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

<span class="kw">let </span><font color="Maroon" title="c1">O</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">E</font> being 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> <br/> for <font color="Olive" title="b2">A</font> being  <a href="group_9.html#NM1" title="GROUP_9:NM.1">Action</a> of <font color="Maroon" title="c1">O</font>,<font color="Olive" title="b1">E</font><br/> for <font color="Olive" title="b3">X</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Olive" title="b1">E</font><br/> for <font color="Olive" title="b4">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">E</font>  st not <font color="Olive" title="b3">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  holds <br/>( <font color="Olive" title="b4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K1" title="GROUP_9:func.1">the_stable_subset_generated_by</a> <font color="Olive" title="b3">X</font>,<font color="Olive" title="b2">A</font> iff  ex <font color="Olive" title="b5">F</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">O</font> ex <font color="Olive" title="b6">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b3">X</font> st <span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Olive" title="b5">F</font>,<font color="Olive" title="b2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">a</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">E</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="group_9.html#NM1" title="GROUP_9:NM.1">Action</a> of <font color="Maroon" title="c1">O</font>,<font color="Maroon" title="c2">E</font><br/> for <font color="Olive" title="b2">X</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">E</font><br/> for <font color="Olive" title="b3">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">E</font>  st not <font color="Olive" title="b2">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  holds <br/>( <font color="Olive" title="b3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K1" title="GROUP_9:func.1">the_stable_subset_generated_by</a> <font color="Olive" title="b2">X</font>,<font color="Olive" title="b1">A</font> iff  ex <font color="Olive" title="b4">F</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">O</font> ex <font color="Olive" title="b5">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b2">X</font> st <span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Olive" title="b4">F</font>,<font color="Olive" title="b1">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">A</font> be   <a href="group_9.html#NM1" title="GROUP_9:NM.1">Action</a> of <font color="Maroon" title="c1">O</font>,<font color="Maroon" title="c2">E</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">X</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">E</font><br/> for <font color="Olive" title="b2">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">E</font>  st not <font color="Olive" title="b1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  holds <br/>( <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K1" title="GROUP_9:func.1">the_stable_subset_generated_by</a> <font color="Olive" title="b1">X</font>,<font color="Maroon" title="c3">A</font> iff  ex <font color="Olive" title="b3">F</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">O</font> ex <font color="Olive" title="b4">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">X</font> st <span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Olive" title="b3">F</font>,<font color="Maroon" title="c3">A</font></span>)</span> <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> <font color="Olive" title="b2">a</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">X</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">E</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">E</font>  st not <font color="Maroon" title="c4">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  holds <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K1" title="GROUP_9:func.1">the_stable_subset_generated_by</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">A</font> iff  ex <font color="Olive" title="b2">F</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">O</font> ex <font color="Olive" title="b3">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font> st <span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Olive" title="b2">F</font>,<font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">E</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not <font color="Maroon" title="c4">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies ( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K1" title="GROUP_9:func.1">the_stable_subset_generated_by</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">A</font> iff  ex <font color="Olive" title="b1">F</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">O</font> ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font> st <span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Olive" title="b1">F</font>,<font color="Maroon" title="c3">A</font></span>)</span> <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> <font color="Maroon" title="c5">a</font> ) )</span><br/>









<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">F</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">O</font> ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font> st <span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Olive" title="b1">F</font>,<font color="Maroon" title="c3">A</font></span>)</span> <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> $1;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">B</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">e</font> where <font color="Olive" title="b1">e</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">E</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">e</font>] </span>}</span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">B</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">e</font> where <font color="Olive" title="b1">e</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">E</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">e</font>] </span>}</span>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">E</font> <span class="kw">from</span> <span class="lab"><a class="ref" href="domain_1.html#S7" title="DOMAIN_1:sch.7">DOMAIN_1:sch 7</a>();<br/></span>
<span class="kw">assume </span><a NAME="E2:94"/><span class="lab"><font color="Green" title="E72">A1</font></span>: 
not <font color="Maroon" title="c4">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K1" title="GROUP_9:func.1">the_stable_subset_generated_by</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">A</font> iff  ex <font color="Olive" title="b1">F</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">O</font> ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font> st <span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Olive" title="b1">F</font>,<font color="Maroon" title="c3">A</font></span>)</span> <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> <font color="Maroon" title="c5">a</font> )</span><br/>

<div><span class="lab"><font color="Green" title="E73">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">Y</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">E</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">Y</font> <a href="group_9.html#R1" title="GROUP_9:pred.1">is_stable_under_the_action_of</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c4">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">Y</font> implies <font color="Maroon" title="c7">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">Y</font> )</span><br/><span class="kw">assume </span><a NAME="E1:94_1"/><span class="lab"><font color="Green" title="E74">A3</font></span>: 
<font color="Maroon" title="c8">Y</font> <a href="group_9.html#R1" title="GROUP_9:pred.1">is_stable_under_the_action_of</a> <font color="Maroon" title="c3">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">Y</font> implies <font color="Maroon" title="c7">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">Y</font> )</span><br/><span class="kw">assume </span><a NAME="E2:94_1"/><span class="lab"><font color="Green" title="E75">A4</font></span>: 
<font color="Maroon" title="c4">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">Y</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> implies <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font> )</span><br/><span class="kw">assume </span><a NAME="E1:94_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">e</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">E</font><span class="kw"> such that </span><br/><a NAME="E3:94_1_1"/><span class="lab"><font color="Green" title="E76">A5</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">e</font>
 <span class="kw">and </span><br/><a NAME="E4:94_1_1"/><span class="lab"><font color="Green" title="E77">A6</font></span>: 
 ex <font color="Olive" title="b1">F</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">O</font> ex <font color="Olive" title="b2">x9</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font> st <span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Olive" title="b1">F</font>,<font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">e</font>
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c11">F</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">O</font>, <font color="Maroon" title="c12">x9</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font><span class="kw"> such that </span><br/><a NAME="E6:94_1_1"/><span class="lab"><font color="Green" title="E78">A7</font></span>: 
<span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c11">F</font>,<font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:94_1_1"><span class="lab"><font color="Green" title="E77">A6</font></span></a></span>;<br/><a NAME="E7:94_1_1"/><span class="lab"><font color="Green" title="E79">A8</font></span>: 
<font color="Maroon" title="c12">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:94"><span class="lab"><font color="Green" title="E72">A1</font></span></a></span>;<br/><a NAME="E8:94_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E</font>
 ;<br/><a NAME="E9:94_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c11">F</font>,<font color="Maroon" title="c3">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E10:94_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A9</font></span>: 
<span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c11">F</font>,<font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c11">F</font>,<font color="Maroon" title="c3">A</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:94_1"><span class="lab"><font color="Green" title="E75">A4</font></span></a>, <a class="txt" href="#E7:94_1_1"><span class="lab"><font color="Green" title="E79">A8</font></span></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></span>;<br/><a NAME="E11:94_1_1"/>
<span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c11">F</font>,<font color="Maroon" title="c3">A</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:94_1"><span class="lab"><font color="Green" title="E74">A3</font></span></a>, <a class="txt" href="group_9.html#E84"><span class="lab"><font color="Green" title="E70">Lm30</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E12:94_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:94_1_1"><span class="lab"><font color="Green" title="E76">A5</font></span></a>, <a class="txt" href="#E6:94_1_1"><span class="lab"><font color="Green" title="E78">A7</font></span></a>, <a class="txt" href="#E10:94_1_1"><span class="lab"><font color="Green" title="E80">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:94_1"/>
<font color="Maroon" title="c7">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">o</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">O</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c2">E</font>  st <font color="Maroon" title="c8">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">O</font> &amp; <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">o</font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c7">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">B</font></span><br/><span class="kw">let </span><font color="Maroon" title="c9">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c2">E</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">O</font> &amp; <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">o</font> implies <font color="Olive">b<sub>2</sub></font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c7">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">B</font> )</span><br/><span class="kw">assume </span><a NAME="E1:94_2"/><span class="lab"><font color="Green" title="E74">A10</font></span>: 
<font color="Maroon" title="c8">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">O</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">o</font> implies <font color="Olive">b<sub>2</sub></font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c7">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">B</font> )</span><br/><span class="kw">assume </span><a NAME="E2:94_2"/><span class="lab"><font color="Green" title="E75">A11</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">o</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>2</sub></font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c7">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">B</font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="94_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:94_2_1"/>
( <font color="Maroon" title="c1">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c1">O</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="94_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:94_2_1_1"/>
<font color="Maroon" title="c1">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>2</sub></font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c7">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">B</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:94_2_1_1"/>
<font color="Maroon" title="c9">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c7">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:94_2"><span class="lab"><font color="Green" title="E74">A10</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="94_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:94_2_1_2"/><span class="lab"><font color="Green" title="E76">A12</font></span>: 
<font color="Maroon" title="c1">O</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>2</sub></font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c7">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">B</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_2_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c10">o</font> = <font color="Maroon" title="c8">o</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">O</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">F99</font> = <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c10">o</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">O</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:94_2_1_2"><span class="lab"><font color="Green" title="E76">A12</font></span></a>, <a class="ref" href="finseq_1.html#T95" title="FINSEQ_1:th.95">FINSEQ_1:95</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c12">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c7">B</font> implies <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> )</span><br/><span class="kw">assume </span><a NAME="E3:94_2_1_2_1"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c7">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:94_2_1_2_1"/><span class="lab"><font color="Green" title="E77">A13</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">f</font>
 <span class="kw">and </span><br/><a NAME="E6:94_2_1_2_1"/><span class="lab"><font color="Green" title="E78">A14</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 <span class="kw">and </span><br/><a NAME="E7:94_2_1_2_1"/><span class="lab"><font color="Green" title="E79">A15</font></span>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></span>;<br/><a NAME="E8:94_2_1_2_1"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:94_2_1_2_1"><span class="lab"><font color="Green" title="E77">A13</font></span></a>, <a class="txt" href="#E7:94_2_1_2_1"><span class="lab"><font color="Green" title="E79">A15</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">e</font> = <font color="Maroon" title="c12">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">E</font> ;<br/><span class="kw">consider </span><font color="Maroon" title="c15">e9</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">E</font><span class="kw"> such that </span><br/><a NAME="E11:94_2_1_2_1"/><span class="lab"><font color="Green" title="E80">A16</font></span>: 
<font color="Maroon" title="c15">e9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font>
 <span class="kw">and </span><br/><a NAME="E12:94_2_1_2_1"/><span class="lab"><font color="Green" title="E81">A17</font></span>: 
 ex <font color="Olive" title="b1">F9</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">O</font> ex <font color="Olive" title="b2">x9</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font> st <span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Olive" title="b1">F9</font>,<font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">e9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:94_2_1_2_1"><span class="lab"><font color="Green" title="E78">A14</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c16">F9</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">O</font>, <font color="Maroon" title="c17">x9</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font><span class="kw"> such that </span><br/><a NAME="E14:94_2_1_2_1"/><span class="lab"><font color="Green" title="E82">A18</font></span>: 
<span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c16">F9</font>,<font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">e9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:94_2_1_2_1"><span class="lab"><font color="Green" title="E81">A17</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c18">F</font> = <font color="Maroon" title="c11">F99</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">F9</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">O</font> ;<br/><a NAME="E16:94_2_1_2_1"/>
<font color="Maroon" title="c17">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:94"><span class="lab"><font color="Green" title="E72">A1</font></span></a></span>;<br/><a NAME="E17:94_2_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E</font>
 ;<br/><a NAME="E18:94_2_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A19</font></span>: 
<font color="Maroon" title="c17">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c16">F9</font>,<font color="Maroon" title="c3">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c18">F</font>,<font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x9</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c11">F99</font>,<font color="Maroon" title="c3">A</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c16">F9</font>,<font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:94_2_1_2"><span class="lab"><font color="Green" title="E76">A12</font></span></a>, <a class="txt" href="group_9.html#E83"><span class="lab"><font color="Green" title="E69">Lm29</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c11">F99</font>,<font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c16">F9</font>,<font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:94_2_1_2_1"><span class="lab"><font color="Green" title="E83">A19</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
<br/>.= 
<font color="Maroon" title="c14">e</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:94_2"><span class="lab"><font color="Green" title="E75">A11</font></span></a>, <a class="txt" href="#E1:94_2_1_2"><span class="lab"><font color="Green" title="E76">A12</font></span></a>, <a class="txt" href="#E7:94_2_1_2_1"><span class="lab"><font color="Green" title="E79">A15</font></span></a>, <a class="txt" href="#E11:94_2_1_2_1"><span class="lab"><font color="Green" title="E80">A16</font></span></a>, <a class="txt" href="#E14:94_2_1_2_1"><span class="lab"><font color="Green" title="E82">A18</font></span></a>, <a class="txt" href="group_9.html#E80"><span class="lab"><font color="Green" title="E66">Lm26</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E20:94_2_1_2_1"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:94_2_1_2"/>
<font color="Maroon" title="c9">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c7">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">B</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E5:94"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A20</font></span>: 
<font color="Maroon" title="c7">B</font> <a href="group_9.html#R1" title="GROUP_9:pred.1">is_stable_under_the_action_of</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D1" target="_self" title="GROUP_9:def.1">Def1</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_3"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c8">F</font> =  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">O</font>;<br/><span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> implies <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> )</span><br/><span class="kw">assume </span><a NAME="E1:94_3"/><span class="lab"><font color="Green" title="E75">A21</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">e</font> = <font color="Maroon" title="c9">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">E</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">x9</font> = <font color="Maroon" title="c10">e</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:94_3"><span class="lab"><font color="Green" title="E75">A21</font></span></a></span>;<br/><a NAME="E4:94_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">O</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><span class="kw">then </span><span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">O</font></span>)</span>,<font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> = 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c2">E</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D3" target="_self" title="GROUP_9:def.3">Def3</a></span>
<br/>.= 
<font color="Maroon" title="c9">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:94_3"><span class="lab"><font color="Green" title="E75">A21</font></span></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></span>
;<br/><a NAME="E6:94_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">O</font></span>)</span>,<font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">e</font>
 ;<br/><span class="kw">hence </span><a NAME="E7:94_3"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E7:94"/><span class="kw">then </span>
<font color="Maroon" title="c4">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E8:94"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A22</font></span>: 
<font color="Maroon" title="c7">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K1" title="GROUP_9:func.1">the_stable_subset_generated_by</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:94"><span class="lab"><font color="Green" title="E74">A20</font></span></a>, <a class="txt" href="#E3:94"><span class="lab"><font color="Green" title="E73">A2</font></span></a>, <a class="ref" href="group_9.html#D2" target="_self" title="GROUP_9:def.2">Def2</a></span>;<br/>
<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">F</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">O</font> ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font> st <span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Olive" title="b1">F</font>,<font color="Maroon" title="c3">A</font></span>)</span> <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> <font color="Maroon" title="c5">a</font> implies <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K1" title="GROUP_9:func.1">the_stable_subset_generated_by</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">A</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:94_4"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K1" title="GROUP_9:func.1">the_stable_subset_generated_by</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">F</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">O</font> ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font> st <span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Olive" title="b1">F</font>,<font color="Maroon" title="c3">A</font></span>)</span> <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> <font color="Maroon" title="c5">a</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">e</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">E</font><span class="kw"> such that </span><br/><a NAME="E3:94_4"/><span class="lab"><font color="Green" title="E76">A23</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">e</font>
 <span class="kw">and </span><br/><a NAME="E4:94_4"/><span class="lab"><font color="Green" title="E77">A24</font></span>: 
 ex <font color="Olive" title="b1">F</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">O</font> ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font> st <span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Olive" title="b1">F</font>,<font color="Maroon" title="c3">A</font></span>)</span> <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> <font color="Maroon" title="c8">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:94"><span class="lab"><font color="Green" title="E75">A22</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">F</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">O</font>, <font color="Maroon" title="c10">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font><span class="kw"> such that </span><br/><a NAME="E6:94_4"/><span class="lab"><font color="Green" title="E78">A25</font></span>: 
<span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c9">F</font>,<font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:94_4"><span class="lab"><font color="Green" title="E77">A24</font></span></a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c11">F</font> = <font color="Maroon" title="c9">F</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font> st <span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c11">F</font>,<font color="Maroon" title="c3">A</font></span>)</span> <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> <font color="Maroon" title="c5">a</font></span><br/><span class="kw">take </span><font color="Maroon" title="c12">x</font> = <font color="Maroon" title="c10">x</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c11">F</font>,<font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font></span><br/><span class="kw">thus </span><a NAME="E7:94_4"/>
<span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c11">F</font>,<font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:94_4"><span class="lab"><font color="Green" title="E76">A23</font></span></a>, <a class="txt" href="#E6:94_4"><span class="lab"><font color="Green" title="E78">A25</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">given </span><font color="Maroon" title="c8">F</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">O</font>, <font color="Maroon" title="c9">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font><span class="kw"> such that </span><a NAME="E11:94"/><span class="lab"><font color="Green" title="E76">A26</font></span>: 
<span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K1" title="GROUP_9:func.1">the_stable_subset_generated_by</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">A</font></span><br/>

<span class="kw">thus </span><a NAME="E12:94"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K1" title="GROUP_9:func.1">the_stable_subset_generated_by</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:94"><span class="lab"><font color="Green" title="E75">A22</font></span></a>, <a class="txt" href="#E11:94"><span class="lab"><font color="Green" title="E76">A26</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
