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<b>let </b><font color="Maroon" title="c1">R</font> be   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>;<br/><b>let </b><font color="Maroon" title="c2">V</font> be   <a href="vectsp_2.html#NM7" title="VECTSP_2:NM.7">RightMod</a> of <font color="Maroon" title="c1">R</font>;<br/><b>let </b><font color="Maroon" title="c3">W1</font>, <font color="Maroon" title="c4">W2</font> be    <a href="rmod_2.html#M1" title="RMOD_2:mode.1">Submodule</a> of <font color="Maroon" title="c2">V</font>;<br/><b>let </b><font color="Maroon" title="c5">v</font>, <font color="Maroon" title="c6">v1</font>, <font color="Maroon" title="c7">v2</font>, <font color="Maroon" title="c8">u1</font>, <font color="Maroon" title="c9">u2</font> be   <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font>;<br/>







<b>assume </b><a NAME="E1:50"/><i><font color="Green" title="E38">A1</font></i>: 
<font color="Maroon" title="c2">V</font> <a href="rmod_3.html#R1" title="RMOD_3:pred.1">is_the_direct_sum_of</a> <font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font>
 ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:50"/><i><font color="Green" title="E39">A2</font></i>: 
( <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">v2</font> &amp; <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">u1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">u2</font> )
 <b>and </b><br/><a NAME="E3:50"/><i><font color="Green" title="E40">A3</font></i>: 
( <font color="Maroon" title="c6">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</font> &amp; <font color="Maroon" title="c8">u1</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="E4:50"/><i><font color="Green" title="E41">A4</font></i>: 
( <font color="Maroon" title="c7">v2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W2</font> &amp; <font color="Maroon" title="c9">u2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W2</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c10">C2</font> = <font color="Maroon" title="c6">v1</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c4">W2</font> as    <a href="rmod_2.html#M2" title="RMOD_2:mode.2">Coset</a> of <font color="Maroon" title="c4">W2</font> <b>by </b><i><a class="ref" href="rmod_2.html#D6" title="RMOD_2:def.6">RMOD_2:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">C1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">W1</font> as    <a href="rmod_2.html#M2" title="RMOD_2:mode.2">Coset</a> of <font color="Maroon" title="c3">W1</font> <b>by </b><i><a class="ref" href="rmod_2.html#T86" title="RMOD_2:th.86">RMOD_2:86</a></i>;<br/>
<i><font color="Green" title="E42">A5</font></i>: <font color="Maroon" title="c8">u1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">v2</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c9">u2</font>
<b>by </b><i><a class="txt" href="rmod_3.html#E2:50"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="rmod_3.html#E63"><i><font color="Green" title="E33">Lm15</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">v2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c9">u2</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
;<br/>
<a NAME="E8:50"/>
<font color="Maroon" title="c7">v2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c9">u2</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="txt" href="rmod_3.html#E4:50"><i><font color="Green" title="E41">A4</font></i></a>, <a class="ref" href="rmod_2.html#T31" title="RMOD_2:th.31">RMOD_2:31</a></i>;<br/>
<a NAME="E9:50"/><b>then </b>
( <font color="Maroon" title="c6">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">C1</font> &amp; <font color="Maroon" title="c6">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">C2</font> &amp; <font color="Maroon" title="c8">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">C1</font> &amp; <font color="Maroon" title="c8">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">C2</font> )
 
<b>by </b><i><a class="txt" href="rmod_3.html#E3:50"><i><font color="Green" title="E40">A3</font></i></a>, <a class="txt" href="rmod_3.html#E7:50"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="rmod_2.html#T59" title="RMOD_2:th.59">RMOD_2:59</a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E10:50"/><b>then </b><i><font color="Green" title="E43">A6</font></i>: 
( <font color="Maroon" title="c6">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">C1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c10">C2</font> &amp; <font color="Maroon" title="c8">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">C1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c10">C2</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">u</font> being   <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E12:50"/><i><font color="Green" title="E44">A7</font></i>: 
<font color="Maroon" title="c11">C1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c10">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="c12">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="rmod_3.html#E1:50"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="rmod_3.html#T49" target="_self" title="RMOD_3:th.49">Th49</a></i>;<br/>
<a NAME="E13:50"/><i><font color="Green" title="E45">A8</font></i>: 
( <font color="Maroon" title="c6">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">u</font> &amp; <font color="Maroon" title="c8">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">u</font> )
 
<b>by </b><i><a class="txt" href="rmod_3.html#E10:50"><i><font color="Green" title="E43">A6</font></i></a>, <a class="txt" href="rmod_3.html#E12:50"><i><font color="Green" title="E44">A7</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E14:50"/>
<font color="Maroon" title="c6">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">u1</font>
 ;<br/>

<b>thus </b><a NAME="E15:50"/>
<font color="Maroon" title="c7">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">u2</font>
 <b>by </b><i><a class="txt" href="rmod_3.html#E2:50"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="rmod_3.html#E13:50"><i><font color="Green" title="E45">A8</font></i></a>, <a class="ref" href="rlvect_1.html#T21" title="RLVECT_1:th.21">RLVECT_1:21</a></i>;<br/>


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