" title="c2">W1</font>, <font color="Maroon" title="c3">W2</font> be    <a href="rusub_1.html#M1" title="RUSUB_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="E1:62"/><i><font color="Green" title="E48">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="rusub_1.html#M2" title="RUSUB_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="rusub_1.html#M2" title="RUSUB_1:mode.2">Coset</a> of <font color="Maroon" title="c3">W2</font> )
 
<b>by </b><i><a class="ref" href="rusub_1.html#T68" title="RUSUB_1:th.68">RUSUB_1:68</a></i>;<br/>
<b>thus </b><a NAME="E2:62"/>
( <font color="Maroon" title="c1">V</font> <a href="rusub_2.html#R1" title="RUSUB_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="rusub_1.html#M2" title="RUSUB_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="rusub_1.html#M2" title="RUSUB_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="62_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:62_1"/><i><font color="Green" title="E49">A2</font></i>: 
<font color="Maroon" title="c1">V</font> <a href="rusub_2.html#R1" title="RUSUB_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="rusub_1.html#M2" title="RUSUB_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="rusub_1.html#M2" title="RUSUB_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="E3:62_1"/><i><font color="Green" title="E50">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="rusub_1.html#K3" title="RUSUB_1:func.3">+</a> <font color="Maroon" title="c2">W1</font>
 <b>by </b><i><a class="ref" href="rusub_1.html#D5" title="RUSUB_1:def.5">RUSUB_1:def 5</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="E5:62_1"/><i><font color="Green" title="E51">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="rusub_1.html#K3" title="RUSUB_1:func.3">+</a> <font color="Maroon" title="c3">W2</font>
 <b>by </b><i><a class="ref" href="rusub_1.html#D5" title="RUSUB_1:def.5">RUSUB_1:def 5</a></i>;<br/>
<a NAME="E6:62_1"/><i><font color="Green" title="E52">A5</font></i>: 
 <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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">Zero</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>,the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</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="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c3">W2</font>
 
<b>by </b><i><a class="txt" href="rusub_2.html#E1:62_1"><i><font color="Green" title="E49">A2</font></i></a>, <a class="ref" href="rusub_2.html#D4" target="_self" title="RUSUB_2:def.4">Def4</a></i>;<br/>
<a NAME="E7:62_1"/>
<font color="Maroon" title="c8">v1</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a>  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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">Zero</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>,the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</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="E9:62_1"/><i><font color="Green" title="E53">A6</font></i>: 
<font color="Maroon" title="c10">v11</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">W1</font>
 <b>and </b><br/><a NAME="E10:62_1"/><i><font color="Green" title="E54">A7</font></i>: 
<font color="Maroon" title="c11">v12</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">W2</font>
 <b>and </b><br/><a NAME="E11:62_1"/><i><font color="Green" title="E55">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#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c11">v12</font>
 <b>by </b><i><a class="txt" href="rusub_2.html#E6:62_1"><i><font color="Green" title="E52">A5</font></i></a>, <a class="ref" href="rusub_2.html#T1" target="_self" title="RUSUB_2:th.1">Th1</a></i>;<br/>
<a NAME="E12:62_1"/>
<font color="Maroon" title="c9">v2</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a>  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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">Zero</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>,the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</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="E14:62_1"/><i><font color="Green" title="E56">A9</font></i>: 
<font color="Maroon" title="c12">v21</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">W1</font>
 <b>and </b><br/><a NAME="E15:62_1"/><i><font color="Green" title="E57">A10</font></i>: 
<font color="Maroon" title="c13">v22</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">W2</font>
 <b>and </b><br/><a NAME="E16:62_1"/><i><font color="Green" title="E58">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#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c13">v22</font>
 <b>by </b><i><a class="txt" href="rusub_2.html#E6:62_1"><i><font color="Green" title="E52">A5</font></i></a>, <a class="ref" href="rusub_2.html#T1" target="_self" title="RUSUB_2:th.1">Th1</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#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c12">v21</font>;<br/>

<a NAME="E17:62_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="62_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:62_1_1"/><i><font color="Green" title="E59">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="62_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="E1:62_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="E2:62_1_1_1"/><b>then </b><i><font color="Green" title="E60">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="E3:62_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#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c10">v11</font>
 
<b>by </b><i><a class="txt" href="rusub_2.html#E11:62_1"><i><font color="Green" title="E55">A8</font></i></a>, <a class="ref" href="rlsub_2.html#T78" title="RLSUB_2:th.78">RLSUB_2:78</a></i>;<br/>
<a NAME="E4:62_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="rusub_2.html#E3:62_1"><i><font color="Green" title="E50">A3</font></i></a>, <a class="txt" href="rusub_2.html#E9:62_1"><i><font color="Green" title="E53">A6</font></i></a>, <a class="ref" href="rusub_1.html#T58" title="RUSUB_1:th.58">RUSUB_1:58</a></i>;<br/>
<a NAME="E5:62_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="rusub_1.html#K3" title="RUSUB_1:func.3">+</a> <font color="Maroon" title="c2">W1</font>
 
<b>by </b><i><a class="ref" href="rusub_1.html#T72" title="RUSUB_1:th.72">RUSUB_1:72</a></i>;<br/>
<a NAME="E6:62_1_1_1"/><b>then </b><i><font color="Green" title="E61">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="rusub_2.html#E14:62_1"><i><font color="Green" title="E56">A9</font></i></a>, <a class="txt" href="rusub_2.html#E2:62_1_1_1"><i><font color="Green" title="E60">A13</font></i></a>, <a class="ref" href="rusub_1.html#T57" title="RUSUB_1:th.57">RUSUB_1:57</a></i>;<br/>
<a NAME="E7:62_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#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c13">v22</font>
 
<b>by </b><i><a class="txt" href="rusub_2.html#E16:62_1"><i><font color="Green" title="E58">A11</font></i></a>, <a class="ref" href="rlsub_2.html#T78" title="RLSUB_2:th.78">RLSUB_2:78</a></i>;<br/>
<a NAME="E8:62_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="rusub_2.html#E5:62_1"><i><font color="Green" title="E51">A4</font></i></a>, <a class="txt" href="rusub_2.html#E15:62_1"><i><font color="Green" title="E57">A10</font></i></a>, <a class="ref" href="rusub_1.html#T58" title="RUSUB_1:th.58">RUSUB_1:58</a></i>;<br/>
<a NAME="E9:62_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="rusub_1.html#K3" title="RUSUB_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#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c11">v12</font> )
 
<b>by </b><i><a class="ref" href="rusub_1.html#T72" title="RUSUB_1:th.72">RUSUB_1:72</a></i>;<br/>
<a NAME="E10:62_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="rusub_2.html#E10:62_1"><i><font color="Green" title="E54">A7</font></i></a>, <a class="txt" href="rusub_2.html#E2:62_1_1_1"><i><font color="Green" title="E60">A13</font></i></a>, <a class="ref" href="rusub_1.html#T57" title="RUSUB_1:th.57">RUSUB_1:57</a></i>;<br/>
<b>hence </b><a NAME="E11:62_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="rusub_2.html#E6:62_1_1_1"><i><font color="Green" title="E61">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="E2:62_1_1"/><i><font color="Green" title="E60">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="E3:62_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="rusub_1.html#M2" title="RUSUB_1:mode.2">Coset</a> of <font color="Maroon" title="c2">W1</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c3">W2</font> <b>by </b><i><a class="ref" href="rusub_2.html#T42" target="_self" title="RUSUB_2:th.42">Th42</a></i>;<br/>
<a NAME="E5:62_1_1"/>
<font color="Maroon" title="c2">W1</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c3">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rusub_1.html#K1" title="RUSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font>
 
<b>by </b><i><a class="txt" href="rusub_2.html#E1:62_1"><i><font color="Green" title="E49">A2</font></i></a>, <a class="ref" href="rusub_2.html#D4" target="_self" title="RUSUB_2:def.4">Def4</a></i>;<br/>
<a NAME="E6:62_1_1"/><b>then </b><i><font color="Green" title="E61">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="rusub_1.html#T67" title="RUSUB_1:th.67">RUSUB_1:67</a></i>;<br/>
<a NAME="E7:62_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="E8:62_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="rusub_2.html#E1:62_1_1"><i><font color="Green" title="E59">A12</font></i></a>, <a class="txt" href="rusub_2.html#E2:62_1_1"><i><font color="Green" title="E60">A15</font></i></a>, <a class="txt" href="rusub_2.html#E6:62_1_1"><i><font color="Green" title="E61">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="E9:62_1_1"/>
verum
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E18:62_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="E3:62"/><i><font color="Green" title="E49">A17</font></i>: 
for <font color="Olive" title="b1">C1</font> being   <a href="rusub_1.html#M2" title="RUSUB_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="rusub_1.html#M2" title="RUSUB_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="62_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="rusub_1.html#M2" title="RUSUB_1:mode.2">Coset</a> of <font color="Maroon" title="c2">W1</font><b> such that </b><br/><a NAME="E2:62_2"/><i><font color="Green" title="E50">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="rusub_2.html#E64"><i><font color="Green" title="E46">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="E4:62_2"/><i><font color="Green" title="E51">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="rusub_2.html#E1:62"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="rusub_2.html#E3:62"><i><font color="Green" title="E49">A17</font></i></a></i>;<br/><a NAME="E5:62_2"/><i><font color="Green" title="E52">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="E6:62_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="rusub_2.html#E4:62_2"><i><font color="Green" title="E51">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="E7:62_2"/><b>then </b><i><font color="Green" title="E53">A21</font></i>: 
<font color="Maroon" title="c8">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">W2</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/><a NAME="E8:62_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="rusub_2.html#E4:62_2"><i><font color="Green" title="E51">A19</font></i></a>, <a class="txt" href="rusub_2.html#E5:62_2"><i><font color="Green" title="E52">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="E10:62_2"/><i><font color="Green" title="E54">A22</font></i>: 
<font color="Maroon" title="c9">v1</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">W1</font>
 <b>and </b><br/><a NAME="E11:62_2"/><i><font color="Green" title="E55">A23</font></i>: 
<font color="Maroon" title="c6">u</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</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="rusub_2.html#E2:62_2"><i><font color="Green" title="E50">A18</font></i></a>, <a class="ref" href="rusub_1.html#T74" title="RUSUB_1:th.74">RUSUB_1:74</a></i>;<br/><a NAME="E12:62_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#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c8">v</font>
 <b>by </b><i><a class="txt" href="rusub_2.html#E11:62_2"><i><font color="Green" title="E55">A23</font></i></a>, <a class="ref" href="rlsub_2.html#T78" title="RLSUB_2:th.78">RLSUB_2:78</a></i>;<br/><b>hence </b><a NAME="E13:62_2"/>
<font color="Maroon" title="c6">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <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="txt" href="rusub_2.html#E7:62_2"><i><font color="Green" title="E53">A21</font></i></a>, <a class="txt" href="rusub_2.html#E10:62_2"><i><font color="Green" title="E54">A22</font></i></a>, <a class="ref" href="rusub_2.html#T1" target="_self" title="RUSUB_2:th.1">Th1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:62"/>
 <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</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">Zero</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>,the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</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="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c3">W2</font>
 <b>by </b><i><a class="txt" href="rusub_2.html#E49"><i><font color="Green" title="E35">Lm12</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="rusub_2.html#D4" target="_self" title="RUSUB_2:def.4">RUSUB_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="E7:62"/><i><font color="Green" title="E50">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="rusub_2.html#E1:62"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="rusub_2.html#E3:62"><i><font color="Green" title="E49">A17</font></i></a></i>;<br/>
<a NAME="E8:62"/>
(  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">V</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">W1</font> &amp;  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">V</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">W2</font> )
 
<b>by </b><i><a class="ref" href="rusub_1.html#T11" title="RUSUB_1:th.11">RUSUB_1:11</a></i>;<br/>
<a NAME="E9:62"/><b>then </b>
(  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">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="rlvect_1.html#K1" title="RLVECT_1:func.1">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="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E10:62"/><b>then </b><i><font color="Green" title="E51">A25</font></i>: 
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">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="rusub_2.html#E7:62"><i><font color="Green" title="E50">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="rusub_1.html#K1" title="RUSUB_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="rlvect_1.html#K1" title="RLVECT_1:func.1">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="rusub_1.html#D2" title="RUSUB_1:def.2">RUSUB_1:def 2</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="rusub_2.html#E7:62"><i><font color="Green" title="E50">A24</font></i></a>, <a class="txt" href="rusub_2.html#E10:62"><i><font color="Green" title="E51">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="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c3">W2</font></span>)</span>
<b>by </b><i><a class="ref" href="rusub_2.html#D2" target="_self" title="RUSUB_2:def.2">Def2</a></i>
;<br/>
<b>hence </b><a NAME="E12:62"/>
<font color="Maroon" title="c2">W1</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c3">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rusub_1.html#K1" title="RUSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="ref" href="rusub_1.html#T24" title="RUSUB_1:th.24">RUSUB_1:24</a></i>;<br/>


</div>
