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<span class="kw">let </span><font color="Maroon" title="c1">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Ring</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">r</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Scalar</a> of <font color="Maroon" title="c1">K</font><br/>  for <font color="Olive" title="b2">V</font> being   <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">LeftMod</a> of <font color="Maroon" title="c1">K</font><br/>  for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Olive" title="b2">V</font><br/>  for <font color="Olive" title="b5">W</font> being    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Olive" title="b2">V</font> holds <br/> ( <span class="p1">(<span class="default"><font color="Olive" title="b3">a</font> <a href="lmod_7.html#K18" title="LMOD_7:func.18">/</a> <font color="Olive" title="b5">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="b4">b</font> <a href="lmod_7.html#K18" title="LMOD_7:func.18">/</a> <font color="Olive" title="b5">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b4">b</font></span>)</span> <a href="lmod_7.html#K18" title="LMOD_7:func.18">/</a> <font color="Olive" title="b5">W</font> &amp; <font color="Olive" title="b1">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">a</font> <a href="lmod_7.html#K18" title="LMOD_7:func.18">/</a> <font color="Olive" title="b5">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">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive" title="b3">a</font></span>)</span> <a href="lmod_7.html#K18" title="LMOD_7:func.18">/</a> <font color="Olive" title="b5">W</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">r</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Scalar</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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#NM5" title="VECTSP_2:NM.5">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#NM8" title="VECTSP_1:NM.8">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#K18" title="LMOD_7:func.18">/</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#K18" title="LMOD_7:func.18">/</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#K18" title="LMOD_7:func.18">/</a> <font color="Olive" title="b4">W</font> &amp; <font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">a</font> <a href="lmod_7.html#K18" title="LMOD_7:func.18">/</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="Maroon" title="c2">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive" title="b2">a</font></span>)</span> <a href="lmod_7.html#K18" title="LMOD_7:func.18">/</a> <font color="Olive" title="b4">W</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">V</font> be   <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">LeftMod</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c3">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="c3">V</font> holds <br/> ( <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lmod_7.html#K18" title="LMOD_7:func.18">/</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#K18" title="LMOD_7:func.18">/</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#K18" title="LMOD_7:func.18">/</a> <font color="Olive" title="b3">W</font> &amp; <font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lmod_7.html#K18" title="LMOD_7:func.18">/</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="Maroon" title="c2">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="lmod_7.html#K18" title="LMOD_7:func.18">/</a> <font color="Olive" title="b3">W</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> be   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c3">V</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c3">V</font> holds <br/> ( <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lmod_7.html#K18" title="LMOD_7:func.18">/</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="c5">b</font> <a href="lmod_7.html#K18" title="LMOD_7:func.18">/</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="c4">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="lmod_7.html#K18" title="LMOD_7:func.18">/</a> <font color="Olive" title="b1">W</font> &amp; <font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lmod_7.html#K18" title="LMOD_7:func.18">/</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="c2">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="lmod_7.html#K18" title="LMOD_7:func.18">/</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="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c3">V</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lmod_7.html#K18" title="LMOD_7:func.18">/</a> <font color="Maroon" title="c6">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="c5">b</font> <a href="lmod_7.html#K18" title="LMOD_7:func.18">/</a> <font color="Maroon" title="c6">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="lmod_7.html#K18" title="LMOD_7:func.18">/</a> <font color="Maroon" title="c6">W</font> &amp; <font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lmod_7.html#K18" title="LMOD_7:func.18">/</a> <font color="Maroon" title="c6">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="lmod_7.html#K18" title="LMOD_7:func.18">/</a> <font color="Maroon" title="c6">W</font> )</span><br/>











<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lmod_7.html#K18" title="LMOD_7:func.18">/</a> <font color="Maroon" title="c6">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="c5">b</font> <a href="lmod_7.html#K18" title="LMOD_7:func.18">/</a> <font color="Maroon" title="c6">W</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Maroon" title="c6">W</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Maroon" title="c6">W</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="lmod_7.html#K18" title="LMOD_7:func.18">/</a> <font color="Maroon" title="c6">W</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lmod_7.html#T17" target="_self" onmouseover="rs('lmod_7/T17')" onmouseout="rh()">Th17</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lmod_7.html#K18" title="LMOD_7:func.18">/</a> <font color="Maroon" title="c6">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="lmod_7.html#K18" title="LMOD_7:func.18">/</a> <font color="Maroon" title="c6">W</font></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lmod_7.html#K18" title="LMOD_7:func.18">/</a> <font color="Maroon" title="c6">W</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="lmod_7.html#K16" title="LMOD_7:func.16">LMULT</a> <font color="Maroon" title="c6">W</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c2">r</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Maroon" title="c6">W</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D12" onmouseover="rs('vectsp_1/D12')" onmouseout="rh()">VECTSP_1:def 12</a></span>
<br/>.= 
<font color="Maroon" title="c2">r</font> <a href="lmod_7.html#K15" title="LMOD_7:func.15">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Maroon" title="c6">W</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lmod_7.html#D21" target="_self" onmouseover="rs('lmod_7/D21')" onmouseout="rh()">Def21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="lmod_7.html#K18" title="LMOD_7:func.18">/</a> <font color="Maroon" title="c6">W</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lmod_7.html#D20" target="_self" onmouseover="rs('lmod_7/D20')" onmouseout="rh()">Def20</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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