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

<b>set </b><font color="Maroon" title="c1">X</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>() <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">k</font>] </span>}</span> ;<br/>
<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>for <font color="Olive" title="b2">j</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()  st <font color="Olive" title="b1">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>() <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">j</font>] implies <font color="Maroon">F<sub>2</sub></font>() <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>() <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">k</font>] </span>}</span>  )</span>
<div class="add"><b>assume </b><a NAME="E1:32_1_1"/>
<font color="Maroon">F<sub>2</sub></font>() <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>() <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">k</font>] </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>for <font color="Olive" title="b2">j</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()  st <font color="Olive" title="b1">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>() <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">j</font>]</span><br/><b>then consider </b><font color="Maroon" title="c2">i</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E3:32_1_1"/><i><font color="Green" title="E11">A1</font></i>: 
for <font color="Olive" title="b1">j</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()  st <font color="Maroon" title="c2">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> holds <br/><font color="Maroon">F<sub>2</sub></font>() <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">k</font></span>)</span> where <font color="Olive" title="b2">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>() <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">k</font>] </span>}</span> 
 <b>by </b><i><a class="ref" href="waybel_0.html#D11" target="_self" title="WAYBEL_0:def.11">Def11</a></i>;<br/><b>take </b><font color="Maroon" title="c3">i</font> = <font color="Maroon" title="c2">i</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">j</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()  st <font color="Maroon" title="c3">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>() <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">j</font>]</span><br/><b>let </b><font color="Maroon" title="c4">j</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">j</font> implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>() <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c4">j</font>] )</span><br/><b>assume </b><a NAME="E4:32_1_1"/>
<font color="Maroon" title="c3">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>() <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c4">j</font>]</span><br/><a NAME="E5:32_1_1"/><b>then </b>
<font color="Maroon">F<sub>2</sub></font>() <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>() <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">k</font>] </span>}</span> 
 <b>by </b><i><a class="txt" href="#E3:32_1_1"><i><font color="Green" title="E11">A1</font></i></a></i>;<br/><a NAME="E6:32_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">k</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Maroon">F<sub>2</sub></font>() <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">k</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>() <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">k</font>] )
 ;<br/><b>hence </b><a NAME="E7:32_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>() <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c4">j</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon" title="c2">i</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><a NAME="E3:32_1"/><i><font color="Green" title="E11">A2</font></i>: 
for <font color="Olive" title="b1">j</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()  st <font color="Maroon" title="c2">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>() <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">j</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">F<sub>2</sub></font>() <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>() <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">k</font>] </span>}</span> </span><br/>

<b>take </b>
<font color="Maroon" title="c2">i</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D11" target="_self" title="WAYBEL_0:def.11">WAYBEL_0:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">j</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()  st <font color="Maroon" title="c2">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> holds <br/><font color="Maroon">F<sub>2</sub></font>() <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">k</font></span>)</span> where <font color="Olive" title="b2">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>() <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">k</font>] </span>}</span> </span><br/>

<b>let </b><font color="Maroon" title="c3">j</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">j</font> implies <font color="Maroon">F<sub>2</sub></font>() <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>() <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">k</font>] </span>}</span>  )</span><br/>

<b>assume </b><a NAME="E4:32_1"/>
<font color="Maroon" title="c2">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">F<sub>2</sub></font>() <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>() <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">k</font>] </span>}</span> </span><br/>

<a NAME="E5:32_1"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>() <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c3">j</font>]
 
<b>by </b><i><a class="txt" href="#E3:32_1"><i><font color="Green" title="E11">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:32_1"/>
<font color="Maroon">F<sub>2</sub></font>() <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>() <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">k</font>] </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
