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<b>let </b><font color="Maroon">Q</font> be  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V2">unital</a>  <a href="quantal1.html#L2" target="_self">QuasiNetStr</a> ;<br/>

<b>assume </b><a NAME="E1:29"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">Q</font> is   <a href="quantal1.html#NM2">Quantale</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">Q'</font> = <font color="Maroon">Q</font> as   <a href="quantal1.html#NM2">Quantale</a> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a> ;<br/>
<a NAME="E3:29"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q'</font>  holds  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">c</font>]
 
;<br/>
<a NAME="E4:29"/><i><font color="Green">E28</font></i>: 
 <a href="lattices.html#K5">Bottom</a> <font color="Maroon">Q'</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K15">"\/"</a> <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">Q'</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T50">LATTICE3:50</a></i>;<br/>
<a NAME="E5:29"/>
<font color="Maroon">Q'</font> is <a href="quantal1.html#V6" target="_self">with_zero</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="quantal1.html#D2" target="_self">QUANTAL1:def 2</a>,<a class="ref" href="quantal1.html#D4" target="_self">QUANTAL1:def 4</a>
<div class="add"><b>reconsider </b><font color="Maroon">a</font> =  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">Q'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q'</font> ;<br/><b>take </b><font color="Maroon">a</font> = <font color="Maroon">a</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q'</font>;<br/><b>deffunc </b><font color="Maroon">H<sub>5</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q'</font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Q'</font> = <font color="Maroon">a<sub>1</sub></font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">b</font>;<br/><a NAME="E2:29_1_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive">c</font>) where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q'</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">c</font>] </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>from </b><i><a class="ref" href="quantal1.html#S2" target="_self">QUANTAL1:sch 2</a>();<br/></i><b>hence </b><a NAME="E3:29_1_1"/>
<font color="Maroon">a</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i>, </i>;<br/></div>
<b>end;</b></div>

<b>take </b><font color="Maroon">b</font> =  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">Q'</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="quantal1.html#D3" target="_self">QUANTAL1:def 3</a><br/>

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q'</font>;<br/>

<b>deffunc </b><font color="Maroon">H<sub>5</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q'</font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Q'</font> = <font color="Maroon">a</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E2:29_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive">c</font>) where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q'</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">c</font>] </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>from </b><i><a class="ref" href="quantal1.html#S2" target="_self">QUANTAL1:sch 2</a>();<br/></i>
<b>hence </b><a NAME="E3:29_1"/>
<font color="Maroon">a</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:29"/><b>then </b>
( <font color="Maroon">Q</font> is  non <a href="struct_0.html#V3">empty</a> <a href="quantal1.html#V6" target="_self">with_zero</a>  <a href="group_1.html#L1">HGrStr</a>  &amp; <font color="Maroon">Q</font> is  non <a href="struct_0.html#V3">empty</a> <a href="lattices.html#V10">Lattice-like</a> <a href="lattice3.html#V4">complete</a> <a href="quantal1.html#V7" target="_self">right-distributive</a> <a href="quantal1.html#V8" target="_self">left-distributive</a>  <a href="quantal1.html#L2" target="_self">QuasiNetStr</a>  )
 
;<br/>
<b>hence </b><a NAME="E7:29"/>
<font color="Maroon">Q</font> is   <a href="quantal1.html#NM4">BlikleNet</a>
 <b>by </b><i/>;<br/>


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