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<b>let </b><font color="Maroon" title="c1">R</font> be   <a href="vectsp_2.html#NM2" title="VECTSP_2:NM.2">domRing</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">V</font> being  <a href="vectsp_2.html#NM7" title="VECTSP_2:NM.7">RightMod</a> of <font color="Maroon" title="c1">R</font><br/> for <font color="Olive" title="b2">A</font>, <font color="Olive" title="b3">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">V</font>  st <font color="Olive" title="b2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">B</font> holds <br/> <a href="rmod_5.html#K1" title="RMOD_5:func.1">Lin</a> <font color="Olive" title="b2">A</font> is    <a href="rmod_2.html#M1" title="RMOD_2:mode.1">Submodule</a> of  <a href="rmod_5.html#K1" title="RMOD_5:func.1">Lin</a> <font color="Olive" title="b3">B</font></span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font>  st <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">B</font> holds <br/> <a href="rmod_5.html#K1" title="RMOD_5:func.1">Lin</a> <font color="Olive" title="b1">A</font> is    <a href="rmod_2.html#M1" title="RMOD_2:mode.1">Submodule</a> of  <a href="rmod_5.html#K1" title="RMOD_5:func.1">Lin</a> <font color="Olive" title="b2">B</font></span><br/><b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B</font> implies  <a href="rmod_5.html#K1" title="RMOD_5:func.1">Lin</a> <font color="Maroon" title="c3">A</font> is    <a href="rmod_2.html#M1" title="RMOD_2:mode.1">Submodule</a> of  <a href="rmod_5.html#K1" title="RMOD_5:func.1">Lin</a> <font color="Maroon" title="c4">B</font> )</span><br/>







<b>assume </b><a NAME="E1:14"/><i><font color="Green" title="E9">A1</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="rmod_5.html#K1" title="RMOD_5:func.1">Lin</a> <font color="Maroon" title="c3">A</font> is    <a href="rmod_2.html#M1" title="RMOD_2:mode.1">Submodule</a> of  <a href="rmod_5.html#K1" title="RMOD_5:func.1">Lin</a> <font color="Maroon" title="c4">B</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">v</font> be   <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="rmod_5.html#K1" title="RMOD_5:func.1">Lin</a> <font color="Maroon" title="c3">A</font> implies <font color="Maroon" title="c5">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="rmod_5.html#K1" title="RMOD_5:func.1">Lin</a> <font color="Maroon" title="c4">B</font> )</span><br/><b>assume </b><a NAME="E1:14_1"/>
<font color="Maroon" title="c5">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="rmod_5.html#K1" title="RMOD_5:func.1">Lin</a> <font color="Maroon" title="c3">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="rmod_5.html#K1" title="RMOD_5:func.1">Lin</a> <font color="Maroon" title="c4">B</font></span><br/><b>then consider </b><font color="Maroon" title="c6">l</font> being    <a href="rmod_4.html#M2" title="RMOD_4:mode.2">Linear_Combination</a> of <font color="Maroon" title="c3">A</font><b> such that </b><br/><a NAME="E3:14_1"/><i><font color="Green" title="E10">A2</font></i>: 
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Maroon" title="c6">l</font>
 <b>by </b><i><a class="ref" href="rmod_5.html#T9" target="_self" title="RMOD_5:th.9">Th9</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">l</font> = <font color="Maroon" title="c6">l</font> as    <a href="rmod_4.html#M2" title="RMOD_4:mode.2">Linear_Combination</a> of <font color="Maroon" title="c4">B</font> <b>by </b><i><a class="txt" href="rmod_5.html#E1:14"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="rmod_4.html#T25" title="RMOD_4:th.25">RMOD_4:25</a></i>;<br/><a NAME="E5:14_1"/>
 <a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Maroon" title="c7">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font>
 <b>by </b><i><a class="txt" href="rmod_5.html#E3:14_1"><i><font color="Green" title="E10">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E6:14_1"/>
<font color="Maroon" title="c5">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="rmod_5.html#K1" title="RMOD_5:func.1">Lin</a> <font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="ref" href="rmod_5.html#T9" target="_self" title="RMOD_5:th.9">Th9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:14"/>
 <a href="rmod_5.html#K1" title="RMOD_5:func.1">Lin</a> <font color="Maroon" title="c3">A</font> is    <a href="rmod_2.html#M1" title="RMOD_2:mode.1">Submodule</a> of  <a href="rmod_5.html#K1" title="RMOD_5:func.1">Lin</a> <font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="ref" href="rmod_2.html#T36" title="RMOD_2:th.36">RMOD_2:36</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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