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

<b>let </b><font color="Maroon" title="c5">F1</font>, <font color="Maroon" title="c6">F2</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> <font color="Maroon" title="c1">A1</font>,<font color="Maroon" title="c3">B1</font>, <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> <font color="Maroon" title="c2">A2</font>,<font color="Maroon" title="c4">B2</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">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> <font color="Maroon" title="c1">A1</font>,<font color="Maroon" title="c3">B1</font> holds <br/> ex <font color="Olive" title="b2">Bo</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">B1</font> ex <font color="Olive" title="b3">yo1</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b2">Bo</font>,<font color="Maroon" title="c1">A1</font> ex <font color="Olive" title="b4">yo2</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b2">Bo</font>,<font color="Maroon" title="c2">A2</font> st <br/>( <font color="Olive" title="b2">Bo</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b3">yo1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b4">yo2</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="petri_2.html#K1" title="PETRI_2:func.1">cylinder0</a> <font color="Maroon" title="c1">A1</font>,<font color="Maroon" title="c3">B1</font>,<font color="Olive" title="b2">Bo</font>,<font color="Olive" title="b3">yo1</font> &amp; <font color="Maroon" title="c5">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="petri_2.html#K1" title="PETRI_2:func.1">cylinder0</a> <font color="Maroon" title="c2">A2</font>,<font color="Maroon" title="c4">B2</font>,<font color="Olive" title="b2">Bo</font>,<font color="Olive" title="b4">yo2</font> ) ) &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> <font color="Maroon" title="c1">A1</font>,<font color="Maroon" title="c3">B1</font> holds <br/> ex <font color="Olive" title="b2">Bo</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">B1</font> ex <font color="Olive" title="b3">yo1</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b2">Bo</font>,<font color="Maroon" title="c1">A1</font> ex <font color="Olive" title="b4">yo2</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b2">Bo</font>,<font color="Maroon" title="c2">A2</font> st <br/>( <font color="Olive" title="b2">Bo</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b3">yo1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b4">yo2</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="petri_2.html#K1" title="PETRI_2:func.1">cylinder0</a> <font color="Maroon" title="c1">A1</font>,<font color="Maroon" title="c3">B1</font>,<font color="Olive" title="b2">Bo</font>,<font color="Olive" title="b3">yo1</font> &amp; <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="petri_2.html#K1" title="PETRI_2:func.1">cylinder0</a> <font color="Maroon" title="c2">A2</font>,<font color="Maroon" title="c4">B2</font>,<font color="Olive" title="b2">Bo</font>,<font color="Olive" title="b4">yo2</font> ) ) implies <font color="Maroon" title="c5">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F2</font> )</span><br/>

<b>assume </b><a NAME="E1:13_1_1"/><i><font color="Green" title="E12">AS1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> <font color="Maroon" title="c1">A1</font>,<font color="Maroon" title="c3">B1</font> holds <br/> ex <font color="Olive" title="b2">Bo</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">B1</font> ex <font color="Olive" title="b3">yo1</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b2">Bo</font>,<font color="Maroon" title="c1">A1</font> ex <font color="Olive" title="b4">yo2</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b2">Bo</font>,<font color="Maroon" title="c2">A2</font> st <br/>( <font color="Olive" title="b2">Bo</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b3">yo1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b4">yo2</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="petri_2.html#K1" title="PETRI_2:func.1">cylinder0</a> <font color="Maroon" title="c1">A1</font>,<font color="Maroon" title="c3">B1</font>,<font color="Olive" title="b2">Bo</font>,<font color="Olive" title="b3">yo1</font> &amp; <font color="Maroon" title="c5">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="petri_2.html#K1" title="PETRI_2:func.1">cylinder0</a> <font color="Maroon" title="c2">A2</font>,<font color="Maroon" title="c4">B2</font>,<font color="Olive" title="b2">Bo</font>,<font color="Olive" title="b4">yo2</font> )
 ; <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">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> <font color="Maroon" title="c1">A1</font>,<font color="Maroon" title="c3">B1</font> &amp; ( for <font color="Olive" title="b2">Bo</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">B1</font><br/> for <font color="Olive" title="b3">yo1</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b2">Bo</font>,<font color="Maroon" title="c1">A1</font><br/> for <font color="Olive" title="b4">yo2</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b2">Bo</font>,<font color="Maroon" title="c2">A2</font>  holds <br/>(  not <font color="Olive" title="b2">Bo</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> or  not <font color="Olive" title="b3">yo1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b4">yo2</font> or  not <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="petri_2.html#K1" title="PETRI_2:func.1">cylinder0</a> <font color="Maroon" title="c1">A1</font>,<font color="Maroon" title="c3">B1</font>,<font color="Olive" title="b2">Bo</font>,<font color="Olive" title="b3">yo1</font> or  not <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="petri_2.html#K1" title="PETRI_2:func.1">cylinder0</a> <font color="Maroon" title="c2">A2</font>,<font color="Maroon" title="c4">B2</font>,<font color="Olive" title="b2">Bo</font>,<font color="Olive" title="b4">yo2</font> ) ) ) or <font color="Maroon" title="c5">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F2</font> )</span><br/>

<b>assume </b><a NAME="E2:13_1_1"/><i><font color="Green" title="E13">AS2</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> <font color="Maroon" title="c1">A1</font>,<font color="Maroon" title="c3">B1</font> holds <br/> ex <font color="Olive" title="b2">Bo</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">B1</font> ex <font color="Olive" title="b3">yo1</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b2">Bo</font>,<font color="Maroon" title="c1">A1</font> ex <font color="Olive" title="b4">yo2</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b2">Bo</font>,<font color="Maroon" title="c2">A2</font> st <br/>( <font color="Olive" title="b2">Bo</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b3">yo1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b4">yo2</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="petri_2.html#K1" title="PETRI_2:func.1">cylinder0</a> <font color="Maroon" title="c1">A1</font>,<font color="Maroon" title="c3">B1</font>,<font color="Olive" title="b2">Bo</font>,<font color="Olive" title="b3">yo1</font> &amp; <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="petri_2.html#K1" title="PETRI_2:func.1">cylinder0</a> <font color="Maroon" title="c2">A2</font>,<font color="Maroon" title="c4">B2</font>,<font color="Olive" title="b2">Bo</font>,<font color="Olive" title="b4">yo2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F2</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> <font color="Maroon" title="c1">A1</font>,<font color="Maroon" title="c3">B1</font> implies <font color="Maroon" title="c5">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> )</span><br/><b>assume </b><a NAME="E1:13_1_1_1"/><i><font color="Green" title="E14">AS3</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> <font color="Maroon" title="c1">A1</font>,<font color="Maroon" title="c3">B1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span><br/><b>consider </b><font color="Maroon" title="c8">Bo1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">B1</font>, <font color="Maroon" title="c9">yo11</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c8">Bo1</font>,<font color="Maroon" title="c1">A1</font>, <font color="Maroon" title="c10">yo21</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c8">Bo1</font>,<font color="Maroon" title="c2">A2</font><b> such that </b><br/><a NAME="E3:13_1_1_1"/><i><font color="Green" title="E15">P1</font></i>: 
( <font color="Maroon" title="c8">Bo1</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c9">yo11</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c10">yo21</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="petri_2.html#K1" title="PETRI_2:func.1">cylinder0</a> <font color="Maroon" title="c1">A1</font>,<font color="Maroon" title="c3">B1</font>,<font color="Maroon" title="c8">Bo1</font>,<font color="Maroon" title="c9">yo11</font> &amp; <font color="Maroon" title="c5">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="petri_2.html#K1" title="PETRI_2:func.1">cylinder0</a> <font color="Maroon" title="c2">A2</font>,<font color="Maroon" title="c4">B2</font>,<font color="Maroon" title="c8">Bo1</font>,<font color="Maroon" title="c10">yo21</font> )
 <b>by </b><i><a class="txt" href="#E1:13_1_1_1"><i><font color="Green" title="E14">AS3</font></i></a>, <a class="txt" href="#E1:13_1_1"><i><font color="Green" title="E12">AS1</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c11">Bo2</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">B1</font>, <font color="Maroon" title="c12">yo12</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c11">Bo2</font>,<font color="Maroon" title="c1">A1</font>, <font color="Maroon" title="c13">yo22</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c11">Bo2</font>,<font color="Maroon" title="c2">A2</font><b> such that </b><br/><a NAME="E5:13_1_1_1"/><i><font color="Green" title="E16">P2</font></i>: 
( <font color="Maroon" title="c11">Bo2</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c12">yo12</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c13">yo22</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="petri_2.html#K1" title="PETRI_2:func.1">cylinder0</a> <font color="Maroon" title="c1">A1</font>,<font color="Maroon" title="c3">B1</font>,<font color="Maroon" title="c11">Bo2</font>,<font color="Maroon" title="c12">yo12</font> &amp; <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="petri_2.html#K1" title="PETRI_2:func.1">cylinder0</a> <font color="Maroon" title="c2">A2</font>,<font color="Maroon" title="c4">B2</font>,<font color="Maroon" title="c11">Bo2</font>,<font color="Maroon" title="c13">yo22</font> )
 <b>by </b><i><a class="txt" href="#E1:13_1_1_1"><i><font color="Green" title="E14">AS3</font></i></a>, <a class="txt" href="#E2:13_1_1"><i><font color="Green" title="E13">AS2</font></i></a></i>;<br/><a NAME="E6:13_1_1_1"/><i><font color="Green" title="E17">P3</font></i>: 
( <font color="Maroon" title="c8">Bo1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Bo2</font> &amp; <font color="Maroon" title="c9">yo11</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c12">yo12</font> )
 <b>by </b><i><a class="ref" href="petri_2.html#T3" target="_self" title="PETRI_2:th.3">LM3</a>, <a class="txt" href="#E3:13_1_1_1"><i><font color="Green" title="E15">P1</font></i></a>, <a class="txt" href="#E5:13_1_1_1"><i><font color="Green" title="E16">P2</font></i></a></i>;<br/><b>thus </b><a NAME="E7:13_1_1_1"/>
<font color="Maroon" title="c5">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
 <b>by </b><i><a class="txt" href="#E3:13_1_1_1"><i><font color="Green" title="E15">P1</font></i></a>, <a class="txt" href="#E5:13_1_1_1"><i><font color="Green" title="E16">P2</font></i></a>, <a class="txt" href="#E6:13_1_1_1"><i><font color="Green" title="E17">P3</font></i></a></i>; <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>hence </b><a NAME="E4:13_1_1"/>
<font color="Maroon" title="c5">F1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">F2</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
