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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="group_1.html#V1" title="GROUP_1:attr.1">unital</a>  <a href="quantal1.html#L2" title="QUANTAL1:struct.2">QuasiNetStr</a> ;<br/>

<b>assume </b><a NAME="E1:29"/><i><font color="Green" title="E9">A1</font></i>: 
<font color="Maroon" title="c1">Q</font> is   <a href="quantal1.html#NM2" title="QUANTAL1:NM.2">Quantale</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c2">Q'</font> = <font color="Maroon" title="c1">Q</font> as   <a href="quantal1.html#NM2" title="QUANTAL1:NM.2">Quantale</a> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ;<br/>
<a NAME="E3:29"/><i><font color="Green" title="E10">A2</font></i>: 
for <font color="Olive" title="b1">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q'</font>  holds  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">c</font>]
 
;<br/>
<a NAME="E4:29"/><i><font color="Green" title="E11">A3</font></i>: 
 <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c2">Q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c2">Q'</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T50" title="LATTICE3:th.50">LATTICE3:50</a></i>;<br/>
<a NAME="E5:29"/>
<font color="Maroon" title="c2">Q'</font> is <a href="quantal1.html#V6" title="QUANTAL1:attr.6">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" title="QUANTAL1:def.2">QUANTAL1:def 2</a>,<a class="ref" href="quantal1.html#D4" target="_self" title="QUANTAL1:def.4">QUANTAL1:def 4</a>
<div class="add"><b>reconsider </b><font color="Maroon" title="c3">a</font> =  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c2">Q'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q'</font> ;<br/><b>take </b><font color="Maroon" title="c4">a</font> = <font color="Maroon" title="c3">a</font>;<br/><b>let </b><font color="Maroon" title="c5">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q'</font>;<br/><b>deffunc </b><font color="Maroon">H<sub>5</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q'</font>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Q'</font> = $1 <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c5">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" title="b1">c</font>) where <font color="Olive" title="b1">c</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q'</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">c</font>] </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>from </b><i><a class="ref" href="quantal1.html#S2" target="_self" title="QUANTAL1:sch.2">QUANTAL1:sch 2</a>(<a class="txt" href="quantal1.html#E3:29"><i><font color="Green" title="E10">A2</font></i></a>);<br/></i><b>hence </b><a NAME="E3:29_1_1"/>
<font color="Maroon" title="c4">a</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="txt" href="quantal1.html#E4:29"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="quantal1.html#D6" target="_self" title="QUANTAL1:def.6">Def6</a></i>;<br/></div>
<b>end;</b></div>

<b>take </b><font color="Maroon" title="c3">b</font> =  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c2">Q'</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="quantal1.html#D3" target="_self" title="QUANTAL1:def.3">QUANTAL1:def 3</a><br/>

<b>let </b><font color="Maroon" title="c4">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q'</font>;<br/>

<b>deffunc </b><font color="Maroon">H<sub>5</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q'</font>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Q'</font> = <font color="Maroon" title="c4">a</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> $1;<br/>
<a NAME="E2:29_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive" title="b1">c</font>) where <font color="Olive" title="b1">c</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q'</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">c</font>] </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>from </b><i><a class="ref" href="quantal1.html#S2" target="_self" title="QUANTAL1:sch.2">QUANTAL1:sch 2</a>(<a class="txt" href="quantal1.html#E3:29"><i><font color="Green" title="E10">A2</font></i></a>);<br/></i>
<b>hence </b><a NAME="E3:29_1"/>
<font color="Maroon" title="c4">a</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="txt" href="quantal1.html#E4:29"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="quantal1.html#D5" target="_self" title="QUANTAL1:def.5">Def5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:29"/><b>then </b>
( <font color="Maroon" title="c1">Q</font> is  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="quantal1.html#V6" title="QUANTAL1:attr.6">with_zero</a>  <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a>  &amp; <font color="Maroon" title="c1">Q</font> is  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#V7" title="QUANTAL1:attr.7">right-distributive</a> <a href="quantal1.html#V8" title="QUANTAL1:attr.8">left-distributive</a>  <a href="quantal1.html#L2" title="QUANTAL1:struct.2">QuasiNetStr</a>  )
 
;<br/>
<b>hence </b><a NAME="E7:29"/>
<font color="Maroon" title="c1">Q</font> is   <a href="quantal1.html#NM4" title="QUANTAL1:NM.4">BlikleNet</a>
 <b>by </b><i><a class="txt" href="quantal1.html#E1:29"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/>


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