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<span class="kw">let </span><font color="Maroon" title="c1">R</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">V</font> being   <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">LeftMod</a> of <font color="Maroon" title="c1">R</font><br/>  for <font color="Olive" title="b2">W</font> being    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Olive" title="b1">V</font><br/>  for <font color="Olive" title="b3">Ws</font> being   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Olive" title="b1">V</font>  st <font color="Olive" title="b3">Ws</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Olive" title="b2">W</font> holds <br/> <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Olive" title="b1">V</font>,<font color="Olive" title="b3">Ws</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">V</font> be   <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">LeftMod</a> of <font color="Maroon" title="c1">R</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="c2">V</font><br/>  for <font color="Olive" title="b2">Ws</font> being   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font>  st <font color="Olive" title="b2">Ws</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Olive" title="b1">W</font> holds <br/> <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c2">V</font>,<font color="Olive" title="b1">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c2">V</font>,<font color="Olive" title="b2">Ws</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">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" 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">Ws</font> being   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font>  st <font color="Olive" title="b1">Ws</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c3">W</font> holds <br/> <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c2">V</font>,<font color="Olive" title="b1">Ws</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c4">Ws</font> be   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">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"> ( <font color="Maroon" title="c4">Ws</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c3">W</font> implies  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c4">Ws</font>) )</span><br/>







<span class="kw">assume </span><a NAME="E1:62"/><span class="lab"><font color="Green" title="E36">A1</font></span>: 
<font color="Maroon" title="c4">Ws</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c3">W</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">  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c4">Ws</font>)</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">Z1</font> =  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c6">Z2</font> =  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c4">Ws</font>);<br/>
<span class="lab"><font color="Green" title="E37">A2</font></span>:  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font>)</span>)</span> = 
 <a href="vectsp10.html#K2" title="VECTSP10:func.2">CosetSet</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp10.html#D6" onmouseover="rs('vectsp10/D6')" onmouseout="rh()">VECTSP10:def 6</a></span>
<br/>.= 
 <a href="vectsp10.html#K2" title="VECTSP10:func.2">CosetSet</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c4">Ws</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="zmodul07.html#T34" target="_self" onmouseover="rs('zmodul07/T34')" onmouseout="rh()">LmStrict1</a></span>
<br/>.= 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c4">Ws</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp10.html#D6" onmouseover="rs('vectsp10/D6')" onmouseout="rh()">VECTSP10:def 6</a></span>
;<br/>
<span class="lab"><font color="Green" title="E38">A3</font></span>:  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font>)</span>)</span> = 
 <a href="vectsp10.html#K3" title="VECTSP10:func.3">addCoset</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp10.html#D6" onmouseover="rs('vectsp10/D6')" onmouseout="rh()">VECTSP10:def 6</a></span>
<br/>.= 
 <a href="vectsp10.html#K3" title="VECTSP10:func.3">addCoset</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c4">Ws</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="zmodul07.html#T35" target="_self" onmouseover="rs('zmodul07/T35')" onmouseout="rh()">LmStrict2</a></span>
<br/>.= 
 the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c4">Ws</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp10.html#D6" onmouseover="rs('vectsp10/D6')" onmouseout="rh()">VECTSP10:def 6</a></span>
;<br/>
<span class="lab"><font color="Green" title="E39">A4</font></span>:  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font>)</span>)</span> = 
 <a href="vectsp10.html#K4" title="VECTSP10:func.4">zeroCoset</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp10.html#D6" onmouseover="rs('vectsp10/D6')" onmouseout="rh()">VECTSP10:def 6</a></span>
<br/>.= 
 <a href="vectsp10.html#K4" title="VECTSP10:func.4">zeroCoset</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c4">Ws</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E36">A1</font></span></a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c4">Ws</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp10.html#D6" onmouseover="rs('vectsp10/D6')" onmouseout="rh()">VECTSP10:def 6</a></span>
;<br/>
 the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font>)</span>)</span> = 
 <a href="vectsp10.html#K5" title="VECTSP10:func.5">lmultCoset</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp10.html#D6" onmouseover="rs('vectsp10/D6')" onmouseout="rh()">VECTSP10:def 6</a></span>
<br/>.= 
 <a href="vectsp10.html#K5" title="VECTSP10:func.5">lmultCoset</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c4">Ws</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="zmodul07.html#T36" target="_self" onmouseover="rs('zmodul07/T36')" onmouseout="rh()">LmStrict3</a></span>
<br/>.= 
 the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c4">Ws</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp10.html#D6" onmouseover="rs('vectsp10/D6')" onmouseout="rh()">VECTSP10:def 6</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:62"/>
 <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c4">Ws</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="txt" href="#E3:62"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="txt" href="#E4:62"><span class="lab"><font color="Green" title="E39">A4</font></span></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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