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

<b>let </b><font color="Maroon">V</font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">W1</font> be    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">V</font>, <font color="Maroon">W2</font> be    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">V</font>;<br/>



<b>set </b><font color="Maroon">VW1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font>;<br/>
<b>set </b><font color="Maroon">VW2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font>;<br/>
<a NAME="E1:60"/><i><font color="Green">E32</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> is    <a href="rlsub_1.html#M2">Coset</a> of <font color="Maroon">W1</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font> is    <a href="rlsub_1.html#M2">Coset</a> of <font color="Maroon">W2</font> )
 
<b>by </b><i><a class="ref" href="rlsub_1.html#T90">RLSUB_1:90</a></i>;<br/>
<b>thus </b><a NAME="E2:60"/>
( <font color="Maroon">V</font> <a href="rlsub_2.html#R1" target="_self">is_the_direct_sum_of</a> <font color="Maroon">W1</font>,<font color="Maroon">W2</font> implies for <font color="Olive">C1</font> being   <a href="rlsub_1.html#M2">Coset</a> of <font color="Maroon">W1</font><br/> for <font color="Olive">C2</font> being   <a href="rlsub_1.html#M2">Coset</a> of <font color="Maroon">W2</font>  ex <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> st <font color="Olive">C1</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">C2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">v</font></span><a href="domain_1.html#K6">}</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">E35</font></i>: 
<font color="Maroon">V</font> <a href="rlsub_2.html#R1" target="_self">is_the_direct_sum_of</a> <font color="Maroon">W1</font>,<font color="Maroon">W2</font>
 ;<br/>

<b>let </b><font color="Maroon">C1</font> be    <a href="rlsub_1.html#M2">Coset</a> of <font color="Maroon">W1</font>;<br/><b>let </b><font color="Maroon">C2</font> be    <a href="rlsub_1.html#M2">Coset</a> of <font color="Maroon">W2</font>;<br/>



<b>consider </b><font color="Maroon">v1</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:60_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> <a href="rlsub_1.html#K3">+</a> <font color="Maroon">W1</font>
 <b>by </b><i><a class="ref" href="rlsub_1.html#D6">RLSUB_1:def 6</a></i>;<br/>
<b>consider </b><font color="Maroon">v2</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E5:60_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">C2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font> <a href="rlsub_1.html#K3">+</a> <font color="Maroon">W2</font>
 <b>by </b><i><a class="ref" href="rlsub_1.html#D6">RLSUB_1:def 6</a></i>;<br/>
<a NAME="E6:60_1"/><i><font color="Green">E38</font></i>: 
 <a href="rlvect_1.html#G2">RLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="rlsub_2.html#K1" target="_self">+</a> <font color="Maroon">W2</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:60_1"/>
<font color="Maroon">v1</font> <a href="rlvect_1.html#R1">in</a>  <a href="rlvect_1.html#G2">RLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</font> #)
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T3">RLVECT_1:3</a></i>;<br/>
<b>then consider </b><font color="Maroon">v11</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>, <font color="Maroon">v12</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E9:60_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">v11</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font>
 <b>and </b><br/><a NAME="E10:60_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">v12</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font>
 <b>and </b><br/><a NAME="E11:60_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v11</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v12</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E12"><i><font color="Green">Lemma52</font></i></a>, </i>;<br/>
<a NAME="E12:60_1"/>
<font color="Maroon">v2</font> <a href="rlvect_1.html#R1">in</a>  <a href="rlvect_1.html#G2">RLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</font> #)
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T3">RLVECT_1:3</a></i>;<br/>
<b>then consider </b><font color="Maroon">v21</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>, <font color="Maroon">v22</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E14:60_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">v21</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font>
 <b>and </b><br/><a NAME="E15:60_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">v22</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font>
 <b>and </b><br/><a NAME="E16:60_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v21</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v22</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E12"><i><font color="Green">Lemma52</font></i></a>, </i>;<br/>
<b>take </b><font color="Maroon">v</font> = <font color="Maroon">v12</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v21</font>;<br/>

<a NAME="E17:60_1"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">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">E61</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">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">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:60_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E2:60_1_1_1"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:60_1_1_1"/>
<font color="Maroon">v12</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v11</font>
 
<b>by </b><i>, <a class="txt" href="rlsub_2.html#E5:60_1"><i><font color="Green">E37</font></i></a></i>;<br/>
<a NAME="E4:60_1_1_1"/><b>then </b>
<font color="Maroon">v12</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font>
 
<b>by </b><i>, <a class="ref" href="rlsub_2.html#T10" target="_self">Th10</a>, <a class="ref" href="rlsub_1.html#T80">RLSUB_1:80</a></i>;<br/>
<a NAME="E5:60_1_1_1"/><b>then </b>
<font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v12</font> <a href="rlsub_1.html#K3">+</a> <font color="Maroon">W1</font>
 
<b>by </b><i><a class="ref" href="rlsub_1.html#T94">RLSUB_1:94</a></i>;<br/>
<a NAME="E6:60_1_1_1"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E10:60_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="rlsub_2.html#E16:60_1"><i><font color="Green">E60</font></i></a></i>;<br/>
<a NAME="E7:60_1_1_1"/>
<font color="Maroon">v21</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v22</font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E14:60_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="rlsub_2.html#E5:60_1"><i><font color="Green">E37</font></i></a></i>;<br/>
<a NAME="E8:60_1_1_1"/><b>then </b>
<font color="Maroon">v21</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C2</font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E11"><i><font color="Green">Lemma51</font></i></a>, <a class="txt" href="rlsub_2.html#E11:60_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="rlsub_1.html#T80">RLSUB_1:80</a></i>;<br/>
<a NAME="E9:60_1_1_1"/><b>then </b>
( <font color="Maroon">C2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v21</font> <a href="rlsub_1.html#K3">+</a> <font color="Maroon">W2</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v21</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v12</font> )
 
<b>by </b><i><a class="ref" href="rlsub_1.html#T94">RLSUB_1:94</a></i>;<br/>
<a NAME="E10:60_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C2</font>
 
<b>by </b><i>, <a class="txt" href="rlsub_2.html#E16:60_1"><i><font color="Green">E60</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:60_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E1:60_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


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

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

<b>assume </b><a NAME="E2:60_1_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font>
 ;<br/>

<a NAME="E3:60_1_1"/><b>then </b>
<font color="Maroon">C1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C2</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#T4">XBOOLE_0:4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">C</font> = <font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font> as    <a href="rlsub_1.html#M2">Coset</a> of <font color="Maroon">W1</font> <a href="rlsub_2.html#K2" target="_self">/\</a> <font color="Maroon">W2</font> <b>by </b><i><a class="ref" href="rlsub_2.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E5:60_1_1"/>
<font color="Maroon">W1</font> <a href="rlsub_2.html#K2" target="_self">/\</a> <font color="Maroon">W2</font> <a href="hidden.html#R1">=</a>  <a href="rlsub_1.html#K1">(0).</a> <font color="Maroon">V</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:60_1_1"/><b>then </b><i><font color="Green">E65</font></i>: 
 ex <font color="Olive">u</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> st <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">u</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="rlsub_1.html#T89">RLSUB_1:89</a></i>;<br/>
<a NAME="E7:60_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E8:60_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E15:60_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="rlsub_2.html#E16:60_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="rlsub_2.html#E1:60_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="tarski.html#D1">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">C1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>


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

<b>assume </b><a NAME="E3:60"/><i><font color="Green">E105</font></i>: 
for <font color="Olive">C1</font> being   <a href="rlsub_1.html#M2">Coset</a> of <font color="Maroon">W1</font><br/> for <font color="Olive">C2</font> being   <a href="rlsub_1.html#M2">Coset</a> of <font color="Maroon">W2</font>  ex <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> st <font color="Olive">C1</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">C2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">v</font></span><a href="domain_1.html#K6">}</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">u</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>consider </b><font color="Maroon">C1</font> being    <a href="rlsub_1.html#M2">Coset</a> of <font color="Maroon">W1</font><b> such that </b><br/><a NAME="E2:60_2"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font>
 <b>by </b><i><a class="ref" href="rlsub_2.html#T6" target="_self">Th6</a></i>;<br/><b>consider </b><font color="Maroon">v</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E4:60_2"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">C1</font> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E10"><i><font color="Green">Lemma48</font></i></a>, </i>;<br/><a NAME="E5:60_2"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E6:60_2"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E11"><i><font color="Green">Lemma51</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E7:60_2"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E8:60_2"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E11"><i><font color="Green">Lemma51</font></i></a>, <a class="txt" href="rlsub_2.html#E12"><i><font color="Green">Lemma52</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">v1</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E10:60_2"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">v1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font>
 <b>and </b><br/><a NAME="E11:60_2"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 <b>by </b><i>, <a class="ref" href="rlsub_1.html#T96">RLSUB_1:96</a></i>;<br/><a NAME="E12:60_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E13:60_2"/>
<font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> <a href="rlsub_2.html#K1" target="_self">+</a> <font color="Maroon">W2</font>
 <b>by </b><i><a class="ref" href="rlsub_2.html#T10" target="_self">Th10</a>, , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:60"/>
 <a href="rlvect_1.html#G2">RLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="rlsub_2.html#K1" target="_self">+</a> <font color="Maroon">W2</font>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="rlsub_2.html#D4" target="_self">RLSUB_2:def 4</a><br/>

<b>consider </b><font color="Maroon">v</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E7:60"/><i><font color="Green">E115</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E10"><i><font color="Green">Lemma48</font></i></a>, </i>;<br/>
<a NAME="E8:60"/>
(  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> &amp;  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> )
 
<b>by </b><i><a class="ref" href="rlsub_1.html#T25">RLSUB_1:25</a></i>;<br/>
<a NAME="E9:60"/><b>then </b>
(  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> &amp;  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font> )
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E10:60"/><b>then </b><i><font color="Green">E116</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="rlsub_1.html#K1">(0).</a> <font color="Maroon">V</font></span>)</span> = 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i><a class="ref" href="rlsub_1.html#D3">RLSUB_1:def 3</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font>
<b>by </b><i>, <a class="txt" href="rlsub_2.html#E11"><i><font color="Green">Lemma51</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="rlsub_2.html#K2" target="_self">/\</a> <font color="Maroon">W2</font></span>)</span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E12:60"/>
<font color="Maroon">W1</font> <a href="rlsub_2.html#K2" target="_self">/\</a> <font color="Maroon">W2</font> <a href="hidden.html#R1">=</a>  <a href="rlsub_1.html#K1">(0).</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="rlsub_1.html#T38">RLSUB_1:38</a></i>;<br/>


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