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<b>let </b><font color="Maroon" title="c1">A1</font>, <font color="Maroon" title="c2">A2</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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"> for <font color="Olive" title="b1">B1</font>, <font color="Olive" title="b2">B2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   ex <font color="Olive" title="b3">G</font> being  <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="c2">A2</font>,<font color="Olive" title="b2">B2</font>, <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> <font color="Maroon" title="c1">A1</font>,<font color="Olive" title="b1">B1</font> st <br/>for <font color="Olive" title="b4">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b4">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="c2">A2</font>,<font color="Olive" title="b2">B2</font> holds <br/> ex <font color="Olive" title="b5">Bo2</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Olive" title="b2">B2</font> ex <font color="Olive" title="b6">Bo1</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Olive" title="b1">B1</font> ex <font color="Olive" title="b7">yo1</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b6">Bo1</font>,<font color="Maroon" title="c1">A1</font> ex <font color="Olive" title="b8">yo2</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b5">Bo2</font>,<font color="Maroon" title="c2">A2</font> st <br/>( <font color="Olive" title="b6">Bo1</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b5">Bo2</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b6">Bo1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b5">Bo2</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><font color="Olive" title="b8">yo2</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c1">A1</font></span>)</span> &amp; <font color="Olive" title="b7">yo1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">yo2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b6">Bo1</font> &amp; <font color="Olive" title="b4">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="Olive" title="b2">B2</font>,<font color="Olive" title="b5">Bo2</font>,<font color="Olive" title="b8">yo2</font> &amp; <font color="Olive" title="b3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">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="Olive" title="b1">B1</font>,<font color="Olive" title="b6">Bo1</font>,<font color="Olive" title="b7">yo1</font> )</span><br/><b>let </b><font color="Maroon" title="c3">B1</font>, <font color="Maroon" title="c4">B2</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">  ex <font color="Olive" title="b1">G</font> being  <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="c2">A2</font>,<font color="Maroon" title="c4">B2</font>, <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> st <br/>for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">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="c2">A2</font>,<font color="Maroon" title="c4">B2</font> holds <br/> ex <font color="Olive" title="b3">Bo2</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c4">B2</font> ex <font color="Olive" title="b4">Bo1</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="b5">yo1</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b4">Bo1</font>,<font color="Maroon" title="c1">A1</font> ex <font color="Olive" title="b6">yo2</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b3">Bo2</font>,<font color="Maroon" title="c2">A2</font> st <br/>( <font color="Olive" title="b4">Bo1</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b3">Bo2</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b4">Bo1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b3">Bo2</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">yo2</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c1">A1</font></span>)</span> &amp; <font color="Olive" title="b5">yo1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">yo2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b4">Bo1</font> &amp; <font color="Olive" title="b2">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="b3">Bo2</font>,<font color="Olive" title="b6">yo2</font> &amp; <font color="Olive" title="b1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">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="b4">Bo1</font>,<font color="Olive" title="b5">yo1</font> )</span><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">Bo2</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c4">B2</font> ex <font color="Olive" title="b2">Bo1</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">Bo1</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="b1">Bo2</font>,<font color="Maroon" title="c2">A2</font> st <br/>( <font color="Olive" title="b2">Bo1</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b1">Bo2</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b2">Bo1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b1">Bo2</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">yo2</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c1">A1</font></span>)</span> &amp; <font color="Olive" title="b3">yo1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">yo2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b2">Bo1</font> &amp; $1 <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="b1">Bo2</font>,<font color="Olive" title="b4">yo2</font> &amp; $2 <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">Bo1</font>,<font color="Olive" title="b3">yo1</font> );<br/>
<div><i><font color="Green" title="E11">P1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">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="c5">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="c2">A2</font>,<font color="Maroon" title="c4">B2</font> implies  ex <font color="Olive" title="b1">D1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">D1</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; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">D1</font>] ) )</span><br/><b>assume </b><a NAME="E1:12_1"/>
<font color="Maroon" title="c5">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="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">  ex <font color="Olive" title="b1">D1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">D1</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; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">D1</font>] )</span><br/><a NAME="E2:12_1"/><b>then </b>
 ex <font color="Olive" title="b1">D</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c4">B2</font>,<font color="Maroon" title="c2">A2</font></span>)</span> st <br/>( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">D</font> &amp; <font color="Olive" title="b1">D</font> is    <a href="petri_2.html#M1" title="PETRI_2:mode.1">thin_cylinder</a> of <font color="Maroon" title="c2">A2</font>,<font color="Maroon" title="c4">B2</font> )
 ;<br/><b>then reconsider </b><font color="Maroon" title="c6">D2</font> = <font color="Maroon" title="c5">x</font> as    <a href="petri_2.html#M1" title="PETRI_2:mode.1">thin_cylinder</a> of <font color="Maroon" title="c2">A2</font>,<font color="Maroon" title="c4">B2</font> ;<br/><b>consider </b><font color="Maroon" title="c7">Bo2</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c4">B2</font>, <font color="Maroon" title="c8">yo2</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c7">Bo2</font>,<font color="Maroon" title="c2">A2</font><b> such that </b><br/><a NAME="E5:12_1"/><i><font color="Green" title="E12">P2</font></i>: 
( <font color="Maroon" title="c7">Bo2</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c6">D2</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="c7">Bo2</font>,<font color="Maroon" title="c8">yo2</font> )
 <b>by </b><i><a class="ref" href="petri_2.html#D2" target="_self" title="PETRI_2:def.2">Def21</a></i>;<br/><b>set </b><font color="Maroon" title="c9">Bo1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">Bo2</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">yo2</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c1">A1</font></span>)</span>;<br/><a NAME="E6:12_1"/><i><font color="Green" title="E13">XXX</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">Bo2</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">yo2</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c1">A1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">Bo2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><a NAME="E7:12_1"/>
<font color="Maroon" title="c3">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">Bo2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B1</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><a NAME="E8:12_1"/><b>then </b><i><font color="Green" title="E14">D13</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">Bo2</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">yo2</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c1">A1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B1</font>
 <b>by </b><i><a class="txt" href="#E6:12_1"><i><font color="Green" title="E13">XXX</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E9:12_1"/>
<font color="Maroon" title="c3">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">Bo2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Bo2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><a NAME="E10:12_1"/><b>then </b><i><font color="Green" title="E15">YYY</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">Bo2</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">yo2</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c1">A1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Bo2</font>
 <b>by </b><i><a class="txt" href="#E6:12_1"><i><font color="Green" title="E13">XXX</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>set </b><font color="Maroon" title="c10">yo1</font> = <font color="Maroon" title="c8">yo2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">Bo2</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">yo2</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c1">A1</font></span>)</span></span>)</span>;<br/><a NAME="E11:12_1"/><i><font color="Green" title="E16">ZZZ</font></i>: 
<font color="Maroon" title="c8">yo2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">Bo2</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">yo2</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c1">A1</font></span>)</span></span>)</span> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">Bo2</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">yo2</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c1">A1</font></span>)</span>,<font color="Maroon" title="c2">A2</font>
 <b>by </b><i><a class="txt" href="#E10:12_1"><i><font color="Green" title="E15">YYY</font></i></a>, <a class="ref" href="funct_2.html#T38" title="FUNCT_2:th.38">FUNCT_2:38</a></i>;<br/><a NAME="E12:12_1"/><i><font color="Green" title="E17">W1</font></i>: 
<font color="Maroon" title="c8">yo2</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">yo2</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c1">A1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A1</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T145" title="FUNCT_1:th.145">FUNCT_1:145</a></i>;<br/><a NAME="E13:12_1"/><i><font color="Green" title="E18">W2</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">yo2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">Bo2</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">yo2</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c1">A1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">yo2</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">Bo2</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">yo2</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c1">A1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T148" title="RELAT_1:th.148">RELAT_1:148</a></i>;<br/><a NAME="E14:12_1"/>
<font color="Maroon" title="c8">yo2</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">Bo2</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">yo2</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c1">A1</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">yo2</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">yo2</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c1">A1</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><a NAME="E15:12_1"/><b>then </b>
<font color="Maroon" title="c8">yo2</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">Bo2</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">yo2</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c1">A1</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A1</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a>, <a class="txt" href="#E12:12_1"><i><font color="Green" title="E17">W1</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c11">yo1</font> = <font color="Maroon" title="c8">yo2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">Bo2</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">yo2</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c1">A1</font></span>)</span></span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">Bo2</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">yo2</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c1">A1</font></span>)</span>,<font color="Maroon" title="c1">A1</font> <b>by </b><i><a class="ref" href="funct_2.html#T8" title="FUNCT_2:th.8">FUNCT_2:8</a>, <a class="txt" href="#E11:12_1"><i><font color="Green" title="E16">ZZZ</font></i></a>, <a class="txt" href="#E13:12_1"><i><font color="Green" title="E18">W2</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c12">D1</font> =  <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>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">Bo2</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">yo2</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c1">A1</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">yo1</font>;<br/><a NAME="E17:12_1"/><i><font color="Green" title="E19">XXX</font></i>: 
 <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>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">Bo2</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">yo2</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c1">A1</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">yo1</font> is    <a href="petri_2.html#M1" title="PETRI_2:mode.1">thin_cylinder</a> of <font color="Maroon" title="c1">A1</font>,<font color="Maroon" title="c3">B1</font>
 <b>by </b><i><a class="txt" href="#E8:12_1"><i><font color="Green" title="E14">D13</font></i></a>, <a class="txt" href="#E5:12_1"><i><font color="Green" title="E12">P2</font></i></a>, <a class="ref" href="petri_2.html#D2" target="_self" title="PETRI_2:def.2">Def21</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">D1</font> =  <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>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">Bo2</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">yo2</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c1">A1</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">yo1</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><b>take </b><font color="Maroon" title="c14">D1</font> = <font color="Maroon" title="c13">D1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c14">D1</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; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c14">D1</font>] )</span><br/><b>thus </b><a NAME="E19:12_1"/>
( <font color="Maroon" title="c14">D1</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; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c14">D1</font>] )
 <b>by </b><i><a class="txt" href="#E5:12_1"><i><font color="Green" title="E12">P2</font></i></a>, <a class="txt" href="#E17:12_1"><i><font color="Green" title="E19">XXX</font></i></a>, <a class="txt" href="#E8:12_1"><i><font color="Green" title="E14">D13</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>consider </b><font color="Maroon" title="c5">G</font> being   <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="c2">A2</font>,<font color="Maroon" title="c4">B2</font>, <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><b> such that </b><br/><a NAME="E3:12"/><i><font color="Green" title="E12">P2</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="c2">A2</font>,<font color="Maroon" title="c4">B2</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="#E1:12"><i><font color="Green" title="E11">P1</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c5">G</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="c2">A2</font>,<font color="Maroon" title="c4">B2</font> holds <br/> ex <font color="Olive" title="b2">Bo2</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c4">B2</font> ex <font color="Olive" title="b3">Bo1</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="b4">yo1</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b3">Bo1</font>,<font color="Maroon" title="c1">A1</font> ex <font color="Olive" title="b5">yo2</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b2">Bo2</font>,<font color="Maroon" title="c2">A2</font> st <br/>( <font color="Olive" title="b3">Bo1</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b2">Bo2</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b3">Bo1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">Bo2</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">yo2</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c1">A1</font></span>)</span> &amp; <font color="Olive" title="b4">yo1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">yo2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b3">Bo1</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="c2">A2</font>,<font color="Maroon" title="c4">B2</font>,<font color="Olive" title="b2">Bo2</font>,<font color="Olive" title="b5">yo2</font> &amp; <font color="Maroon" title="c5">G</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="c1">A1</font>,<font color="Maroon" title="c3">B1</font>,<font color="Olive" title="b3">Bo1</font>,<font color="Olive" title="b4">yo1</font> )</span><br/>

<b>thus </b><a NAME="E4:12"/>
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="c2">A2</font>,<font color="Maroon" title="c4">B2</font> holds <br/> ex <font color="Olive" title="b2">Bo2</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c4">B2</font> ex <font color="Olive" title="b3">Bo1</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="b4">yo1</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b3">Bo1</font>,<font color="Maroon" title="c1">A1</font> ex <font color="Olive" title="b5">yo2</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b2">Bo2</font>,<font color="Maroon" title="c2">A2</font> st <br/>( <font color="Olive" title="b3">Bo1</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b2">Bo2</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b3">Bo1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">Bo2</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">yo2</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c1">A1</font></span>)</span> &amp; <font color="Olive" title="b4">yo1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">yo2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b3">Bo1</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="c2">A2</font>,<font color="Maroon" title="c4">B2</font>,<font color="Olive" title="b2">Bo2</font>,<font color="Olive" title="b5">yo2</font> &amp; <font color="Maroon" title="c5">G</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="c1">A1</font>,<font color="Maroon" title="c3">B1</font>,<font color="Olive" title="b3">Bo1</font>,<font color="Olive" title="b4">yo1</font> )
 <b>by </b><i><a class="txt" href="#E3:12"><i><font color="Green" title="E12">P2</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/>


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