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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="firebrick">::  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">v</font>, <font color="Olive" title="b3">u</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="b4">W</font> being    <a href="rmod_2.html#M1" title="RMOD_2:mode.1">Submodule</a> of <font color="Olive" title="b1">V</font> holds <br/> ( <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">u</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Olive" title="b4">W</font> iff <font color="Olive" title="b3">u</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="b2">v</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="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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">u</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="b3">W</font> being    <a href="rmod_2.html#M1" title="RMOD_2:mode.1">Submodule</a> of <font color="Maroon" title="c2">V</font> holds <br/> ( <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">u</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Olive" title="b3">W</font> iff <font color="Olive" title="b2">u</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Olive" title="b3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Olive" title="b3">W</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c4">u</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="firebrick">::  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> holds <br/> ( <font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">u</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Olive" title="b1">W</font> iff <font color="Maroon" title="c4">u</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="c3">v</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="c5">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">u</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c5">W</font> iff <font color="Maroon" title="c4">u</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c5">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c5">W</font> )</span><br/>









<span class="kw">thus </span><a NAME="E1:56"/>
( <font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">u</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c5">W</font> implies <font color="Maroon" title="c4">u</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c5">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c5">W</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">u</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c5">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c5">W</font> implies <font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">u</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c5">W</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:56_1"/>
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">u</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c5">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">u</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c5">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c5">W</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">z</font> being   <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E3:56_1"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">u</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">z</font>
 <span class="kw">and </span><br/><a NAME="E4:56_1"/><span class="lab"><font color="Green" title="E40">A2</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">W</font>
 ;<br/>
<span class="kw">thus </span><a NAME="E5:56_1"/>
<font color="Maroon" title="c4">u</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c5">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">v</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c5">W</font>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">v</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c5">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">u</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c5">W</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">u</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c5">W</font> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">v</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c5">W</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">u</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c5">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">v</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c5">W</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">v1</font> being   <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E3:56_1_1"/><span class="lab"><font color="Green" title="E41">A3</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">u</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">v1</font>
 <span class="kw">and </span><br/><a NAME="E4:56_1_1"/><span class="lab"><font color="Green" title="E42">A4</font></span>: 
<font color="Maroon" title="c8">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">W</font>
 ;<br/>
<font color="Maroon" title="c3">v</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c6">z</font> = 
<font color="Maroon" title="c4">u</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c6">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_1"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="rlvect_1.html#D3" title="RLVECT_1:def.3">RLVECT_1:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c4">u</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T19" title="VECTSP_1:th.19">VECTSP_1:19</a></span>
<br/>.= 
<font color="Maroon" title="c4">u</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D4" title="RLVECT_1:def.4">RLVECT_1:def 4</a></span>
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E43">A5</font></span>: <font color="Maroon" title="c7">x</font> = 
<font color="Maroon" title="c3">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c6">z</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_1_1"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="ref" href="rlvect_1.html#D3" title="RLVECT_1:def.3">RLVECT_1:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c3">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">v1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c6">z</font></span>)</span>

;<br/>
<a NAME="E7:56_1_1"/>
<font color="Maroon" title="c8">v1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c6">z</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_1"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E4:56_1_1"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="ref" href="rmod_2.html#T23" target="_self" title="RMOD_2:th.23">Th31</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:56_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">v</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c5">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:56_1_1"><span class="lab"><font color="Green" title="E43">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">v</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c5">W</font> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">u</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c5">W</font> )</span><br/>

<span class="kw">assume </span><a NAME="E6:56_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">v</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c5">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">u</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c5">W</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">v2</font> being   <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E8:56_1"/><span class="lab"><font color="Green" title="E41">A6</font></span>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">v2</font> &amp; <font color="Maroon" title="c8">v2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">W</font> )
 ;<br/>
<a NAME="E9:56_1"/>
( <font color="Maroon" title="c6">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">v2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">W</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">u</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">v2</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_1"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E4:56_1"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E8:56_1"><span class="lab"><font color="Green" title="E41">A6</font></span></a>, <a class="ref" href="rmod_2.html#T20" target="_self" title="RMOD_2:th.20">Th28</a>, <a class="ref" href="rlvect_1.html#D3" title="RLVECT_1:def.3">RLVECT_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:56_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">u</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c5">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E2:56"/>
( <font color="Maroon" title="c4">u</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c5">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c5">W</font> implies <font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">u</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c5">W</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rmod_2.html#T44" target="_self" title="RMOD_2:th.44">Th59</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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