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

<b>set </b><font color="Maroon">x</font> = <font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> ;<br/>
<b>set </b><font color="Maroon">y</font> = <font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> ;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">q</font> <a href="mcart_1.html#K10">`3</a> , <font color="Maroon">b</font> = <font color="Maroon">q</font> <a href="mcart_1.html#K11">`4</a>  as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E2:38_1_1"/><i><font color="Green">E29</font></i>: 
( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:38_1_1_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="enumset1.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>
<a NAME="E2:38_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="enumset1.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:38_1_1_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E4:38_1_1_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="enumset1.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>
<a NAME="E5:38_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="enumset1.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E6:38_1_1_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E7:38_1_1_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="enumset1.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>
<a NAME="E8:38_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="enumset1.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E9:38_1_1_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>
 ;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>,   <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>,  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> implies <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font> ) &amp; ( ( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) ) implies <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> ) &amp; ( ( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) ) implies <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> ) &amp; ( ( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) ) implies <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> ) &amp; ( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> implies <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> implies  ex <font color="Olive">p'</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> st <br/>( <font color="Olive">p'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font> ) ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> implies  ex <font color="Olive">p'</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> st <br/>( <font color="Olive">p'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> ) ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> implies  ex <font color="Olive">p'</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> st <br/>( <font color="Olive">p'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> ) ) &amp; ( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> implies  ex <font color="Olive">q'</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> st <br/>( <font color="Olive">q'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">q'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font> ) ) &amp; ( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> implies  ex <font color="Olive">q'</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> st <br/>( <font color="Olive">q'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">q'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> ) ) &amp; ( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> implies  ex <font color="Olive">q'</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> st <br/>( <font color="Olive">q'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">q'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> ) ) );<br/>
<a NAME="E3:38_1_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>  ex <font color="Olive">r</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">r</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>, <font color="Maroon">q</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>;<br/>

<a NAME="E1:38_1_1_2"/><i><font color="Green">E39</font></i>: 
( ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> or <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="enumset1.html#K1">}</a></span> ) &amp; ( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> or <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="enumset1.html#K1">}</a></span> ) )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E2:38_1_1_2"/><i><font color="Green">E42</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_1_1_2_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:38_1_1_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:38_1_1_2_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:38_1_1_2"/><i><font color="Green">E47</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_1_1_2_2"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:38_1_1_2_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:38_1_1_2_2"/>
contradiction
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:38_1_1_2"/><i><font color="Green">E49</font></i>: 
not <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:38_1_1_2"/><i><font color="Green">E50</font></i>: 
not <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_1_1_2_3"/><i><font color="Green">E51</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:38_1_1_2_3"/>
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:38_1_1_2_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="ordinal1.html#T3">ORDINAL1:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:38_1_1_2"/><i><font color="Green">E52</font></i>: 
not <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_2_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_1_1_2_4"/><i><font color="Green">E53</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:38_1_1_2_4"/>
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:38_1_1_2_4"/>
contradiction
 <b>by </b><i>, <a class="ref" href="ordinal1.html#T4">ORDINAL1:4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:38_1_1_2"/><i><font color="Green">E54</font></i>: 
( ( ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> ) iff ( ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) ) )
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_2_5"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_1_1_2_5"/>
( ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) )
 </span><b>by </b><i>, , <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_2_5_1"><b>suppose </b></a><a NAME="E1:38_1_1_2_5_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font>, <font color="Maroon">q'</font> = <font color="Maroon">q</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> ;<br/><b>take </b>
<font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font>
;<br/><b>thus </b><a NAME="E3:38_1_1_2_5_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font>]
 <b>by </b><i>, <a class="txt" href="lattice5.html#E2:38_1_1"><i><font color="Green">E29</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_2_5_2"><b>suppose </b></a><a NAME="E1:38_1_1_2_5_2"/><i><font color="Green">E55</font></i>: 
( ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) )
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">a</font>
;<br/><b>thus </b><a NAME="E2:38_1_1_2_5_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">a</font>]
 <b>by </b><i>, , <a class="txt" href="lattice5.html#E2:38_1_1"><i><font color="Green">E29</font></i></a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_2_5_3"><b>suppose </b></a><a NAME="E1:38_1_1_2_5_3"/><i><font color="Green">E56</font></i>: 
( ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) )
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">b</font>
;<br/><b>thus </b><a NAME="E2:38_1_1_2_5_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">b</font>]
 <b>by </b><i>, , <a class="txt" href="lattice5.html#E2:38_1_1"><i><font color="Green">E29</font></i></a>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_2_5_4"><b>suppose </b></a><a NAME="E1:38_1_1_2_5_4"/><i><font color="Green">E57</font></i>: 
( ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) )
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
;<br/><b>thus </b><a NAME="E2:38_1_1_2_5_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>]
 <b>by </b><i>, , , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_2_5_5"><b>suppose </b></a><a NAME="E1:38_1_1_2_5_5"/><i><font color="Green">E58</font></i>: 
( ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 ;<br/><div class="add"><b>take </b>
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>
;<br/><b>thus </b><a NAME="E2:38_1_1_2_5_5"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">p</font>,<font color="Maroon">q</font>, <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>]
 <b>by </b><i>, , <a class="txt" href="lattice5.html#E2:38_1_1"><i><font color="Green">E29</font></i></a>, , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_2_5_6"><b>suppose </b></a><a NAME="E1:38_1_1_2_5_6"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> ;<br/><b>take </b>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>
;<br/><b>thus </b><a NAME="E3:38_1_1_2_5_6"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">p</font>,<font color="Maroon">q</font>,<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>]
 <b>by </b><i>, <a class="txt" href="lattice5.html#E2:38_1_1"><i><font color="Green">E29</font></i></a>, , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_2_5_7"><b>suppose </b></a><a NAME="E1:38_1_1_2_5_7"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> ;<br/><b>take </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
;<br/><b>thus </b><a NAME="E3:38_1_1_2_5_7"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">p</font>,<font color="Maroon">q</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>]
 <b>by </b><i>, <a class="txt" href="lattice5.html#E2:38_1_1"><i><font color="Green">E29</font></i></a>, , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_2_5_8"><b>suppose </b></a><a NAME="E1:38_1_1_2_5_8"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> ;<br/><b>take </b>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
;<br/><b>thus </b><a NAME="E3:38_1_1_2_5_8"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">p</font>,<font color="Maroon">q</font>,<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>]
 <b>by </b><i>, , <a class="txt" href="lattice5.html#E2:38_1_1"><i><font color="Green">E29</font></i></a>, , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_2_5_9"><b>suppose </b></a><a NAME="E1:38_1_1_2_5_9"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">q'</font> = <font color="Maroon">q</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> ;<br/><b>take </b>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>
;<br/><b>thus </b><a NAME="E3:38_1_1_2_5_9"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">p</font>,<font color="Maroon">q</font>,<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>]
 <b>by </b><i>, <a class="txt" href="lattice5.html#E2:38_1_1"><i><font color="Green">E29</font></i></a>, , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_2_5_10"><b>suppose </b></a><a NAME="E1:38_1_1_2_5_10"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">q'</font> = <font color="Maroon">q</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> ;<br/><b>take </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
;<br/><b>thus </b><a NAME="E3:38_1_1_2_5_10"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">p</font>,<font color="Maroon">q</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>]
 <b>by </b><i>, <a class="txt" href="lattice5.html#E2:38_1_1"><i><font color="Green">E29</font></i></a>, , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_2_5_11"><b>suppose </b></a><a NAME="E1:38_1_1_2_5_11"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">q'</font> = <font color="Maroon">q</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> ;<br/><b>take </b>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
;<br/><b>thus </b><a NAME="E3:38_1_1_2_5_11"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">p</font>,<font color="Maroon">q</font>,<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>]
 <b>by </b><i>, , <a class="txt" href="lattice5.html#E2:38_1_1"><i><font color="Green">E29</font></i></a>, , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E5:38_1_1"/><i><font color="Green">E65</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Maroon">f</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">p</font>,<font color="Olive">q</font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3">BINOP_1:sch 3</a>();<br/></i>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="lattice5.html#NM3">BiFunction</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">L</font> ;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<a NAME="E7:38_1_1"/><i><font color="Green">E66</font></i>: 
for <font color="Olive">u</font>, <font color="Olive">v</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Olive">u</font>,<font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">u</font>,<font color="Olive">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">v</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>;<br/>

<a NAME="E1:38_1_1_3"/>
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="enumset1.html#K1">}</a></span> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="enumset1.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">u'</font> = <font color="Maroon">u</font>, <font color="Maroon">v'</font> = <font color="Maroon">v</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font> ;<br/>
<b>thus </b><font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">u</font>,<font color="Maroon">v</font> = 
<font color="Maroon">f</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<font color="Maroon">v'</font>

<br/>.= 
<font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u</font>,<font color="Maroon">v</font>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>
<a NAME="E8:38_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:38_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:38_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:38_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E12:38_1_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E13:38_1_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E14:38_1_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E15:38_1_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E16:38_1_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E17:38_1_1"/><i><font color="Green">E80</font></i>: 
for <font color="Olive">u</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds <br/> ( <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Olive">u</font>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">u</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font> &amp; <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Olive">u</font>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">u</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> &amp; <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Olive">u</font>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">u</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>;<br/>

<a NAME="E1:38_1_1_4"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="enumset1.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">u'</font> = <font color="Maroon">u</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font> ;<br/>
<b>consider </b><font color="Maroon">u1</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E4:38_1_1_4"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u'</font> &amp; <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">u'</font>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u1</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font> )
 <b>by </b><i>, </i>;<br/>
<b>thus </b><a NAME="E5:38_1_1_4"/>
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">u</font>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:38_1_1"><i><font color="Green">E37</font></i></a></i>;<br/>

<b>consider </b><font color="Maroon">u2</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E7:38_1_1_4"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">u2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u'</font> &amp; <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">u'</font>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u2</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> )
 <b>by </b><i>, </i>;<br/>
<b>thus </b><a NAME="E8:38_1_1_4"/>
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">u</font>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/>

<b>consider </b><font color="Maroon">u3</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E10:38_1_1_4"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">u3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u'</font> &amp; <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">u'</font>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u3</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> )
 <b>by </b><i>, </i>;<br/>
<b>thus </b><a NAME="E11:38_1_1_4"/>
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">u</font>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E18:38_1_1"/>
for <font color="Olive">u</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds <br/> ( <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span>,<font color="Olive">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">u</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font> &amp; <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<font color="Olive">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">u</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> &amp; <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<font color="Olive">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">u</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>;<br/>

<a NAME="E1:38_1_1_5"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="enumset1.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">u'</font> = <font color="Maroon">u</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font> ;<br/>
<b>consider </b><font color="Maroon">u1</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E4:38_1_1_5"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u'</font> &amp; <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">u'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u1</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font> )
 <b>by </b><i>, </i>;<br/>
<b>thus </b><a NAME="E5:38_1_1_5"/>
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:38_1_1"><i><font color="Green">E37</font></i></a></i>;<br/>

<b>consider </b><font color="Maroon">u2</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E7:38_1_1_5"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">u2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u'</font> &amp; <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">u'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u2</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> )
 <b>by </b><i>, </i>;<br/>
<b>thus </b><a NAME="E8:38_1_1_5"/>
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/>

<b>consider </b><font color="Maroon">u3</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E10:38_1_1_5"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">u3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u'</font> &amp; <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">u'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u3</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> )
 <b>by </b><i>, </i>;<br/>
<b>thus </b><a NAME="E11:38_1_1_5"/>
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E19:38_1_1"/>
( ( for <font color="Olive">u</font>, <font color="Olive">v</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Olive">u</font>,<font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">u</font>,<font color="Olive">v</font> ) &amp; <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> &amp; <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> &amp; <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> &amp; <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="mcart_1.html#K10">`3</a>  &amp; <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="mcart_1.html#K10">`3</a>  &amp; <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="mcart_1.html#K11">`4</a>  &amp; <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="mcart_1.html#K11">`4</a>  &amp; <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K10">`3</a> </span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K11">`4</a> </span>)</span> &amp; <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K10">`3</a> </span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K11">`4</a> </span>)</span> &amp; ( for <font color="Olive">u</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds <br/> ( <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Olive">u</font>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">u</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K10">`3</a> </span>)</span> &amp; <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span>,<font color="Olive">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">u</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K10">`3</a> </span>)</span> &amp; <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Olive">u</font>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">u</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K10">`3</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K11">`4</a> </span>)</span> &amp; <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<font color="Olive">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">u</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K10">`3</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K11">`4</a> </span>)</span> &amp; <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Olive">u</font>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">u</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K11">`4</a> </span>)</span> &amp; <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<font color="Olive">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">u</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K11">`4</a> </span>)</span> ) ) )
 <b>by </b><i>, , , <a class="txt" href="lattice5.html#E2:38_1_1"><i><font color="Green">E29</font></i></a>, , , , , , , </i>;<br/>


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