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

<span class="kw">let </span><font color="Maroon" title="c1">S</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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">R</font> being   <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="relset_1.html#NM1" title="RELSET_1:NM.1">Relation</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font><br/>  for <font color="Olive" title="b2">BASSIGN</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K31" title="MODELC_1:func.31">ModelSP</a> <font color="Maroon" title="c1">S</font></span>)</span><br/>  for <font color="Olive" title="b3">f</font> being   <a href="modelc_1.html#NM2" title="MODELC_1:NM.2">Assign</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K49" title="MODELC_1:func.49">BASSModel</a> (<font color="Olive" title="b1">R</font>,<font color="Olive" title="b2">BASSIGN</font>)</span>)</span> holds   <a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K26" title="MODELC_1:func.26">EX</a> <font color="Olive" title="b3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_1.html#K51" title="MODELC_1:func.51">Pred</a> (<span class="p1">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Olive" title="b3">f</font></span>)</span>,<font color="Olive" title="b1">R</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">R</font> be   <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="relset_1.html#NM1" title="RELSET_1:NM.1">Relation</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">BASSIGN</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K31" title="MODELC_1:func.31">ModelSP</a> <font color="Maroon" title="c1">S</font></span>)</span><br/>  for <font color="Olive" title="b2">f</font> being   <a href="modelc_1.html#NM2" title="MODELC_1:NM.2">Assign</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K49" title="MODELC_1:func.49">BASSModel</a> (<font color="Maroon" title="c2">R</font>,<font color="Olive" title="b1">BASSIGN</font>)</span>)</span> holds   <a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K26" title="MODELC_1:func.26">EX</a> <font color="Olive" title="b2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_1.html#K51" title="MODELC_1:func.51">Pred</a> (<span class="p1">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Olive" title="b2">f</font></span>)</span>,<font color="Maroon" title="c2">R</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">BASSIGN</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K31" title="MODELC_1:func.31">ModelSP</a> <font color="Maroon" title="c1">S</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="modelc_1.html#NM2" title="MODELC_1:NM.2">Assign</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K49" title="MODELC_1:func.49">BASSModel</a> (<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font>)</span>)</span> holds   <a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K26" title="MODELC_1:func.26">EX</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_1.html#K51" title="MODELC_1:func.51">Pred</a> (<span class="p1">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Olive" title="b1">f</font></span>)</span>,<font color="Maroon" title="c2">R</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font> be   <a href="modelc_1.html#NM2" title="MODELC_1:NM.2">Assign</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K49" title="MODELC_1:func.49">BASSModel</a> (<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font>)</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K26" title="MODELC_1:func.26">EX</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_1.html#K51" title="MODELC_1:func.51">Pred</a> (<span class="p1">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c2">R</font>)</span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">g</font> =  <a href="modelc_1.html#K26" title="MODELC_1:func.26">EX</a> <font color="Maroon" title="c4">f</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">H</font> =  <a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">f</font>;<br/>
<a NAME="E1:182"/><span class="lab"><font color="Green" title="E132">A1</font></span>: 
 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="modelc_1.html#K51" title="MODELC_1:func.51">Pred</a> (<span class="p1">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c2">R</font>) holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K26" title="MODELC_1:func.26">EX</a> <font color="Maroon" title="c4">f</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="182_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K51" title="MODELC_1:func.51">Pred</a> (<span class="p1">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c2">R</font>) implies <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K26" title="MODELC_1:func.26">EX</a> <font color="Maroon" title="c4">f</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:182_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K51" title="MODELC_1:func.51">Pred</a> (<span class="p1">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c2">R</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K26" title="MODELC_1:func.26">EX</a> <font color="Maroon" title="c4">f</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">s</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font><span class="kw"> such that </span><br/><a NAME="E3:182_1"/><span class="lab"><font color="Green" title="E133">A2</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font>
 <span class="kw">and </span><br/><a NAME="E4:182_1"/><span class="lab"><font color="Green" title="E134">A3</font></span>: 
 ex <font color="Olive" title="b1">s1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Olive" title="b1">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">f</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">s</font>,<font color="Olive" title="b1">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> )
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">s1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font><span class="kw"> such that </span><br/><a NAME="E6:182_1"/><span class="lab"><font color="Green" title="E135">A4</font></span>: 
<font color="Maroon" title="c9">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">and </span><br/><a NAME="E7:182_1"/><span class="lab"><font color="Green" title="E136">A5</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">s</font>,<font color="Maroon" title="c9">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:182_1"><span class="lab"><font color="Green" title="E134">A3</font></span></a></span>;<br/>
<a NAME="E8:182_1"/>
 ex <font color="Olive" title="b1">s2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Maroon" title="c9">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s2</font> &amp; <font color="Olive" title="b1">s2</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:182_1"><span class="lab"><font color="Green" title="E135">A4</font></span></a></span>;<br/>
<a NAME="E9:182_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K26" title="MODELC_1:func.26">EX</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:182_1"><span class="lab"><font color="Green" title="E136">A5</font></span></a>, <a class="ref" href="modelc_1.html#T28" target="_self" title="MODELC_1:th.28">Th28</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:182_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K26" title="MODELC_1:func.26">EX</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:182_1"><span class="lab"><font color="Green" title="E133">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:182"/>
 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="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K26" title="MODELC_1:func.26">EX</a> <font color="Maroon" title="c4">f</font></span>)</span> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K51" title="MODELC_1:func.51">Pred</a> (<span class="p1">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c2">R</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="182_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K26" title="MODELC_1:func.26">EX</a> <font color="Maroon" title="c4">f</font></span>)</span> implies <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K51" title="MODELC_1:func.51">Pred</a> (<span class="p1">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c2">R</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:182_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K26" title="MODELC_1:func.26">EX</a> <font color="Maroon" title="c4">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K51" title="MODELC_1:func.51">Pred</a> (<span class="p1">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c2">R</font>)</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">s</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font><span class="kw"> such that </span><br/><a NAME="E3:182_2"/><span class="lab"><font color="Green" title="E133">A6</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font>
 <span class="kw">and </span><br/><a NAME="E4:182_2"/><span class="lab"><font color="Green" title="E134">A7</font></span>: 
<font color="Maroon" title="c8">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K26" title="MODELC_1:func.26">EX</a> <font color="Maroon" title="c4">f</font>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">s1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font><span class="kw"> such that </span><br/><a NAME="E6:182_2"/><span class="lab"><font color="Green" title="E135">A8</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">s</font>,<font color="Maroon" title="c9">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 <span class="kw">and </span><br/><a NAME="E7:182_2"/><span class="lab"><font color="Green" title="E136">A9</font></span>: 
<font color="Maroon" title="c9">s1</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:182_2"><span class="lab"><font color="Green" title="E134">A7</font></span></a>, <a class="ref" href="modelc_1.html#T28" target="_self" title="MODELC_1:th.28">Th28</a></span>;<br/>
<a NAME="E8:182_2"/>
<font color="Maroon" title="c9">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:182_2"><span class="lab"><font color="Green" title="E136">A9</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:182_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K51" title="MODELC_1:func.51">Pred</a> (<span class="p1">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c2">R</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:182_2"><span class="lab"><font color="Green" title="E133">A6</font></span></a>, <a class="txt" href="#E6:182_2"><span class="lab"><font color="Green" title="E135">A8</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:182"/>
 <a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K26" title="MODELC_1:func.26">EX</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_1.html#K51" title="MODELC_1:func.51">Pred</a> (<span class="p1">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c2">R</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:182"><span class="lab"><font color="Green" title="E132">A1</font></span></a>, <a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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