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

<span class="kw">set </span><font color="Maroon" title="c1">B</font> =  <a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>;<br/>
<span class="kw">let </span><font color="Maroon" title="c2">Q</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="quantal1.html#L1" title="QUANTAL1:struct.1">QuantaleStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Q</font>, the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c2">Q</font>, the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c2">Q</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> implies ( <font color="Maroon" title="c2">Q</font> is  <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>  &amp; <font color="Maroon" title="c2">Q</font> is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>  &amp; <font color="Maroon" title="c2">Q</font> is  <a href="group_1.html#V1" title="GROUP_1:attr.1">unital</a>  &amp; <font color="Maroon" title="c2">Q</font> is  <a href="quantal1.html#V6" title="QUANTAL1:attr.6">with_zero</a>  &amp; <font color="Maroon" title="c2">Q</font> is  <a href="lattice3.html#V4" title="LATTICE3:attr.4">complete</a>  &amp; <font color="Maroon" title="c2">Q</font> is  <a href="quantal1.html#V7" title="QUANTAL1:attr.7">right-distributive</a>  &amp; <font color="Maroon" title="c2">Q</font> is  <a href="quantal1.html#V8" title="QUANTAL1:attr.8">left-distributive</a>  &amp; <font color="Maroon" title="c2">Q</font> is  <a href="lattices.html#V10" title="LATTICES:attr.10">Lattice-like</a>  ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">a</font> =  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q</font>;<br/>
<span class="kw">assume </span><a NAME="E1:17"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
 <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Q</font>, the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c2">Q</font>, the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c2">Q</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">Q</font> is  <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>  &amp; <font color="Maroon" title="c2">Q</font> is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>  &amp; <font color="Maroon" title="c2">Q</font> is  <a href="group_1.html#V1" title="GROUP_1:attr.1">unital</a>  &amp; <font color="Maroon" title="c2">Q</font> is  <a href="quantal1.html#V6" title="QUANTAL1:attr.6">with_zero</a>  &amp; <font color="Maroon" title="c2">Q</font> is  <a href="lattice3.html#V4" title="LATTICE3:attr.4">complete</a>  &amp; <font color="Maroon" title="c2">Q</font> is  <a href="quantal1.html#V7" title="QUANTAL1:attr.7">right-distributive</a>  &amp; <font color="Maroon" title="c2">Q</font> is  <a href="quantal1.html#V8" title="QUANTAL1:attr.8">left-distributive</a>  &amp; <font color="Maroon" title="c2">Q</font> is  <a href="lattices.html#V10" title="LATTICES:attr.10">Lattice-like</a>  )</span><br/>

<div><span class="lab"><font color="Green" title="E8">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font></span><br/><a NAME="E1:17_1"/><span class="lab"><font color="Green" title="E9">A3</font></span>: 
<font color="Maroon">H<sub>1</sub></font>( <a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#D1" title="LATTICE3:def.1">LATTICE3:def 1</a>, <a class="ref" href="zfmisc_1.html#T1" title="ZFMISC_1:th.1">ZFMISC_1:1</a></span>;<br/><a NAME="E2:17_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:17_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E1:17_1"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E3:17"/>
<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c2">Q</font>) is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> 
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="monoid_0.html#D12" title="MONOID_0:def.12">MONOID_0:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">Q</font> is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>  &amp; <font color="Maroon" title="c2">Q</font> is  <a href="group_1.html#V1" title="GROUP_1:attr.1">unital</a>  &amp; <font color="Maroon" title="c2">Q</font> is  <a href="quantal1.html#V6" title="QUANTAL1:attr.6">with_zero</a>  &amp; <font color="Maroon" title="c2">Q</font> is  <a href="lattice3.html#V4" title="LATTICE3:attr.4">complete</a>  &amp; <font color="Maroon" title="c2">Q</font> is  <a href="quantal1.html#V7" title="QUANTAL1:attr.7">right-distributive</a>  &amp; <font color="Maroon" title="c2">Q</font> is  <a href="quantal1.html#V8" title="QUANTAL1:attr.8">left-distributive</a>  &amp; <font color="Maroon" title="c2">Q</font> is  <a href="lattices.html#V10" title="LATTICES:attr.10">Lattice-like</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="binop_1.html#D14" title="BINOP_1:def.14">BINOP_1:def 14</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c2">Q</font>) <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c4">a</font>,<span class="p1">(<span class="default"><font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c2">Q</font>) <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c2">Q</font>) <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<span class="p1">(<span class="default"><font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c2">Q</font>) <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>)</span>)</span>,<font color="Maroon" title="c6">c</font>)</span><br/>

<span class="kw">thus </span><a NAME="E1:17_2"/>
<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c2">Q</font>) <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c4">a</font>,<span class="p1">(<span class="default"><font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c2">Q</font>) <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c2">Q</font>) <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<span class="p1">(<span class="default"><font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c2">Q</font>) <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>)</span>)</span>,<font color="Maroon" title="c6">c</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E8">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<a NAME="E4:17"/><span class="lab"><font color="Green" title="E9">A4</font></span>: 
( (  for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q</font> holds  <font color="Olive" title="b1">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">q</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b3">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b3">r</font> ) &amp; (  for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q</font> holds  <font color="Olive" title="b1">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E8">A2</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E5:17"/>
<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c2">Q</font>) is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> 
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="monoid_0.html#D11" title="MONOID_0:def.11">MONOID_0:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">Q</font> is  <a href="group_1.html#V1" title="GROUP_1:attr.1">unital</a>  &amp; <font color="Maroon" title="c2">Q</font> is  <a href="quantal1.html#V6" title="QUANTAL1:attr.6">with_zero</a>  &amp; <font color="Maroon" title="c2">Q</font> is  <a href="lattice3.html#V4" title="LATTICE3:attr.4">complete</a>  &amp; <font color="Maroon" title="c2">Q</font> is  <a href="quantal1.html#V7" title="QUANTAL1:attr.7">right-distributive</a>  &amp; <font color="Maroon" title="c2">Q</font> is  <a href="quantal1.html#V8" title="QUANTAL1:attr.8">left-distributive</a>  &amp; <font color="Maroon" title="c2">Q</font> is  <a href="lattices.html#V10" title="LATTICES:attr.10">Lattice-like</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">a</font>, <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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="binop_1.html#D13" title="BINOP_1:def.13">BINOP_1:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c2">Q</font>) <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c2">Q</font>) <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">a</font>)</span><br/>

<span class="kw">thus </span><a NAME="E1:17_3"/>
<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c2">Q</font>) <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c2">Q</font>) <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">a</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E8">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E6:17"/>
<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c2">Q</font>) is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a> 
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="monoid_0.html#D10" title="MONOID_0:def.10">MONOID_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">Q</font> is  <a href="quantal1.html#V6" title="QUANTAL1:attr.6">with_zero</a>  &amp; <font color="Maroon" title="c2">Q</font> is  <a href="lattice3.html#V4" title="LATTICE3:attr.4">complete</a>  &amp; <font color="Maroon" title="c2">Q</font> is  <a href="quantal1.html#V7" title="QUANTAL1:attr.7">right-distributive</a>  &amp; <font color="Maroon" title="c2">Q</font> is  <a href="quantal1.html#V8" title="QUANTAL1:attr.8">left-distributive</a>  &amp; <font color="Maroon" title="c2">Q</font> is  <a href="lattices.html#V10" title="LATTICES:attr.10">Lattice-like</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span>
 the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="setwiseo.html#D2" title="SETWISEO:def.2">SETWISEO:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c2">Q</font>)</span><br/>

<span class="kw">thus </span><a NAME="E1:17_4"/>
 the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q</font> <a href="binop_1.html#R1" title="BINOP_1:pred.1">is_a_left_unity_wrt</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c2">Q</font>)
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="binop_1.html#D7" title="BINOP_1:def.7">BINOP_1:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q</font> <a href="binop_1.html#R2" title="BINOP_1:pred.2">is_a_right_unity_wrt</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c2">Q</font>)</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="binop_1.html#D16" title="BINOP_1:def.16">BINOP_1:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c2">Q</font>) <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> ( the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font></span><br/>

<span class="kw">thus </span><a NAME="E1:17_4_1"/>
<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c2">Q</font>) <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> ( the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E8">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c4">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="binop_1.html#D17" title="BINOP_1:def.17">BINOP_1:def 17</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c2">Q</font>) <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c4">b</font>, the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font></span><br/>

<span class="kw">thus </span><a NAME="E2:17_4"/>
<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c2">Q</font>) <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c4">b</font>, the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E8">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E7:17"/>
<font color="Maroon" title="c2">Q</font> is  <a href="quantal1.html#V6" title="QUANTAL1:attr.6">with_zero</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">Q</font> is  <a href="lattice3.html#V4" title="LATTICE3:attr.4">complete</a>  &amp; <font color="Maroon" title="c2">Q</font> is  <a href="quantal1.html#V7" title="QUANTAL1:attr.7">right-distributive</a>  &amp; <font color="Maroon" title="c2">Q</font> is  <a href="quantal1.html#V8" title="QUANTAL1:attr.8">left-distributive</a>  &amp; <font color="Maroon" title="c2">Q</font> is  <a href="lattices.html#V10" title="LATTICES:attr.10">Lattice-like</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:17_5"/>
<font color="Maroon" title="c2">Q</font> is  <a href="quantal1.html#V4" title="QUANTAL1:attr.4">with_left-zero</a> 
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="quantal1.html#D4" target="_self" title="QUANTAL1:def.4">QUANTAL1:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">Q</font> is  <a href="quantal1.html#V5" title="QUANTAL1:attr.5">with_right-zero</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span>
 the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="quantal1.html#D2" target="_self" title="QUANTAL1:def.2">QUANTAL1:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q</font> holds   the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q</font></span><br/>

<span class="kw">thus </span><a NAME="E1:17_5_1"/>
 for <font color="Olive" title="b1">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q</font> holds   the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E8">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">take </span>
 the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="quantal1.html#D3" target="_self" title="QUANTAL1:def.3">QUANTAL1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q</font> holds  <font color="Olive" title="b1">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a>  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q</font></span><br/>

<span class="kw">thus </span><a NAME="E2:17_5"/>
 for <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q</font> holds  <font color="Olive" title="b1">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a>  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E8">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_6"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q</font> st <br/>( <font color="Maroon" title="c4">X</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Olive" title="b1">p9</font> &amp; (  for <font color="Olive" title="b2">r9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q</font>  st <font color="Maroon" title="c4">X</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Olive" title="b2">r9</font> holds <br/><font color="Olive" title="b1">p9</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Olive" title="b2">r9</font> ) )</span><br/><span class="kw">consider </span><font color="Maroon" title="c5">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span>)</span><span class="kw"> such that </span><br/><a NAME="E2:17_6"/><span class="lab"><font color="Green" title="E10">A5</font></span>: 
<font color="Maroon" title="c4">X</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c5">p</font>
 <span class="kw">and </span><br/><a NAME="E3:17_6"/><span class="lab"><font color="Green" title="E11">A6</font></span>: 
 for <font color="Olive" title="b1">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span>)</span>  st <font color="Maroon" title="c4">X</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Olive" title="b1">r</font> holds <br/><font color="Maroon" title="c5">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#D18" title="LATTICE3:def.18">LATTICE3:def 18</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c6">p9</font> = <font color="Maroon" title="c5">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E7">A1</font></span></a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c7">p9</font> = <font color="Maroon" title="c6">p9</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">X</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c7">p9</font> &amp; (  for <font color="Olive" title="b1">r9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q</font>  st <font color="Maroon" title="c4">X</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Olive" title="b1">r9</font> holds <br/><font color="Maroon" title="c7">p9</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Olive" title="b1">r9</font> ) )</span><br/><span class="kw">thus </span><a NAME="E5:17_6"/>
<font color="Maroon" title="c4">X</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c7">p9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E2:17_6"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="ref" href="quantal1.html#T2" target="_self" title="QUANTAL1:th.2">Th2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">r9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q</font>  st <font color="Maroon" title="c4">X</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Olive" title="b1">r9</font> holds <br/><font color="Maroon" title="c7">p9</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Olive" title="b1">r9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c8">r9</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">X</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c8">r9</font> implies <font color="Maroon" title="c7">p9</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c8">r9</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c9">r</font> = <font color="Maroon" title="c8">r9</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E7">A1</font></span></a></span>;<br/><span class="kw">assume </span><a NAME="E7:17_6"/>
<font color="Maroon" title="c4">X</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c8">r9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">p9</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c8">r9</font></span><br/><a NAME="E8:17_6"/><span class="kw">then </span>
<font color="Maroon" title="c4">X</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c9">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="quantal1.html#T2" target="_self" title="QUANTAL1:th.2">Th2</a></span>;<br/><a NAME="E9:17_6"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c9">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_6"><span class="lab"><font color="Green" title="E11">A6</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E10:17_6"/>
<font color="Maroon" title="c7">p9</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c8">r9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="quantal1.html#T1" target="_self" title="QUANTAL1:th.1">Th1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E9:17"/>
 for <font color="Olive" title="b1">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   ex <font color="Olive" title="b2">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q</font> st <br/>( <font color="Olive" title="b1">X</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Olive" title="b2">p</font> &amp; (  for <font color="Olive" title="b3">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q</font>  st <font color="Olive" title="b1">X</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Olive" title="b3">r</font> holds <br/><font color="Olive" title="b2">p</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Olive" title="b3">r</font> ) )
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="lattice3.html#D18" title="LATTICE3:def.18">LATTICE3:def 18</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">Q</font> is  <a href="quantal1.html#V7" title="QUANTAL1:attr.7">right-distributive</a>  &amp; <font color="Maroon" title="c2">Q</font> is  <a href="quantal1.html#V8" title="QUANTAL1:attr.8">left-distributive</a>  &amp; <font color="Maroon" title="c2">Q</font> is  <a href="lattices.html#V10" title="LATTICES:attr.10">Lattice-like</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E10:17"/>
<font color="Maroon" title="c2">Q</font> is  <a href="quantal1.html#V7" title="QUANTAL1:attr.7">right-distributive</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">Q</font> is  <a href="quantal1.html#V8" title="QUANTAL1:attr.8">left-distributive</a>  &amp; <font color="Maroon" title="c2">Q</font> is  <a href="lattices.html#V10" title="LATTICES:attr.10">Lattice-like</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="quantal1.html#D5" target="_self" title="QUANTAL1:def.5">QUANTAL1:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds  <font color="Maroon" title="c4">a</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> (<font color="Olive" title="b1">X</font>,<font color="Maroon" title="c2">Q</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> (<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Olive" title="b2">b</font></span>)</span> where <font color="Olive" title="b2">b</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="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> </span> } </span> ,<font color="Maroon" title="c2">Q</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c5">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">a</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> (<font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c2">Q</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> (<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Olive" title="b1">b</font></span>)</span> where <font color="Olive" title="b1">b</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="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font> </span> } </span> ,<font color="Maroon" title="c2">Q</font>)</span><br/>



<span class="kw">thus </span><a NAME="E1:17_7"/>
<font color="Maroon" title="c4">a</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> (<font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c2">Q</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> (<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Olive" title="b1">b</font></span>)</span> where <font color="Olive" title="b1">b</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="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font> </span> } </span> ,<font color="Maroon" title="c2">Q</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E8">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E11:17"/>
<font color="Maroon" title="c2">Q</font> is  <a href="quantal1.html#V8" title="QUANTAL1:attr.8">left-distributive</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">Q</font> is  <a href="lattices.html#V10" title="LATTICES:attr.10">Lattice-like</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="quantal1.html#D6" target="_self" title="QUANTAL1:def.6">QUANTAL1:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds  <span class="p1">(<span class="default"><a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> (<font color="Olive" title="b1">X</font>,<font color="Maroon" title="c2">Q</font>)</span>)</span> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> (<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b2">b</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c4">a</font></span>)</span> where <font color="Olive" title="b2">b</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="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> </span> } </span> ,<font color="Maroon" title="c2">Q</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c5">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> (<font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c2">Q</font>)</span>)</span> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> (<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">b</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c4">a</font></span>)</span> where <font color="Olive" title="b1">b</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="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font> </span> } </span> ,<font color="Maroon" title="c2">Q</font>)</span><br/>



<span class="kw">thus </span><a NAME="E1:17_8"/>
<span class="p1">(<span class="default"><a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> (<font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c2">Q</font>)</span>)</span> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> (<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">b</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c4">a</font></span>)</span> where <font color="Olive" title="b1">b</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="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font> </span> } </span> ,<font color="Maroon" title="c2">Q</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E8">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<a NAME="E12:17"/><span class="lab"><font color="Green" title="E10">A7</font></span>: 
( (  for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q</font> ) &amp; (  for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q</font> holds  <font color="Olive" title="b1">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b1">p</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E8">A2</font></span></a></span>;<br/>
<a NAME="E13:17"/>
( (  for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q</font> holds  <font color="Olive" title="b1">p</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b1">p</font> ) &amp; (  for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Q</font> holds  <font color="Olive" title="b1">p</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">q</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b3">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b3">r</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E8">A2</font></span></a></span>;<br/>
<a NAME="E14:17"/><span class="kw">then </span>
( <font color="Maroon" title="c2">Q</font> is  <a href="lattices.html#V4" title="LATTICES:attr.4">join-commutative</a>  &amp; <font color="Maroon" title="c2">Q</font> is  <a href="lattices.html#V5" title="LATTICES:attr.5">join-associative</a>  &amp; <font color="Maroon" title="c2">Q</font> is  <a href="lattices.html#V8" title="LATTICES:attr.8">meet-absorbing</a>  &amp; <font color="Maroon" title="c2">Q</font> is  <a href="lattices.html#V6" title="LATTICES:attr.6">meet-commutative</a>  &amp; <font color="Maroon" title="c2">Q</font> is  <a href="lattices.html#V7" title="LATTICES:attr.7">meet-associative</a>  &amp; <font color="Maroon" title="c2">Q</font> is  <a href="lattices.html#V9" title="LATTICES:attr.9">join-absorbing</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:17"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="txt" href="#E4:17"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="ref" href="lattices.html#D4" title="LATTICES:def.4">LATTICES:def 4</a>, <a class="ref" href="lattices.html#D5" title="LATTICES:def.5">LATTICES:def 5</a>, <a class="ref" href="lattices.html#D6" title="LATTICES:def.6">LATTICES:def 6</a>, <a class="ref" href="lattices.html#D7" title="LATTICES:def.7">LATTICES:def 7</a>, <a class="ref" href="lattices.html#D8" title="LATTICES:def.8">LATTICES:def 8</a>, <a class="ref" href="lattices.html#D9" title="LATTICES:def.9">LATTICES:def 9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:17"/>
<font color="Maroon" title="c2">Q</font> is  <a href="lattices.html#V10" title="LATTICES:attr.10">Lattice-like</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
