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<span class="kw">let </span><font color="Maroon" title="c1">K</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#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c1">K</font><br/> for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Olive" title="b1">V</font><br/> for <font color="Olive" title="b4">W</font> being   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Olive" title="b1">V</font> holds <br/> ( <span class="p1">(<span class="default"><font color="Olive" title="b2">a</font> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Olive" title="b4">W</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">b</font> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Olive" title="b4">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b3">b</font></span>)</span> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Olive" title="b4">W</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">V</font> <a href="lmod_7.html#K13" title="LMOD_7:func.13">.</a> <font color="Olive" title="b4">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b1">V</font></span>)</span> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</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#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c1">K</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">a</font>, <font color="Olive" title="b2">b</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font><br/> for <font color="Olive" title="b3">W</font> being   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font> holds <br/> ( <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Olive" title="b3">W</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Olive" title="b3">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Olive" title="b3">W</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">V</font> <a href="lmod_7.html#K13" title="LMOD_7:func.13">.</a> <font color="Olive" title="b3">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Olive" title="b3">W</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">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="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font> holds <br/> ( <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Olive" title="b1">W</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Olive" title="b1">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Olive" title="b1">W</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">V</font> <a href="lmod_7.html#K13" title="LMOD_7:func.13">.</a> <font color="Olive" title="b1">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</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="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</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"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Maroon" title="c5">W</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Maroon" title="c5">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Maroon" title="c5">W</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">V</font> <a href="lmod_7.html#K13" title="LMOD_7:func.13">.</a> <font color="Maroon" title="c5">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Maroon" title="c5">W</font> )</span><br/>









<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Maroon" title="c5">W</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Maroon" title="c5">W</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="lmod_7.html#K8" title="LMOD_7:func.8">..</a> <font color="Maroon" title="c5">W</font></span>)</span> <a href="lmod_7.html#K10" title="LMOD_7:func.10">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="lmod_7.html#K8" title="LMOD_7:func.8">..</a> <font color="Maroon" title="c5">W</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lmod_7.html#D17" target="_self" title="LMOD_7:def.17">Def17</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Maroon" title="c5">W</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lmod_7.html#D15" target="_self" title="LMOD_7:def.15">Def15</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">V</font> <a href="lmod_7.html#K13" title="LMOD_7:func.13">.</a> <font color="Maroon" title="c5">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Maroon" title="c5">W</font></span><br/>

<span class="kw">thus </span><a NAME="E2:55"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">V</font> <a href="lmod_7.html#K13" title="LMOD_7:func.13">.</a> <font color="Maroon" title="c5">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Maroon" title="c5">W</font>
 ; <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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