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<b>let </b><font color="Maroon" title="c1">Q</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="lattices.html#V10" title="LATTICES:attr.10">Lattice-like</a> <a href="lattice3.html#V4" title="LATTICE3:attr.4">complete</a>  <a href="quantal1.html#L2" title="QUANTAL1:struct.2">QuasiNetStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">Q</font> is <a href="quantal1.html#V8" title="QUANTAL1:attr.8">left-distributive</a> &amp; <font color="Maroon" title="c1">Q</font> is <a href="quantal1.html#V7" title="QUANTAL1:attr.7">right-distributive</a> implies ( <font color="Maroon" title="c1">Q</font> is <a href="quantal1.html#V10" title="QUANTAL1:attr.10">times-continuous</a> &amp; <font color="Maroon" title="c1">Q</font> is <a href="quantal1.html#V9" title="QUANTAL1:attr.9">times-additive</a> ) )</span><br/>

<b>assume </b><a NAME="E1:25_1_1"/>
( <font color="Maroon" title="c1">Q</font> is <a href="quantal1.html#V8" title="QUANTAL1:attr.8">left-distributive</a> &amp; <font color="Maroon" title="c1">Q</font> is <a href="quantal1.html#V7" title="QUANTAL1:attr.7">right-distributive</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">Q</font> is <a href="quantal1.html#V10" title="QUANTAL1:attr.10">times-continuous</a> &amp; <font color="Maroon" title="c1">Q</font> is <a href="quantal1.html#V9" title="QUANTAL1:attr.9">times-additive</a> )</span><br/>

<b>hence </b><a NAME="E2:25_1_1"/>
( ( for <font color="Olive" title="b1">X</font>, <font color="Olive" title="b2">Y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Q</font>  st <font color="Olive" title="b1">X</font> is <a href="quantal1.html#V1" title="QUANTAL1:attr.1">directed</a> &amp; <font color="Olive" title="b2">Y</font> is <a href="quantal1.html#V1" title="QUANTAL1:attr.1">directed</a> holds <br/><span class="p1">(<span class="default"><a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">Q</font></span>)</span> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Olive" title="b2">Y</font>,<font color="Maroon" title="c1">Q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b3">a</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Olive" title="b4">b</font></span>)</span> where <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font> : ( <font color="Olive" title="b3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> &amp; <font color="Olive" title="b4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">Y</font> ) </span>}</span> ,<font color="Maroon" title="c1">Q</font> ) &amp; ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font> holds <br/> ( <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Olive" title="b3">c</font> <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="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Olive" title="b3">c</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Olive" title="b3">c</font></span>)</span> &amp; <font color="Olive" title="b3">c</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">c</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">c</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Olive" title="b2">b</font></span>)</span> ) ) )
 <b>by </b><i><a class="ref" href="quantal1.html#T5" target="_self" title="QUANTAL1:th.5">Th5</a>, <a class="ref" href="quantal1.html#T6" target="_self" title="QUANTAL1:th.6">Th6</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="quantal1.html#D7" target="_self" title="QUANTAL1:def.7">QUANTAL1:def 7</a>,<a class="ref" href="quantal1.html#D8" target="_self" title="QUANTAL1:def.8">QUANTAL1:def 8</a> <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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