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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="vectsp_2.html#NM7">RightMod</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="rmod_2.html#M1">Submodule</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>





<b>set </b><font color="Maroon">c<sub>5</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E1:56"/><i><font color="Green">E40</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> is    <a href="rmod_2.html#M2">Coset</a> of <font color="Maroon">c<sub>3</sub></font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> is    <a href="rmod_2.html#M2">Coset</a> of <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="rmod_2.html#T86">RMOD_2:86</a></i>;<br/>
<b>thus </b><a NAME="E2:56"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="rmod_3.html#R1" target="_self">is_the_direct_sum_of</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> implies for <font color="Olive">b<sub>1</sub></font> being   <a href="rmod_2.html#M2">Coset</a> of <font color="Maroon">c<sub>3</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="rmod_2.html#M2">Coset</a> of <font color="Maroon">c<sub>4</sub></font>ex <font color="Olive">b<sub>3</sub></font> being  <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>3</sub></font></span><a href="domain_1.html#K6">}</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">E41</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="rmod_3.html#R1" target="_self">is_the_direct_sum_of</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="rmod_2.html#M2">Coset</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="rmod_2.html#M2">Coset</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>



<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:56_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="rmod_2.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="rmod_2.html#D6">RMOD_2:def 6</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E5:56_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="rmod_2.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="rmod_2.html#D6">RMOD_2:def 6</a></i>;<br/>
<a NAME="E6:56_1"/><i><font color="Green">E44</font></i>: 
 <a href="vectsp_2.html#G1">RightModStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="vectsp_2.html#U1">rmult</a> of <font color="Maroon">c<sub>2</sub></font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="rmod_3.html#E1:56_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="rmod_3.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E7:56_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="rmod_2.html#K2">(Omega).</a> <font color="Maroon">c<sub>2</sub></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">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> being   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E9:56_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E10:56_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E11:56_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="rmod_3.html#E6:56_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="rmod_3.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E12:56_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="rmod_2.html#K2">(Omega).</a> <font color="Maroon">c<sub>2</sub></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">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E14:56_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E15:56_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E16:56_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="rmod_3.html#E6:56_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="rmod_3.html#T5" target="_self">Th5</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>13</sub></font>;<br/>

<a NAME="E17:56_1"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>8</sub></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">E51</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>8</sub></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="56_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</sub></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="E1:56_1_1_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E2:56_1_1_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:56_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="rmod_3.html#E11:56_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="rmod_3.html#E63"><i><font color="Green">Lemma36</font></i></a></i>;<br/>
<a NAME="E4:56_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="rmod_3.html#E3:56_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="rmod_3.html#E9:56_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="rmod_2.html#T75">RMOD_2:75</a></i>;<br/>
<a NAME="E5:56_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="rmod_2.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="rmod_2.html#T90">RMOD_2:90</a></i>;<br/>
<a NAME="E6:56_1_1_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="rmod_3.html#E14:56_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="rmod_3.html#E2:56_1_1_1"><i><font color="Green">E52</font></i></a></i>;<br/>
<a NAME="E7:56_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="rmod_3.html#E16:56_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="rmod_3.html#E63"><i><font color="Green">Lemma36</font></i></a></i>;<br/>
<a NAME="E8:56_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="rmod_3.html#E5:56_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="rmod_3.html#E15:56_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="rmod_2.html#T75">RMOD_2:75</a></i>;<br/>
<a NAME="E9:56_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="rmod_2.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>12</sub></font> )
 
<b>by </b><i><a class="ref" href="rmod_2.html#T90">RMOD_2:90</a></i>;<br/>
<a NAME="E10:56_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="rmod_3.html#E10:56_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="rmod_3.html#E2:56_1_1_1"><i><font color="Green">E52</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:56_1_1_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="rmod_3.html#E6:56_1_1_1"><i><font color="Green">E53</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">c<sub>16</sub></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="E2:56_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<a NAME="E3:56_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>8</sub></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<sub>17</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>8</sub></font> as    <a href="rmod_2.html#M2">Coset</a> of <font color="Maroon">c<sub>3</sub></font> <a href="rmod_3.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="ref" href="rmod_3.html#T48" target="_self">Th48</a></i>;<br/>
<a NAME="E5:56_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="rmod_3.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="rmod_2.html#K1">(0).</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="rmod_3.html#E1:56_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="rmod_3.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E6:56_1_1"/><b>then </b><i><font color="Green">E53</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="rmod_2.html#T85">RMOD_2:85</a></i>;<br/>
<a NAME="E7:56_1_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>15</sub></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="E8:56_1_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="rmod_3.html#E1:56_1_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="rmod_3.html#E2:56_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="rmod_3.html#E6:56_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="tarski.html#D1">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">c<sub>7</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>


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

<b>assume </b><a NAME="E3:56"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="rmod_2.html#M2">Coset</a> of <font color="Maroon">c<sub>3</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="rmod_2.html#M2">Coset</a> of <font color="Maroon">c<sub>4</sub></font>ex <font color="Olive">b<sub>3</sub></font> being  <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E4:56"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font> is    <a href="rmod_2.html#M1">Submodule</a> of  <a href="rmod_2.html#K2">(Omega).</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="vectsp_2.html#G1">RightModStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="vectsp_2.html#U1">rmult</a> of <font color="Maroon">c<sub>2</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="rmod_2.html#K2">(Omega).</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="rmod_3.html#E20"><i><font color="Green">Lemma13</font></i></a></i>;<br/>
<a NAME="E5:56"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">c<sub>4</sub></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">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
  <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="56_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></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="E1:56_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="rmod_2.html#M2">Coset</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E4:56_2_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="rmod_2.html#T81">RMOD_2:81</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E6:56_2_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="rmod_3.html#E1:56"><i><font color="Green">E40</font></i></a>, <a class="txt" href="rmod_3.html#E3:56"><i><font color="Green">E41</font></i></a></i>;<br/>
<a NAME="E7:56_2_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>10</sub></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="E8:56_2_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="rmod_3.html#E6:56_2_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E9:56_2_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E10:56_2_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="rmod_3.html#E6:56_2_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="rmod_3.html#E7:56_2_1"><i><font color="Green">E45</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">c<sub>11</sub></font> being   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E12:56_2_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E13:56_2_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="rmod_3.html#E4:56_2_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="rmod_2.html#T92">RMOD_2:92</a></i>;<br/>
<a NAME="E14:56_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="rmod_3.html#E13:56_2_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="rmod_3.html#E63"><i><font color="Green">Lemma36</font></i></a></i>;<br/>
<a NAME="E15:56_2_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="rmod_3.html#E9:56_2_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="rmod_3.html#E12:56_2_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="rmod_3.html#T5" target="_self">Th5</a></i>;<br/>
<b>hence </b><a NAME="E16:56_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>


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

<b>thus </b><a NAME="E2:56_2"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="rmod_2.html#D2">RMOD_2:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:56"/>
 <a href="vectsp_2.html#G1">RightModStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="vectsp_2.html#U1">rmult</a> of <font color="Maroon">c<sub>2</sub></font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="rmod_3.html#E4:56"><i><font color="Green">E42</font></i></a>, <a class="ref" href="rmod_2.html#T39">RMOD_2:39</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="rmod_3.html#D4" target="_self">RMOD_3:def 4</a><br/>

<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E8:56"/><i><font color="Green">E43</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="rmod_3.html#E1:56"><i><font color="Green">E40</font></i></a>, <a class="txt" href="rmod_3.html#E3:56"><i><font color="Green">E41</font></i></a></i>;<br/>
<a NAME="E9:56"/>
(  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="rmod_2.html#T25">RMOD_2:25</a></i>;<br/>
<a NAME="E10:56"/><b>then </b>
(  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E11:56"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="rmod_3.html#E8:56"><i><font color="Green">E43</font></i></a>, <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="rmod_2.html#K1">(0).</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i><a class="ref" href="rmod_2.html#D3">RMOD_2:def 3</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="rmod_3.html#E8:56"><i><font color="Green">E43</font></i></a>, <a class="txt" href="rmod_3.html#E11:56"><i><font color="Green">E44</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">c<sub>3</sub></font> <a href="rmod_3.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rmod_3.html#D2" target="_self">Def2</a></i>
;<br/>
<b>hence </b><a NAME="E13:56"/>
<font color="Maroon">c<sub>3</sub></font> <a href="rmod_3.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="rmod_2.html#K1">(0).</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="rmod_2.html#T37">RMOD_2:37</a></i>;<br/>


</div>
