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

<b>let </b><font color="Maroon" title="c1">GF</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">M</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">VectSp-like</a> <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  <a href="vectsp_1.html#L1" title="VECTSP_1:struct.1">VectSpStr</a> of <font color="Maroon" title="c1">GF</font>;<br/><b>let </b><font color="Maroon" title="c3">W1</font>, <font color="Maroon" title="c4">W2</font> be    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">M</font>;<br/>







<b>set </b><font color="Maroon" title="c5">VW1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">W1</font>;<br/>
<b>set </b><font color="Maroon" title="c6">VW2</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">W2</font>;<br/>
<a NAME="E1:56"/><i><font color="Green" title="E43">A1</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">W1</font> is    <a href="vectsp_4.html#M2" title="VECTSP_4:mode.2">Coset</a> of <font color="Maroon" title="c3">W1</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">W2</font> is    <a href="vectsp_4.html#M2" title="VECTSP_4:mode.2">Coset</a> of <font color="Maroon" title="c4">W2</font> )
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T89" title="VECTSP_4:th.89">VECTSP_4:89</a></i>;<br/>
<b>thus </b><a NAME="E2:56"/>
( <font color="Maroon" title="c2">M</font> <a href="vectsp_5.html#R1" title="VECTSP_5:pred.1">is_the_direct_sum_of</a> <font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font> implies for <font color="Olive" title="b1">C1</font> being   <a href="vectsp_4.html#M2" title="VECTSP_4:mode.2">Coset</a> of <font color="Maroon" title="c3">W1</font><br/> for <font color="Olive" title="b2">C2</font> being   <a href="vectsp_4.html#M2" title="VECTSP_4:mode.2">Coset</a> of <font color="Maroon" title="c4">W2</font>  ex <font color="Olive" title="b3">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M</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="56_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:56_1"/><i><font color="Green" title="E44">A2</font></i>: 
<font color="Maroon" title="c2">M</font> <a href="vectsp_5.html#R1" title="VECTSP_5:pred.1">is_the_direct_sum_of</a> <font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c7">C1</font> be    <a href="vectsp_4.html#M2" title="VECTSP_4:mode.2">Coset</a> of <font color="Maroon" title="c3">W1</font>;<br/><b>let </b><font color="Maroon" title="c8">C2</font> be    <a href="vectsp_4.html#M2" title="VECTSP_4:mode.2">Coset</a> of <font color="Maroon" title="c4">W2</font>;<br/>



<b>consider </b><font color="Maroon" title="c9">v1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M</font><b> such that </b><br/><a NAME="E3:56_1"/><i><font color="Green" title="E45">A3</font></i>: 
<font color="Maroon" title="c7">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v1</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c3">W1</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D6" title="VECTSP_4:def.6">VECTSP_4:def 6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">v2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M</font><b> such that </b><br/><a NAME="E5:56_1"/><i><font color="Green" title="E46">A4</font></i>: 
<font color="Maroon" title="c8">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">v2</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c4">W2</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D6" title="VECTSP_4:def.6">VECTSP_4:def 6</a></i>;<br/>
<a NAME="E6:56_1"/><i><font color="Green" title="E47">A5</font></i>: 
 <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">VectSpStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">U6</a> of <font color="Maroon" title="c2">M</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c2">M</font>,the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c2">M</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c4">W2</font>
 
<b>by </b><i><a class="txt" href="vectsp_5.html#E1:56_1"><i><font color="Green" title="E44">A2</font></i></a>, <a class="ref" href="vectsp_5.html#D4" target="_self" title="VECTSP_5:def.4">Def4</a></i>;<br/>
<a NAME="E7:56_1"/>
<font color="Maroon" title="c9">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c2">M</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="c11">v11</font>, <font color="Maroon" title="c12">v12</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M</font><b> such that </b><br/><a NAME="E9:56_1"/><i><font color="Green" title="E48">A6</font></i>: 
<font color="Maroon" title="c11">v11</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</font>
 <b>and </b><br/><a NAME="E10:56_1"/><i><font color="Green" title="E49">A7</font></i>: 
<font color="Maroon" title="c12">v12</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W2</font>
 <b>and </b><br/><a NAME="E11:56_1"/><i><font color="Green" title="E50">A8</font></i>: 
<font color="Maroon" title="c9">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">v11</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c12">v12</font>
 <b>by </b><i><a class="txt" href="vectsp_5.html#E6:56_1"><i><font color="Green" title="E47">A5</font></i></a>, <a class="ref" href="vectsp_5.html#T5" target="_self" title="VECTSP_5:th.5">Th5</a></i>;<br/>
<a NAME="E12:56_1"/>
<font color="Maroon" title="c10">v2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c2">M</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="c13">v21</font>, <font color="Maroon" title="c14">v22</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M</font><b> such that </b><br/><a NAME="E14:56_1"/><i><font color="Green" title="E51">A9</font></i>: 
<font color="Maroon" title="c13">v21</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</font>
 <b>and </b><br/><a NAME="E15:56_1"/><i><font color="Green" title="E52">A10</font></i>: 
<font color="Maroon" title="c14">v22</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W2</font>
 <b>and </b><br/><a NAME="E16:56_1"/><i><font color="Green" title="E53">A11</font></i>: 
<font color="Maroon" title="c10">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">v21</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c14">v22</font>
 <b>by </b><i><a class="txt" href="vectsp_5.html#E6:56_1"><i><font color="Green" title="E47">A5</font></i></a>, <a class="ref" href="vectsp_5.html#T5" target="_self" title="VECTSP_5:th.5">Th5</a></i>;<br/>
<b>take </b><font color="Maroon" title="c15">v</font> = <font color="Maroon" title="c12">v12</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c13">v21</font>;<br/>

<a NAME="E17:56_1"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c15">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="c7">C1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">C2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:56_1_1"/><i><font color="Green" title="E54">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="c15">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="c7">C1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">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="56_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">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="E1:56_1_1_1"/>
<font color="Maroon" title="c16">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="c15">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/>

<a NAME="E2:56_1_1_1"/><b>then </b><i><font color="Green" title="E55">A13</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">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="E3:56_1_1_1"/>
<font color="Maroon" title="c12">v12</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="c11">v11</font>
 
<b>by </b><i><a class="txt" href="vectsp_5.html#E11:56_1"><i><font color="Green" title="E50">A8</font></i></a>, <a class="ref" href="vectsp_2.html#T22" title="VECTSP_2:th.22">VECTSP_2:22</a></i>;<br/>
<a NAME="E4:56_1_1_1"/><b>then </b>
<font color="Maroon" title="c12">v12</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="vectsp_5.html#E3:56_1"><i><font color="Green" title="E45">A3</font></i></a>, <a class="txt" href="vectsp_5.html#E9:56_1"><i><font color="Green" title="E48">A6</font></i></a>, <a class="ref" href="vectsp_4.html#T78" title="VECTSP_4:th.78">VECTSP_4:78</a></i>;<br/>
<a NAME="E5:56_1_1_1"/><b>then </b>
<font color="Maroon" title="c7">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">v12</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c3">W1</font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T93" title="VECTSP_4:th.93">VECTSP_4:93</a></i>;<br/>
<a NAME="E6:56_1_1_1"/><b>then </b><i><font color="Green" title="E56">A14</font></i>: 
<font color="Maroon" title="c16">x</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="vectsp_5.html#E14:56_1"><i><font color="Green" title="E51">A9</font></i></a>, <a class="txt" href="vectsp_5.html#E2:56_1_1_1"><i><font color="Green" title="E55">A13</font></i></a></i>;<br/>
<a NAME="E7:56_1_1_1"/>
<font color="Maroon" title="c13">v21</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">v2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c14">v22</font>
 
<b>by </b><i><a class="txt" href="vectsp_5.html#E16:56_1"><i><font color="Green" title="E53">A11</font></i></a>, <a class="ref" href="vectsp_2.html#T22" title="VECTSP_2:th.22">VECTSP_2:22</a></i>;<br/>
<a NAME="E8:56_1_1_1"/><b>then </b>
<font color="Maroon" title="c13">v21</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">C2</font>
 
<b>by </b><i><a class="txt" href="vectsp_5.html#E5:56_1"><i><font color="Green" title="E46">A4</font></i></a>, <a class="txt" href="vectsp_5.html#E15:56_1"><i><font color="Green" title="E52">A10</font></i></a>, <a class="ref" href="vectsp_4.html#T78" title="VECTSP_4:th.78">VECTSP_4:78</a></i>;<br/>
<a NAME="E9:56_1_1_1"/><b>then </b>
( <font color="Maroon" title="c8">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">v21</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c4">W2</font> &amp; <font color="Maroon" title="c15">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">v21</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c12">v12</font> )
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T93" title="VECTSP_4:th.93">VECTSP_4:93</a></i>;<br/>
<a NAME="E10:56_1_1_1"/><b>then </b>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">C2</font>
 
<b>by </b><i><a class="txt" href="vectsp_5.html#E10:56_1"><i><font color="Green" title="E49">A7</font></i></a>, <a class="txt" href="vectsp_5.html#E2:56_1_1_1"><i><font color="Green" title="E55">A13</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:56_1_1_1"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">C2</font>
 <b>by </b><i><a class="txt" href="vectsp_5.html#E6:56_1_1_1"><i><font color="Green" title="E56">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="c16">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="E2:56_1_1"/><i><font color="Green" title="E55">A15</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">C2</font>
 ;<br/>

<a NAME="E3:56_1_1"/><b>then </b>
<font color="Maroon" title="c7">C1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c8">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="c17">C</font> = <font color="Maroon" title="c7">C1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">C2</font> as    <a href="vectsp_4.html#M2" title="VECTSP_4:mode.2">Coset</a> of <font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c4">W2</font> <b>by </b><i><a class="ref" href="vectsp_5.html#T55" target="_self" title="VECTSP_5:th.55">Th55</a></i>;<br/>
<a NAME="E5:56_1_1"/>
<font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c4">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="txt" href="vectsp_5.html#E1:56_1"><i><font color="Green" title="E44">A2</font></i></a>, <a class="ref" href="vectsp_5.html#D4" target="_self" title="VECTSP_5:def.4">Def4</a></i>;<br/>
<a NAME="E6:56_1_1"/><b>then </b><i><font color="Green" title="E56">A16</font></i>: 
 ex <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M</font> st <font color="Maroon" title="c17">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="vectsp_4.html#T88" title="VECTSP_4:th.88">VECTSP_4:88</a></i>;<br/>
<a NAME="E7:56_1_1"/>
<font color="Maroon" title="c15">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="c15">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="E8:56_1_1"/>
<font color="Maroon" title="c16">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="c15">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="vectsp_5.html#E1:56_1_1"><i><font color="Green" title="E54">A12</font></i></a>, <a class="txt" href="vectsp_5.html#E2:56_1_1"><i><font color="Green" title="E55">A15</font></i></a>, <a class="txt" href="vectsp_5.html#E6:56_1_1"><i><font color="Green" title="E56">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="E9:56_1_1"/>
verum
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E18:56_1"/>
<font color="Maroon" title="c7">C1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">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="c15">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="E3:56"/><i><font color="Green" title="E44">A17</font></i>: 
for <font color="Olive" title="b1">C1</font> being   <a href="vectsp_4.html#M2" title="VECTSP_4:mode.2">Coset</a> of <font color="Maroon" title="c3">W1</font><br/> for <font color="Olive" title="b2">C2</font> being   <a href="vectsp_4.html#M2" title="VECTSP_4:mode.2">Coset</a> of <font color="Maroon" title="c4">W2</font>  ex <font color="Olive" title="b3">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M</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/>

<a NAME="E4:56"/><i><font color="Green" title="E45">A18</font></i>: 
( <font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c4">W2</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c2">M</font> &amp;  <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">VectSpStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">U6</a> of <font color="Maroon" title="c2">M</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c2">M</font>,the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c2">M</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c2">M</font> )
 
<b>by </b><i><a class="txt" href="vectsp_5.html#E21"><i><font color="Green" title="E15">Lm6</font></i></a></i>;<br/>
<a NAME="E5:56"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 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="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:56_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 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="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c4">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="56_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">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="E1:56_2_1"/>
<font color="Maroon" title="c7">x</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">M</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c8">u</font> = <font color="Maroon" title="c7">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M</font> ;<br/>
<b>consider </b><font color="Maroon" title="c9">C1</font> being    <a href="vectsp_4.html#M2" title="VECTSP_4:mode.2">Coset</a> of <font color="Maroon" title="c3">W1</font><b> such that </b><br/><a NAME="E4:56_2_1"/><i><font color="Green" title="E46">A19</font></i>: 
<font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C1</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T84" title="VECTSP_4:th.84">VECTSP_4:84</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M</font><b> such that </b><br/><a NAME="E6:56_2_1"/><i><font color="Green" title="E47">A20</font></i>: 
<font color="Maroon" title="c9">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="c4">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="c10">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="vectsp_5.html#E1:56"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="vectsp_5.html#E3:56"><i><font color="Green" title="E44">A17</font></i></a></i>;<br/>
<a NAME="E7:56_2_1"/><i><font color="Green" title="E48">A21</font></i>: 
<font color="Maroon" title="c10">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="c10">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="E8:56_2_1"/><b>then </b>
<font color="Maroon" title="c10">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="c4">W2</font>
 
<b>by </b><i><a class="txt" href="vectsp_5.html#E6:56_2_1"><i><font color="Green" title="E47">A20</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="E9:56_2_1"/><b>then </b><i><font color="Green" title="E49">A22</font></i>: 
<font color="Maroon" title="c10">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">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="E10:56_2_1"/>
<font color="Maroon" title="c10">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C1</font>
 
<b>by </b><i><a class="txt" href="vectsp_5.html#E6:56_2_1"><i><font color="Green" title="E47">A20</font></i></a>, <a class="txt" href="vectsp_5.html#E7:56_2_1"><i><font color="Green" title="E48">A21</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">v1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M</font><b> such that </b><br/><a NAME="E12:56_2_1"/><i><font color="Green" title="E50">A23</font></i>: 
<font color="Maroon" title="c11">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</font>
 <b>and </b><br/><a NAME="E13:56_2_1"/><i><font color="Green" title="E51">A24</font></i>: 
<font color="Maroon" title="c8">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c11">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">v</font>
 <b>by </b><i><a class="txt" href="vectsp_5.html#E4:56_2_1"><i><font color="Green" title="E46">A19</font></i></a>, <a class="ref" href="vectsp_4.html#T95" title="VECTSP_4:th.95">VECTSP_4:95</a></i>;<br/>
<a NAME="E14:56_2_1"/>
<font color="Maroon" title="c8">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">v</font>
 
<b>by </b><i><a class="txt" href="vectsp_5.html#E13:56_2_1"><i><font color="Green" title="E51">A24</font></i></a>, <a class="ref" href="vectsp_2.html#T22" title="VECTSP_2:th.22">VECTSP_2:22</a></i>;<br/>
<a NAME="E15:56_2_1"/><b>then </b>
<font color="Maroon" title="c7">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c4">W2</font>
 
<b>by </b><i><a class="txt" href="vectsp_5.html#E9:56_2_1"><i><font color="Green" title="E49">A22</font></i></a>, <a class="txt" href="vectsp_5.html#E12:56_2_1"><i><font color="Green" title="E50">A23</font></i></a>, <a class="ref" href="vectsp_5.html#T5" target="_self" title="VECTSP_5:th.5">Th5</a></i>;<br/>
<b>hence </b><a NAME="E16:56_2_1"/>
<font color="Maroon" title="c7">x</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 <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span>
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>


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

<b>thus </b><a NAME="E2:56_2"/>
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="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D2" title="VECTSP_4:def.2">VECTSP_4:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:56"/>
 <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">VectSpStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">U6</a> of <font color="Maroon" title="c2">M</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c2">M</font>,the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c2">M</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c4">W2</font>
 <b>by </b><i><a class="txt" href="vectsp_5.html#E4:56"><i><font color="Green" title="E45">A18</font></i></a>, <a class="ref" href="vectsp_4.html#T39" title="VECTSP_4:th.39">VECTSP_4:39</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_5.html#D4" target="_self" title="VECTSP_5:def.4">VECTSP_5:def 4</a><br/>

<b>consider </b><font color="Maroon" title="c7">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M</font><b> such that </b><br/><a NAME="E8:56"/><i><font color="Green" title="E46">A25</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">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="c4">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="c7">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="vectsp_5.html#E1:56"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="vectsp_5.html#E3:56"><i><font color="Green" title="E44">A17</font></i></a></i>;<br/>
<a NAME="E9:56"/>
(  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">M</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">M</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W2</font> )
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T25" title="VECTSP_4:th.25">VECTSP_4:25</a></i>;<br/>
<a NAME="E10:56"/><b>then </b>
(  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">M</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">W1</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">M</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="c4">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="E11:56"/><b>then </b><i><font color="Green" title="E47">A26</font></i>: 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">M</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="c7">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="vectsp_5.html#E8:56"><i><font color="Green" title="E46">A25</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="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">M</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="c2">M</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="vectsp_4.html#D3" title="VECTSP_4:def.3">VECTSP_4:def 3</a></i>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">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="c4">W2</font>
<b>by </b><i><a class="txt" href="vectsp_5.html#E8:56"><i><font color="Green" title="E46">A25</font></i></a>, <a class="txt" href="vectsp_5.html#E11:56"><i><font color="Green" title="E47">A26</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="c3">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c4">W2</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_5.html#D2" target="_self" title="VECTSP_5:def.2">Def2</a></i>
;<br/>
<b>hence </b><a NAME="E13:56"/>
<font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c4">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T37" title="VECTSP_4:th.37">VECTSP_4:37</a></i>;<br/>


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