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<b>let </b><font color="Maroon">Q</font> be   <a href="quantal1.html#NM2">Quantale</a>;<br/><b>let </b><font color="Maroon">s</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font>;<br/><b>let </b><font color="Maroon">j</font> be   <a href="quantal1.html#NM1">UnOp</a> of <font color="Maroon">Q</font>;<br/>





<b>assume </b><a NAME="E1:42"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> holds  <font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <font color="Maroon">s</font></span>)</span> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <font color="Maroon">s</font>
 ;<br/>

<b>thus </b><a NAME="E2:42"/>
<font color="Maroon">j</font> is <a href="quantal1.html#V14" target="_self">monotone</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="quantal1.html#D12" target="_self">QUANTAL1:def 12</a><br/>

<b>assume </b><a NAME="E1:42_1"/>
<font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:42_1"/><b>then </b>
<font color="Maroon">b</font> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <font color="Maroon">s</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <font color="Maroon">s</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:42_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <font color="Maroon">s</font></span>)</span> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <font color="Maroon">s</font> <a href="lattices.html#R3">[=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <font color="Maroon">s</font></span>)</span> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <font color="Maroon">s</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <font color="Maroon">s</font></span>)</span> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:42_1"/>
<font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="lattices.html#R1">[=</a> <font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>


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