<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">Q</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="quantal1.html#L1" target="_self">QuantaleStr</a> ;<br/>

<b>set </b><font color="Maroon">B</font> =  <a href="lattice3.html#K1">BooleLatt</a> <a href="xboole_0.html#K1">{}</a> ;<br/>
<b>assume </b><a NAME="E1:17"/><i><font color="Green">E26</font></i>: 
 <a href="lattices.html#G3">LattStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Q</font>,the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">Q</font>,the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">Q</font> #) <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K1">BooleLatt</a> <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<div><i><font color="Green">E27</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font>;<br/><a NAME="E1:17_1"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="lattice3.html#K1">BooleLatt</a> <a href="xboole_0.html#K1">{}</a> ) <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="lattice3.html#D1">LATTICE3:def 1</a>, <a class="ref" href="zfmisc_1.html#T1">ZFMISC_1:1</a></i>;<br/><a NAME="E2:17_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:17_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E3:17"/>
<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">Q</font>) is <a href="binop_1.html#V2">associative</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D12">MONOID_0:def 12</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><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>, <font color="Maroon">c</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="binop_1.html#D14">BINOP_1:def 14</a><br/>

<b>thus </b><a NAME="E1:17_2"/>
<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">Q</font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">Q</font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">Q</font>) <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">Q</font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>,<font color="Maroon">c</font>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E4:17"/>
<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">Q</font>) is <a href="binop_1.html#V1">commutative</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D11">MONOID_0:def 11</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_3"><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="binop_1.html#D13">BINOP_1:def 13</a><br/>

<b>thus </b><a NAME="E1:17_3"/>
<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">Q</font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">Q</font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>
 <b>by </b><i/>;<br/>


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

<b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font>;<br/>
<b>thus </b><a NAME="E6:17"/>
<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">Q</font>) <a href="setwiseo.html#NR1" target="_self">has_a_unity</a> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D10">MONOID_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_4"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">a</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="setwiseo.html#D2">SETWISEO:def 2</a><br/>

<b>thus </b><a NAME="E1:17_4"/>
<font color="Maroon">a</font> <a href="binop_1.html#R1">is_a_left_unity_wrt</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">Q</font>)
  <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D7">BINOP_1:def 7</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_4_1"><b>proof </b></a><div class="add">

<b>let </b><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="binop_1.html#D16">BINOP_1:def 16</a><br/>

<b>thus </b><a NAME="E1:17_4_1"/>
<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">Q</font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/>


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

<b>let </b><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="binop_1.html#D17">BINOP_1:def 17</a><br/>

<b>thus </b><a NAME="E2:17_4"/>
<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">Q</font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E7:17"/>
<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="17_5"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:17_5"/>
<font color="Maroon">Q</font> is <a href="quantal1.html#V4" target="_self">with_left-zero</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="quantal1.html#D4" target="_self">QUANTAL1:def 4</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_5_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">a</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="quantal1.html#D2" target="_self">QUANTAL1:def 2</a><br/>

<b>thus </b><a NAME="E1:17_5_1"/>
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> holds  <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/>


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

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

<b>thus </b><a NAME="E2:17_5"/>
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> holds  <font color="Olive">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/>


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

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>consider </b><font color="Maroon">p</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> <a href="xboole_0.html#K1">{}</a> </span>)</span><b> such that </b><br/><a NAME="E2:17_6"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">X</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">p</font> &amp; ( for <font color="Olive">r</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> <a href="xboole_0.html#K1">{}</a> </span>)</span>  st <font color="Maroon">X</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Olive">r</font> holds <br/><font color="Maroon">p</font> <a href="lattices.html#R3">[=</a> <font color="Olive">r</font> ) )
 <b>by </b><i><a class="ref" href="lattice3.html#D18">LATTICE3:def 18</a></i>;<br/><b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> <b>by </b><i/>;<br/><b>take </b><font color="Maroon">p'</font> = <font color="Maroon">p'</font>;<br/><b>thus </b><a NAME="E4:17_6"/>
<font color="Maroon">X</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">p'</font>
 <b>by </b><i>, , </i>;<br/><b>let </b><font color="Maroon">r'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font>;<br/><b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r'</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <b>by </b><i/>;<br/><b>assume </b><a NAME="E6:17_6"/>
<font color="Maroon">X</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">r'</font>
 ;<br/><a NAME="E7:17_6"/><b>then </b>
<font color="Maroon">X</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">r</font>
 <b>by </b><i>, </i>;<br/><a NAME="E8:17_6"/><b>then </b>
<font color="Maroon">p</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E9:17_6"/>
<font color="Maroon">p'</font> <a href="lattices.html#R1">[=</a> <font color="Maroon">r'</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:17"/>
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   ex <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> st <br/>( <font color="Olive">X</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Olive">p</font> &amp; ( for <font color="Olive">r</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font>  st <font color="Olive">X</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Olive">r</font> holds <br/><font color="Olive">p</font> <a href="lattices.html#R1">[=</a> <font color="Olive">r</font> ) )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D18">LATTICE3:def 18</a><br/>

<b>thus </b><a NAME="E10:17"/>
<font color="Maroon">Q</font> is <a href="quantal1.html#V7" target="_self">right-distributive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_7"><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>; <i><font color="Red">:: according to </font></i><a class="ref" href="quantal1.html#D5" target="_self">QUANTAL1:def 5</a><br/><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>thus </b><a NAME="E1:17_7"/>
<font color="Maroon">a</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">Q</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Olive">b</font></span>)</span> where <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span> ,<font color="Maroon">Q</font>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E11:17"/>
<font color="Maroon">Q</font> is <a href="quantal1.html#V8" target="_self">left-distributive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_8"><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>; <i><font color="Red">:: according to </font></i><a class="ref" href="quantal1.html#D6" target="_self">QUANTAL1:def 6</a><br/><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>thus </b><a NAME="E1:17_8"/>
<span class="p1">(<span class="default"><a href="lattice3.html#K15">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">Q</font></span>)</span> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">a</font></span>)</span> where <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span> ,<font color="Maroon">Q</font>
 <b>by </b><i/>;<br/>


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

<a NAME="E12:17"/>
( ( for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> holds  <font color="Olive">p</font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">p</font> ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font>, <font color="Olive">r</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> holds  <font color="Olive">p</font> <a href="lattices.html#K1">"\/"</a> <span class="p1">(<span class="default"><font color="Olive">q</font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">r</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">q</font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Olive">r</font> ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> holds  <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">q</font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> holds  <font color="Olive">p</font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">p</font> ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font>, <font color="Olive">r</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> holds  <font color="Olive">p</font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Olive">q</font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">r</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">q</font></span>)</span> <a href="lattices.html#K2">"/\"</a> <font color="Olive">r</font> ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> holds  <font color="Olive">p</font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">q</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> ) )
 
<b>by </b><i/>;<br/>
<a NAME="E13:17"/><b>then </b>
( <font color="Maroon">Q</font> is <a href="lattices.html#V4">join-commutative</a> &amp; <font color="Maroon">Q</font> is <a href="lattices.html#V5">join-associative</a> &amp; <font color="Maroon">Q</font> is <a href="lattices.html#V8">meet-absorbing</a> &amp; <font color="Maroon">Q</font> is <a href="lattices.html#V6">meet-commutative</a> &amp; <font color="Maroon">Q</font> is <a href="lattices.html#V7">meet-associative</a> &amp; <font color="Maroon">Q</font> is <a href="lattices.html#V9">join-absorbing</a> )
 
<b>by </b><i><a class="ref" href="lattices.html#D4">LATTICES:def 4</a>, <a class="ref" href="lattices.html#D5">LATTICES:def 5</a>, <a class="ref" href="lattices.html#D6">LATTICES:def 6</a>, <a class="ref" href="lattices.html#D7">LATTICES:def 7</a>, <a class="ref" href="lattices.html#D8">LATTICES:def 8</a>, <a class="ref" href="lattices.html#D9">LATTICES:def 9</a></i>;<br/>
<b>hence </b><a NAME="E14:17"/>
<font color="Maroon">Q</font> is <a href="lattices.html#V10">Lattice-like</a>
 <b>by </b><i><a class="ref" href="lattices.html#D10">LATTICES:def 10</a></i>;<br/>


</div>
