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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#K5" title="VECTSP10:func.5">lmultCoset</a> (<font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">W</font>) <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="vectsp10.html#K5" title="VECTSP10:func.5">lmultCoset</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#K5" title="VECTSP10:func.5">lmultCoset</a> (<font color="Maroon" title="c2">V</font>,<font color="Olive" title="b1">W</font>) <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="vectsp10.html#K5" title="VECTSP10:func.5">lmultCoset</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#K5" title="VECTSP10:func.5">lmultCoset</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font>) <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="vectsp10.html#K5" title="VECTSP10:func.5">lmultCoset</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#K5" title="VECTSP10:func.5">lmultCoset</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font>) <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <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><br/>







<span class="kw">assume </span><a NAME="E1:61"/><span class="lab"><font color="Green" title="E35">AS</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#K5" title="VECTSP10:func.5">lmultCoset</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font>) <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <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><br/>

<span class="kw">set </span><font color="Maroon" title="c5">f1</font> =  <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>);<br/>
<span class="kw">set </span><font color="Maroon" title="c6">f2</font> =  <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>);<br/>
<span class="kw">set </span><font color="Maroon" title="c7">C</font> =  <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>);<br/>
<span class="kw">set </span><font color="Maroon" title="c8">Cs</font> =  <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>);<br/>
<a NAME="E2:61"/><span class="lab"><font color="Green" title="E36">A14</font></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>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="#E1:61"><span class="lab"><font color="Green" title="E35">AS</font></span></a>, <a class="ref" href="zmodul07.html#T34" target="_self" onmouseover="rs('zmodul07/T34')" onmouseout="rh()">LmStrict1</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="61_1"><span class="kw">now </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">  for <font color="Olive" title="b1">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><br/>  for <font color="Olive" title="b2">A</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <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>) holds  <span class="p1">(<span class="default"><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>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><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>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">A</font>)</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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">A</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <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>) holds  <span class="p1">(<span class="default"><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>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c9">z</font>,<font color="Olive" title="b1">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><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>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c9">z</font>,<font color="Olive" title="b1">A</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c10">A</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <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>); <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"><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>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c9">z</font>,<font color="Maroon" title="c10">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><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>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c9">z</font>,<font color="Maroon" title="c10">A</font>)</span><br/><a NAME="E1:61_1"/>
<font color="Maroon" title="c10">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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>)
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">A1</font> being    <a href="vectsp_4.html#M2" title="VECTSP_4:mode.2">Coset</a> of <font color="Maroon" title="c3">W</font><span class="kw"> such that </span><br/><a NAME="E3:61_1"/><span class="lab"><font color="Green" title="E37">A17</font></span>: 
<font color="Maroon" title="c11">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">A</font>
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c12">a</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E5:61_1"/><span class="lab"><font color="Green" title="E38">A18</font></span>: 
<font color="Maroon" title="c11">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#D6" onmouseover="rs('vectsp_4/D6')" onmouseout="rh()">VECTSP_4:def 6</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">As</font> = <font color="Maroon" title="c10">A</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <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="#E1:61"><span class="lab"><font color="Green" title="E35">AS</font></span></a>, <a class="ref" href="zmodul07.html#T34" target="_self" onmouseover="rs('zmodul07/T34')" onmouseout="rh()">LmStrict1</a></span>;<br/><a NAME="E7:61_1"/><span class="lab"><font color="Green" title="E39">A21</font></span>: 
<font color="Maroon" title="c13">As</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c4">Ws</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E35">AS</font></span></a>, <a class="txt" href="#E3:61_1"><span class="lab"><font color="Green" title="E37">A17</font></span></a>, <a class="txt" href="#E5:61_1"><span class="lab"><font color="Green" title="E38">A18</font></span></a>, <a class="ref" href="zmodul07.html#T32" target="_self" onmouseover="rs('zmodul07/T32')" onmouseout="rh()">LmStrict11a</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><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>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c9">z</font>,<font color="Maroon" title="c10">A</font>) = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c3">W</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1"><span class="lab"><font color="Green" title="E37">A17</font></span></a>, <a class="txt" href="#E5:61_1"><span class="lab"><font color="Green" title="E38">A18</font></span></a>, <a class="ref" href="vectsp10.html#D5" onmouseover="rs('vectsp10/D5')" onmouseout="rh()">VECTSP10:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c4">Ws</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul07.html#T32" target="_self" onmouseover="rs('zmodul07/T32')" onmouseout="rh()">LmStrict11a</a>, <a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E35">AS</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><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>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c9">z</font>,<font color="Maroon" title="c10">A</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:61_1"><span class="lab"><font color="Green" title="E39">A21</font></span></a>, <a class="ref" href="vectsp10.html#D5" onmouseover="rs('vectsp10/D5')" onmouseout="rh()">VECTSP10:def 5</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/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:61"/>
 <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>) <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <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="#E2:61"><span class="lab"><font color="Green" title="E36">A14</font></span></a>, <a class="ref" href="binop_1.html#T2" onmouseover="rs('binop_1/T2')" onmouseout="rh()">BINOP_1:2</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/>


</div>
