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

<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="waybel_1.html#V11" title="WAYBEL_1:attr.11">Boolean</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:37"/><i><font color="Green" title="E28">A1</font></i>: 
<font color="Maroon" title="c1">L</font> is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>
 <b>and </b><br/><a NAME="E2:37"/><i><font color="Green" title="E29">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  ex <font color="Olive" title="b2">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel15.html#K1" title="WAYBEL15:func.1">ATOM</a> <font color="Maroon" title="c1">L</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b2">X</font> )
 ;<br/>

<b>take </b><font color="Maroon" title="c2">Y</font> =  <a href="waybel15.html#K1" title="WAYBEL15:func.1">ATOM</a> <font color="Maroon" title="c1">L</font>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">x'</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">x'</font> );<br/>
<a NAME="E3:37"/><i><font color="Green" title="E30">A3</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Maroon" title="c2">Y</font></span>)</span>  ex <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Maroon" title="c2">Y</font></span>)</span>;<br/>

<a NAME="E1:37_1"/>
<font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="ref" href="waybel_8.html#T28" title="WAYBEL_8:th.28">WAYBEL_8:28</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">x'</font> = <font color="Maroon" title="c3">x</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>take </b><font color="Maroon" title="c5">y</font> =  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c4">x'</font>;<br/>

<b>take </b>
<font color="Maroon" title="c4">x'</font>
;<br/>

<b>thus </b><a NAME="E3:37_1"/>
( <font color="Maroon" title="c4">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c4">x'</font> )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c3">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E5:37"/><i><font color="Green" title="E31">A4</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Maroon" title="c2">Y</font></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S10" title="FUNCT_2:sch.10">FUNCT_2:sch 10</a>(<a class="txt" href="waybel15.html#E3:37"><i><font color="Green" title="E30">A3</font></i></a>);<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">z</font>, <font color="Maroon" title="c5">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Maroon" title="c2">Y</font></span>)</span>;<br/><b>consider </b><font color="Maroon" title="c6">z'</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E2:37_2"/><i><font color="Green" title="E32">A5</font></i>: 
<font color="Maroon" title="c6">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z</font>
 <b>and </b><br/><a NAME="E3:37_2"/><i><font color="Green" title="E33">A6</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c6">z'</font>
 <b>by </b><i><a class="txt" href="waybel15.html#E5:37"><i><font color="Green" title="E31">A4</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c7">v'</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E5:37_2"/><i><font color="Green" title="E34">A7</font></i>: 
<font color="Maroon" title="c7">v'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font>
 <b>and </b><br/><a NAME="E6:37_2"/><i><font color="Green" title="E35">A8</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c7">v'</font>
 <b>by </b><i><a class="txt" href="waybel15.html#E5:37"><i><font color="Green" title="E31">A4</font></i></a></i>;<br/><a NAME="E7:37_2"/><i><font color="Green" title="E36">A9</font></i>: 
<font color="Maroon" title="c6">z'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel15.html#K1" title="WAYBEL15:func.1">ATOM</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="waybel15.html#E2:37_2"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="waybel_8.html#T28" title="WAYBEL_8:th.28">WAYBEL_8:28</a></i>;<br/><a NAME="E8:37_2"/><i><font color="Green" title="E37">A10</font></i>: 
<font color="Maroon" title="c7">v'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel15.html#K1" title="WAYBEL15:func.1">ATOM</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="waybel15.html#E5:37_2"><i><font color="Green" title="E34">A7</font></i></a>, <a class="ref" href="waybel_8.html#T28" title="WAYBEL_8:th.28">WAYBEL_8:28</a></i>;<br/><b>assume </b><a NAME="E9:37_2"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">v</font>
 ;<br/><a NAME="E10:37_2"/><b>then </b>
( <font color="Maroon" title="c4">z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">v</font> &amp; <font color="Maroon" title="c5">v</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">z</font> )
 <b>by </b><i><a class="txt" href="waybel15.html#E1:37"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="waybel15.html#E2:37_2"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="waybel15.html#E3:37_2"><i><font color="Green" title="E33">A6</font></i></a>, <a class="txt" href="waybel15.html#E5:37_2"><i><font color="Green" title="E34">A7</font></i></a>, <a class="txt" href="waybel15.html#E6:37_2"><i><font color="Green" title="E35">A8</font></i></a>, <a class="txt" href="waybel15.html#E7:37_2"><i><font color="Green" title="E36">A9</font></i></a>, <a class="txt" href="waybel15.html#E8:37_2"><i><font color="Green" title="E37">A10</font></i></a>, <a class="ref" href="waybel15.html#T29" target="_self" title="WAYBEL15:th.29">Th29</a></i>;<br/><b>hence </b><a NAME="E11:37_2"/>
<font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:37"/><b>then </b><i><font color="Green" title="E32">A11</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="waybel_1.html#D1" title="WAYBEL_1:def.1">WAYBEL_1:def 1</a></i>;<br/>
<a NAME="E8:37"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">k</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:37_3"/>
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">K</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E3:37_3"/><i><font color="Green" title="E33">A12</font></i>: 
<font color="Maroon" title="c5">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel15.html#K1" title="WAYBEL15:func.1">ATOM</a> <font color="Maroon" title="c1">L</font>
 <b>and </b><br/><a NAME="E4:37_3"/><i><font color="Green" title="E34">A13</font></i>: 
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">K</font>
 <b>by </b><i><a class="txt" href="waybel15.html#E2:37"><i><font color="Green" title="E29">A2</font></i></a></i>;<br/>
<a NAME="E5:37_3"/><i><font color="Green" title="E35">A14</font></i>: 
<font color="Maroon" title="c5">K</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel15.html#E3:37_3"><i><font color="Green" title="E33">A12</font></i></a>, <a class="ref" href="waybel_8.html#T28" title="WAYBEL_8:th.28">WAYBEL_8:28</a></i>;<br/>
<a NAME="E6:37_3"/><b>then </b>
( <font color="Maroon" title="c5">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Maroon" title="c2">Y</font></span>)</span> &amp; not the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 
;<br/>
<a NAME="E7:37_3"/><b>then </b><i><font color="Green" title="E36">A15</font></i>: 
<font color="Maroon" title="c5">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">K'</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E9:37_3"/><i><font color="Green" title="E37">A16</font></i>: 
<font color="Maroon" title="c6">K'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">K</font>
 <b>and </b><br/><a NAME="E10:37_3"/><i><font color="Green" title="E38">A17</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c6">K'</font>
 <b>by </b><i><a class="txt" href="waybel15.html#E5:37"><i><font color="Green" title="E31">A4</font></i></a>, <a class="txt" href="waybel15.html#E5:37_3"><i><font color="Green" title="E35">A14</font></i></a></i>;<br/>
<b>thus </b><a NAME="E11:37_3"/>
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="waybel15.html#E4:37_3"><i><font color="Green" title="E34">A13</font></i></a>, <a class="txt" href="waybel15.html#E7:37_3"><i><font color="Green" title="E36">A15</font></i></a>, <a class="txt" href="waybel15.html#E9:37_3"><i><font color="Green" title="E37">A16</font></i></a>, <a class="txt" href="waybel15.html#E10:37_3"><i><font color="Green" title="E38">A17</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:37"/><b>then </b><i><font color="Green" title="E33">A18</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">z</font>, <font color="Maroon" title="c5">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Maroon" title="c2">Y</font></span>)</span>;<br/><b>consider </b><font color="Maroon" title="c6">z'</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E2:37_4"/><i><font color="Green" title="E34">A19</font></i>: 
<font color="Maroon" title="c6">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z</font>
 <b>and </b><br/><a NAME="E3:37_4"/><i><font color="Green" title="E35">A20</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c6">z'</font>
 <b>by </b><i><a class="txt" href="waybel15.html#E5:37"><i><font color="Green" title="E31">A4</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c7">v'</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E5:37_4"/><i><font color="Green" title="E36">A21</font></i>: 
<font color="Maroon" title="c7">v'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font>
 <b>and </b><br/><a NAME="E6:37_4"/><i><font color="Green" title="E37">A22</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c7">v'</font>
 <b>by </b><i><a class="txt" href="waybel15.html#E5:37"><i><font color="Green" title="E31">A4</font></i></a></i>;<br/><a NAME="E7:37_4"/><i><font color="Green" title="E38">A23</font></i>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c6">z'</font>,<font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="waybel15.html#E1:37"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a></i>;<br/><a NAME="E8:37_4"/><i><font color="Green" title="E39">A24</font></i>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c7">v'</font>,<font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="waybel15.html#E1:37"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a></i>;<br/><a NAME="E9:37_4"/><i><font color="Green" title="E40">A25</font></i>: 
<font color="Maroon" title="c6">z'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel15.html#K1" title="WAYBEL15:func.1">ATOM</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="waybel15.html#E2:37_4"><i><font color="Green" title="E34">A19</font></i></a>, <a class="ref" href="waybel_8.html#T28" title="WAYBEL_8:th.28">WAYBEL_8:28</a></i>;<br/><a NAME="E10:37_4"/><i><font color="Green" title="E41">A26</font></i>: 
<font color="Maroon" title="c7">v'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel15.html#K1" title="WAYBEL15:func.1">ATOM</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="waybel15.html#E5:37_4"><i><font color="Green" title="E36">A21</font></i></a>, <a class="ref" href="waybel_8.html#T28" title="WAYBEL_8:th.28">WAYBEL_8:28</a></i>;<br/><b>thus </b><a NAME="E11:37_4"/>
( <font color="Maroon" title="c4">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">v</font> implies <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">v</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_4_1"/>
<font color="Maroon" title="c4">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">v</font>
 ;<br/>

<a NAME="E2:37_4_1"/><b>then </b>
<font color="Maroon" title="c4">z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">v</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T2" title="YELLOW_1:th.2">YELLOW_1:2</a></i>;<br/>
<b>hence </b><a NAME="E3:37_4_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">v</font>
 <b>by </b><i><a class="txt" href="waybel15.html#E2:37_4"><i><font color="Green" title="E34">A19</font></i></a>, <a class="txt" href="waybel15.html#E3:37_4"><i><font color="Green" title="E35">A20</font></i></a>, <a class="txt" href="waybel15.html#E5:37_4"><i><font color="Green" title="E36">A21</font></i></a>, <a class="txt" href="waybel15.html#E6:37_4"><i><font color="Green" title="E37">A22</font></i></a>, <a class="txt" href="waybel15.html#E7:37_4"><i><font color="Green" title="E38">A23</font></i></a>, <a class="txt" href="waybel15.html#E8:37_4"><i><font color="Green" title="E39">A24</font></i></a>, <a class="ref" href="yellow_0.html#T34" title="YELLOW_0:th.34">YELLOW_0:34</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E12:37_4"/>
( <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">v</font> implies <font color="Maroon" title="c4">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">v</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_4_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_4_2"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">v</font>
 ;<br/>

<a NAME="E2:37_4_2"/><b>then </b>
<font color="Maroon" title="c4">z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">v</font>
 
<b>by </b><i><a class="txt" href="waybel15.html#E1:37"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="waybel15.html#E2:37_4"><i><font color="Green" title="E34">A19</font></i></a>, <a class="txt" href="waybel15.html#E3:37_4"><i><font color="Green" title="E35">A20</font></i></a>, <a class="txt" href="waybel15.html#E5:37_4"><i><font color="Green" title="E36">A21</font></i></a>, <a class="txt" href="waybel15.html#E6:37_4"><i><font color="Green" title="E37">A22</font></i></a>, <a class="txt" href="waybel15.html#E9:37_4"><i><font color="Green" title="E40">A25</font></i></a>, <a class="txt" href="waybel15.html#E10:37_4"><i><font color="Green" title="E41">A26</font></i></a>, <a class="ref" href="waybel15.html#T29" target="_self" title="WAYBEL15:th.29">Th29</a></i>;<br/>
<b>hence </b><a NAME="E3:37_4_2"/>
<font color="Maroon" title="c4">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">v</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T2" title="YELLOW_1:th.2">YELLOW_1:2</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E11:37"/><b>then </b>
<font color="Maroon" title="c3">f</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a>
 
<b>by </b><i><a class="txt" href="waybel15.html#E7:37"><i><font color="Green" title="E32">A11</font></i></a>, <a class="txt" href="waybel15.html#E9:37"><i><font color="Green" title="E33">A18</font></i></a>, <a class="ref" href="waybel_0.html#T66" title="WAYBEL_0:th.66">WAYBEL_0:66</a></i>;<br/>
<a NAME="E12:37"/><b>then </b>
 <a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c1">L</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="waybel_1.html#D8" title="WAYBEL_1:def.8">WAYBEL_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E13:37"/>
<font color="Maroon" title="c1">L</font>, <a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Maroon" title="c2">Y</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a> 
 <b>by </b><i><a class="ref" href="waybel_1.html#T7" title="WAYBEL_1:th.7">WAYBEL_1:7</a></i>;<br/>


</div>
