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<b>let </b><font color="Maroon">R</font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="vectsp_2.html#NM7">RightMod</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">W1</font> be    <a href="rmod_2.html#M1">Submodule</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">W2</font> be    <a href="rmod_2.html#M1">Submodule</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">C1</font> be    <a href="rmod_2.html#M2">Coset</a> of <font color="Maroon">W1</font>;<br/><b>let </b><font color="Maroon">C2</font> be    <a href="rmod_2.html#M2">Coset</a> of <font color="Maroon">W2</font>;<br/>











<b>assume </b><a NAME="E1:55"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/>

<b>consider </b><font color="Maroon">v</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font>;<br/>
<b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">v</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>set </b><font color="Maroon">C</font> = <font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font>;<br/>
<a NAME="E4:55"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:55"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="rmod_2.html#K3">+</a> <font color="Maroon">W1</font>
 
<b>by </b><i><a class="ref" href="rmod_2.html#T90">RMOD_2:90</a></i>;<br/>
<a NAME="E6:55"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C2</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E7:55"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">C2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="rmod_2.html#K3">+</a> <font color="Maroon">W2</font>
 
<b>by </b><i><a class="ref" href="rmod_2.html#T90">RMOD_2:90</a></i>;<br/>
<a NAME="E8:55"/>
<font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font> is    <a href="rmod_2.html#M2">Coset</a> of <font color="Maroon">W1</font> <a href="rmod_3.html#K2" target="_self">/\</a> <font color="Maroon">W2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">v</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="rmod_2.html#D6">RMOD_2:def 6</a><br/>

<b>thus </b><a NAME="E1:55_1"/>
<font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">v</font> <a href="rmod_2.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="rmod_3.html#K2" target="_self">/\</a> <font color="Maroon">W2</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="55_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="E1:55_1_1"/><i><font color="Green">E35</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="E2:55_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">u1</font> being   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E4:55_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">u1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font>
 <b>and </b><br/><a NAME="E5:55_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u1</font>
 <b>by </b><i>, <a class="ref" href="rmod_2.html#T57">RMOD_2:57</a></i>;<br/>
<a NAME="E6:55_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C2</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">u2</font> being   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E8:55_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">u2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font>
 <b>and </b><br/><a NAME="E9:55_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u2</font>
 <b>by </b><i>, <a class="ref" href="rmod_2.html#T57">RMOD_2:57</a></i>;<br/>
<a NAME="E10:55_1_1"/>
<font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u2</font>
 
<b>by </b><i><a class="ref" href="rmod_3.html#T7" target="_self">Th7</a>, , <a class="ref" href="rlvect_1.html#T21">RLVECT_1:21</a></i>;<br/>
<a NAME="E11:55_1_1"/><b>then </b>
<font color="Maroon">u1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> <a href="rmod_3.html#K2" target="_self">/\</a> <font color="Maroon">W2</font>
 
<b>by </b><i><a class="ref" href="rmod_3.html#T5" target="_self">Th5</a>, <a class="txt" href="rmod_3.html#E10"><i><font color="Green">Lemma45</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E12:55_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="rmod_2.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="rmod_3.html#K2" target="_self">/\</a> <font color="Maroon">W2</font></span>)</span>
 <b>by </b><i><a class="ref" href="rmod_3.html#T7" target="_self">Th7</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="E2:55_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="rmod_2.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="rmod_3.html#K2" target="_self">/\</a> <font color="Maroon">W2</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">u</font> being   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E4:55_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> <a href="rmod_3.html#K2" target="_self">/\</a> <font color="Maroon">W2</font>
 <b>and </b><br/><a NAME="E5:55_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="ref" href="rmod_2.html#T57">RMOD_2:57</a></i>;<br/>
<a NAME="E6:55_1"/>
( <font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:55_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C2</font> )
 
<b>by </b><i>, , <a class="ref" href="rmod_3.html#T5" target="_self">Th5</a></i>;<br/>
<b>hence </b><a NAME="E8:55_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="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:55"/>
<font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font> is    <a href="rmod_2.html#M2">Coset</a> of <font color="Maroon">W1</font> <a href="rmod_3.html#K2" target="_self">/\</a> <font color="Maroon">W2</font>
 ;<br/>


</div>
