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<span class="kw">let </span><font color="Maroon" title="c1">R</font> be   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">u</font>, <font color="Olive" title="b3">v1</font>, <font color="Olive" title="b4">v2</font> being  <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">Vector</a> of <font color="Olive" title="b1">V</font><br/> for <font color="Olive" title="b5">W</font> being   <a href="rmod_2.html#M1" title="RMOD_2:mode.1">Submodule</a> of <font color="Olive" title="b1">V</font>  st <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">v1</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Olive" title="b5">W</font> &amp; <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">v2</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Olive" title="b5">W</font> holds <br/><font color="Olive" title="b3">v1</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Olive" title="b5">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">v2</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Olive" title="b5">W</font></span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v1</font>, <font color="Olive" title="b3">v2</font> being  <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font><br/> for <font color="Olive" title="b4">W</font> being   <a href="rmod_2.html#M1" title="RMOD_2:mode.1">Submodule</a> of <font color="Maroon" title="c2">V</font>  st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">v1</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Olive" title="b4">W</font> &amp; <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">v2</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Olive" title="b4">W</font> holds <br/><font color="Olive" title="b2">v1</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Olive" title="b4">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v2</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Olive" title="b4">W</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">u</font>, <font color="Maroon" title="c4">v1</font>, <font color="Maroon" title="c5">v2</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">W</font> being   <a href="rmod_2.html#M1" title="RMOD_2:mode.1">Submodule</a> of <font color="Maroon" title="c2">V</font>  st <font color="Maroon" title="c3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v1</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Olive" title="b1">W</font> &amp; <font color="Maroon" title="c3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">v2</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Olive" title="b1">W</font> holds <br/><font color="Maroon" title="c4">v1</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Olive" title="b1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v2</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Olive" title="b1">W</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">W</font> be    <a href="rmod_2.html#M1" title="RMOD_2:mode.1">Submodule</a> of <font color="Maroon" title="c2">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v1</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c6">W</font> &amp; <font color="Maroon" title="c3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">v2</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c6">W</font> implies <font color="Maroon" title="c4">v1</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c6">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v2</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c6">W</font> )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:72"/><span class="lab"><font color="Green" title="E40">A1</font></span>: 
<font color="Maroon" title="c3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v1</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c6">W</font>
 <span class="kw">and </span><br/><a NAME="E2:72"/><span class="lab"><font color="Green" title="E41">A2</font></span>: 
<font color="Maroon" title="c3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">v2</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c6">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">v1</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c6">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v2</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c6">W</font></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c4">v1</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c6">W</font> = 
<font color="Maroon" title="c3">u</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c6">W</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:72"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="rmod_2.html#T68" target="_self" title="RMOD_2:th.68">Th68</a></span>
<br/>.= 
<font color="Maroon" title="c5">v2</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c6">W</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:72"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="ref" href="rmod_2.html#T68" target="_self" title="RMOD_2:th.68">Th68</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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