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<b>let </b><font color="Maroon" title="c1">K</font>, <font color="Maroon" title="c2">L</font> be   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">V</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="vectsp_1.html#L1" title="VECTSP_1:struct.1">VectSpStr</a> of <font color="Maroon" title="c1">K</font><br/> for <font color="Olive" title="b2">W</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="vectsp_2.html#L1" title="VECTSP_2:struct.1">RightModStr</a> of <font color="Maroon" title="c2">L</font><br/> for <font color="Olive" title="b3">x</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font><br/> for <font color="Olive" title="b4">y</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c2">L</font><br/> for <font color="Olive" title="b5">v</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="b6">w</font> being  <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">Vector</a> of <font color="Olive" title="b2">W</font>  st <font color="Maroon" title="c2">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_4.html#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c1">K</font> &amp; <font color="Olive" title="b2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_4.html#K3" title="MOD_4:func.3">opp</a> <font color="Olive" title="b1">V</font> &amp; <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">y</font> &amp; <font color="Olive" title="b5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">w</font> holds <br/><font color="Olive" title="b6">w</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <font color="Olive" title="b4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b5">v</font></span><br/><b>let </b><font color="Maroon" title="c3">V</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="vectsp_1.html#L1" title="VECTSP_1:struct.1">VectSpStr</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">W</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="vectsp_2.html#L1" title="VECTSP_2:struct.1">RightModStr</a> of <font color="Maroon" title="c2">L</font><br/> for <font color="Olive" title="b2">x</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font><br/> for <font color="Olive" title="b3">y</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c2">L</font><br/> for <font color="Olive" title="b4">v</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">V</font><br/> for <font color="Olive" title="b5">w</font> being  <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">Vector</a> of <font color="Olive" title="b1">W</font>  st <font color="Maroon" title="c2">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_4.html#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c1">K</font> &amp; <font color="Olive" title="b1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_4.html#K3" title="MOD_4:func.3">opp</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y</font> &amp; <font color="Olive" title="b4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">w</font> holds <br/><font color="Olive" title="b5">w</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <font color="Olive" title="b3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b4">v</font></span><br/><b>let </b><font color="Maroon" title="c4">W</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="vectsp_2.html#L1" title="VECTSP_2:struct.1">RightModStr</a> of <font color="Maroon" title="c2">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font><br/> for <font color="Olive" title="b2">y</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c2">L</font><br/> for <font color="Olive" title="b3">v</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">V</font><br/> for <font color="Olive" title="b4">w</font> being  <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">Vector</a> of <font color="Maroon" title="c4">W</font>  st <font color="Maroon" title="c2">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_4.html#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c1">K</font> &amp; <font color="Maroon" title="c4">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_4.html#K3" title="MOD_4:func.3">opp</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">w</font> holds <br/><font color="Olive" title="b4">w</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b3">v</font></span><br/><b>let </b><font color="Maroon" title="c5">x</font> be   <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">y</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c2">L</font><br/> for <font color="Olive" title="b2">v</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">V</font><br/> for <font color="Olive" title="b3">w</font> being  <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">Vector</a> of <font color="Maroon" title="c4">W</font>  st <font color="Maroon" title="c2">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_4.html#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c1">K</font> &amp; <font color="Maroon" title="c4">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_4.html#K3" title="MOD_4:func.3">opp</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> &amp; <font color="Olive" title="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">w</font> holds <br/><font color="Olive" title="b3">w</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b2">v</font></span><br/><b>let </b><font color="Maroon" title="c6">y</font> be   <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c2">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">v</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">V</font><br/> for <font color="Olive" title="b2">w</font> being  <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">Vector</a> of <font color="Maroon" title="c4">W</font>  st <font color="Maroon" title="c2">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_4.html#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c1">K</font> &amp; <font color="Maroon" title="c4">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_4.html#K3" title="MOD_4:func.3">opp</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> &amp; <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">w</font> holds <br/><font color="Olive" title="b2">w</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b1">v</font></span><br/><b>let </b><font color="Maroon" title="c7">v</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">w</font> being  <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">Vector</a> of <font color="Maroon" title="c4">W</font>  st <font color="Maroon" title="c2">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_4.html#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c1">K</font> &amp; <font color="Maroon" title="c4">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_4.html#K3" title="MOD_4:func.3">opp</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> &amp; <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">w</font> holds <br/><font color="Olive" title="b1">w</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c7">v</font></span><br/><b>let </b><font color="Maroon" title="c8">w</font> be   <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">Vector</a> of <font color="Maroon" title="c4">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_4.html#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c1">K</font> &amp; <font color="Maroon" title="c4">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_4.html#K3" title="MOD_4:func.3">opp</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> &amp; <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">w</font> implies <font color="Maroon" title="c8">w</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c7">v</font> )</span><br/>













<b>assume </b><a NAME="E1:35"/><i><font color="Green" title="E15">A1</font></i>: 
( <font color="Maroon" title="c2">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_4.html#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c1">K</font> &amp; <font color="Maroon" title="c4">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_4.html#K3" title="MOD_4:func.3">opp</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> &amp; <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">w</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">w</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c7">v</font></span><br/>

<b>set </b><font color="Maroon" title="c9">o</font> = the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c3">V</font>;<br/>
<a NAME="E2:35"/>
 <a href="mod_4.html#K4" title="MOD_4:func.4">opp</a> the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c3">V</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> the <a href="vectsp_2.html#U1" title="VECTSP_2:sel.1">rmult</a> of <span class="p1">(<span class="default"><a href="mod_4.html#K3" title="MOD_4:func.3">opp</a> <font color="Maroon" title="c3">V</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mod_4.html#T10" target="_self" title="MOD_4:th.10">Th10</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c8">w</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <font color="Maroon" title="c6">y</font> = 
<span class="p1">(<span class="default"><a href="mod_4.html#K4" title="MOD_4:func.4">opp</a> the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c3">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">w</font>,<font color="Maroon" title="c6">y</font>
<b>by </b><i><a class="txt" href="#E1:35"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="vectsp_2.html#D15" title="VECTSP_2:def.15">VECTSP_2:def 15</a></i>
<br/>.= 
<font color="Maroon" title="c5">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c7">v</font>
<b>by </b><i><a class="txt" href="#E1:35"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="mod_4.html#T1" target="_self" title="MOD_4:th.1">Th1</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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