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<b>let </b><font color="Maroon" title="c1">Q</font> be   <a href="quantal1.html#NM2" title="QUANTAL1:NM.2">Quantale</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">s</font>, <font color="Olive" title="b2">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font>  st <font color="Olive" title="b1">s</font> is <a href="quantal1.html#V18" title="QUANTAL1:attr.18">cyclic</a> holds <br/>( <font color="Olive" title="b2">a</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">a</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Olive" title="b1">s</font></span>)</span> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Olive" title="b1">s</font></span>)</span> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b2">a</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">a</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Olive" title="b1">s</font></span>)</span> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Olive" title="b1">s</font></span>)</span> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Olive" title="b1">s</font> )</span><br/><b>let </b><font color="Maroon" title="c2">s</font>, <font color="Maroon" title="c3">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</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">s</font> is <a href="quantal1.html#V18" title="QUANTAL1:attr.18">cyclic</a> implies ( <font color="Maroon" title="c3">a</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">s</font> &amp; <font color="Maroon" title="c3">a</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">s</font> ) )</span><br/>



<b>assume </b><a NAME="E1:46"/>
<font color="Maroon" title="c2">s</font> is <a href="quantal1.html#V18" title="QUANTAL1:attr.18">cyclic</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">a</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">s</font> &amp; <font color="Maroon" title="c3">a</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">s</font> )</span><br/>

<a NAME="E2:46"/><b>then </b><i><font color="Green" title="E21">A1</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">s</font> &amp; <font color="Maroon" title="c3">a</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">s</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">s</font> &amp; <font color="Maroon" title="c3">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">s</font> &amp; <font color="Maroon" title="c3">a</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">s</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">s</font> )
 
<b>by </b><i><a class="ref" href="quantal1.html#T16" target="_self" title="QUANTAL1:th.16">Th16</a></i>;<br/>
<a NAME="E3:46"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">s</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">a</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">s</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">s</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">a</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">s</font> )
 
<b>by </b><i><a class="ref" href="quantal1.html#T13" target="_self" title="QUANTAL1:th.13">Th13</a></i>;<br/>
<b>hence </b><a NAME="E4:46"/>
( <font color="Maroon" title="c3">a</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">s</font> &amp; <font color="Maroon" title="c3">a</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">s</font> )
 <b>by </b><i><a class="txt" href="quantal1.html#E2:46"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="lattices.html#T26" title="LATTICES:th.26">LATTICES:26</a></i>; <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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