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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="lattice5.html#NM3">BiFunction</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>





<b>assume </b><a NAME="E1:21"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="lattice5.html#V1">symmetric</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K4">:]</a></span>;<br/>

<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="mcart_1.html#K8">`1</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="mcart_1.html#K9">`2</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="mcart_1.html#K10">`3</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="mcart_1.html#K11">`4</a> ;<br/>
<b>let </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K1" target="_self">new_set2</a> <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice5.html#D6">LATTICE5:def 6</a><br/>

<a NAME="E2:21"/><i><font color="Green">E16</font></i>: 
( ( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span> ) &amp; ( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span> ) )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1"/>
( ( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> ) or ( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>11</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">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> ) or ( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>11</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">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">c<sub>10</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">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> ) or ( <font color="Maroon">c<sub>10</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">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">c<sub>10</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">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>11</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">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) )
 </span><b>by </b><i><a class="txt" href="lattice8.html#E2:21"><i><font color="Green">E16</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1"><b>suppose </b></a><a NAME="E1:21_1_1"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>thus </b><span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> = 
<font color="Maroon">c<sub>3</sub></font> <a href="lattice5.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="lattice8.html#D5" target="_self">Def5</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="lattice5.html#K4">.</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="lattice8.html#E1:21"><i><font color="Green">E15</font></i></a>, <a class="ref" href="lattice5.html#D6">LATTICE5:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="lattice8.html#D5" target="_self">Def5</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2"><b>suppose </b></a><a NAME="E1:21_1_2"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>thus </b><span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice5.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font> <a href="mcart_1.html#K10">`3</a> </span>)</span>
<b>by </b><i><a class="txt" href="lattice8.html#E1:21_1_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="lattice8.html#E1:21_1_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self">Def5</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_3"><b>suppose </b></a><a NAME="E1:21_1_3"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>11</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">c<sub>1</sub></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">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>thus </b><span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice5.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font> <a href="mcart_1.html#K10">`3</a> </span>)</span>
<b>by </b><i><a class="txt" href="lattice8.html#E1:21_1_3"><i><font color="Green">E17</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="lattice8.html#E1:21_1_3"><i><font color="Green">E17</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self">Def5</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_4"><b>suppose </b></a><a NAME="E1:21_1_4"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>thus </b><span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice5.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font> <a href="mcart_1.html#K10">`3</a> </span>)</span>
<b>by </b><i><a class="txt" href="lattice8.html#E1:21_1_4"><i><font color="Green">E17</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="lattice8.html#E1:21_1_4"><i><font color="Green">E17</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self">Def5</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_5"><b>suppose </b></a><a NAME="E1:21_1_5"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_5"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_6"><b>suppose </b></a><a NAME="E1:21_1_6"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>11</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">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice5.html#K4">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="mcart_1.html#K10">`3</a> </span>)</span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="mcart_1.html#K11">`4</a> </span>)</span>
<b>by </b><i><a class="ref" href="lattice8.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="lattice8.html#E1:21_1_6"><i><font color="Green">E17</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self">Def5</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_7"><b>suppose </b></a><a NAME="E1:21_1_7"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>10</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">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>thus </b><span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice5.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font> <a href="mcart_1.html#K10">`3</a> </span>)</span>
<b>by </b><i><a class="txt" href="lattice8.html#E1:21_1_7"><i><font color="Green">E17</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="lattice8.html#E1:21_1_7"><i><font color="Green">E17</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self">Def5</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_8"><b>suppose </b></a><a NAME="E1:21_1_8"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>10</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">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice5.html#K4">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="mcart_1.html#K10">`3</a> </span>)</span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="mcart_1.html#K11">`4</a> </span>)</span>
<b>by </b><i><a class="ref" href="lattice8.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="lattice8.html#E1:21_1_8"><i><font color="Green">E17</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self">Def5</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_9"><b>suppose </b></a><a NAME="E1:21_1_9"/>
( <font color="Maroon">c<sub>10</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">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>11</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">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_9"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
