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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>, <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">v</font> be   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font>, <font color="Maroon">v1</font> be   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font>, <font color="Maroon">v2</font> be   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font>, <font color="Maroon">u1</font> be   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font>, <font color="Maroon">u2</font> be   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font>;<br/>







<b>assume </b><a NAME="E1:57"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">V</font> <a href="rmod_3.html#R1" target="_self">is_the_direct_sum_of</a> <font color="Maroon">W1</font>,<font color="Maroon">W2</font>
 ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:57"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u2</font> )
 <b>and </b><br/><a NAME="E3:57"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">v1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">u1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> )
 <b>and </b><br/><a NAME="E4:57"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">v2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> &amp; <font color="Maroon">u2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">C2</font> = <font color="Maroon">v1</font> <a href="rmod_2.html#K3">+</a> <font color="Maroon">W2</font> as    <a href="rmod_2.html#M2">Coset</a> of <font color="Maroon">W2</font> <b>by </b><i><a class="ref" href="rmod_2.html#D6">RMOD_2:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">C1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> as    <a href="rmod_2.html#M2">Coset</a> of <font color="Maroon">W1</font> <b>by </b><i><a class="ref" href="rmod_2.html#T86">RMOD_2:86</a></i>;<br/>
<i><font color="Green">E36</font></i>: <font color="Maroon">u1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u2</font>
<b>by </b><i>, <a class="txt" href="rmod_3.html#E7:57"><i><font color="Green">E36</font></i></a></i>
<br/>.= 
<font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u2</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
;<br/>
<a NAME="E8:57"/>
<font color="Maroon">v2</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font>
 
<b>by </b><i><a class="ref" href="rmod_3.html#T7" target="_self">Th7</a>, <a class="ref" href="rmod_2.html#T31">RMOD_2:31</a></i>;<br/>
<a NAME="E9:57"/><b>then </b>
( <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> &amp; <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C2</font> &amp; <font color="Maroon">u1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> &amp; <font color="Maroon">u1</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>, <a class="txt" href="rmod_3.html#E10"><i><font color="Green">Lemma45</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a>, <a class="ref" href="rmod_2.html#T59">RMOD_2:59</a></i>;<br/>
<a NAME="E10:57"/><b>then </b><i><font color="Green">E37</font></i>: 
( <font color="Maroon">v1</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> &amp; <font color="Maroon">u1</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/>
<b>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="E12:57"/><i><font color="Green">E53</font></i>: 
<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">u</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, </i>;<br/>
<a NAME="E13:57"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> &amp; <font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> )
 
<b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E14:57"/>
<font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u1</font>
 ;<br/>

<b>thus </b><a NAME="E15:57"/>
<font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u2</font>
 <b>by </b><i>, <a class="txt" href="rmod_3.html#E11"><i><font color="Green">Lemma48</font></i></a>, <a class="ref" href="rlvect_1.html#T21">RLVECT_1:21</a></i>;<br/>


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