<?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="Red">::  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#K32" title="MODELC_1:func.32">ModelSP</a> <font color="Maroon" title="c1">S</font></span>)</span><br/> for <font color="Olive" title="b3">T</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font> holds   <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K54" title="MODELC_1:func.54">Tau</a> <font color="Olive" title="b3">T</font>,<font color="Olive" title="b1">R</font>,<font color="Olive" title="b2">BASSIGN</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">T</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="Red">::  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#K32" title="MODELC_1:func.32">ModelSP</a> <font color="Maroon" title="c1">S</font></span>)</span><br/> for <font color="Olive" title="b2">T</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font> holds   <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K54" title="MODELC_1:func.54">Tau</a> <font color="Olive" title="b2">T</font>,<font color="Maroon" title="c2">R</font>,<font color="Olive" title="b1">BASSIGN</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">T</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#K32" title="MODELC_1:func.32">ModelSP</a> <font color="Maroon" title="c1">S</font></span>)</span>; <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">T</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font> holds   <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K54" title="MODELC_1:func.54">Tau</a> <font color="Olive" title="b1">T</font>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">T</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">T</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K54" title="MODELC_1:func.54">Tau</a> <font color="Maroon" title="c4">T</font>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">T</font></span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">f</font> =  <a href="modelc_1.html#K54" title="MODELC_1:func.54">Tau</a> <font color="Maroon" title="c4">T</font>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">U</font> =  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K54" title="MODELC_1:func.54">Tau</a> <font color="Maroon" title="c4">T</font>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font></span>)</span>;<br/>
<a NAME="E1:153"/><span class="lab"><font color="Green" title="E115">A1</font></span>: 
 <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K54" title="MODELC_1:func.54">Tau</a> <font color="Maroon" title="c4">T</font>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> : <span class="p1">(<span class="default"><a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K54" title="MODELC_1:func.54">Tau</a> <font color="Maroon" title="c4">T</font>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font></span>)</span>,<font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="modelc_1.html#E137"><span class="lab"><font color="Green" title="E109">Lm46</font></span></a></span>;<br/>
<a NAME="E2:153"/>
for <font color="Olive" title="b1">s</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K54" title="MODELC_1:func.54">Tau</a> <font color="Maroon" title="c4">T</font>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font></span>)</span> iff <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">T</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="153_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">s</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="c7">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K54" title="MODELC_1:func.54">Tau</a> <font color="Maroon" title="c4">T</font>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font></span>)</span> iff <font color="Maroon" title="c7">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">T</font> )</span><br/>

<span class="kw">thus </span><a NAME="E1:153_1"/>
( <font color="Maroon" title="c7">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K54" title="MODELC_1:func.54">Tau</a> <font color="Maroon" title="c4">T</font>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font></span>)</span> implies <font color="Maroon" title="c7">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">T</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">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">T</font> implies <font color="Maroon" title="c7">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K54" title="MODELC_1:func.54">Tau</a> <font color="Maroon" title="c4">T</font>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font></span>)</span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="153_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:153_1_1"/>
<font color="Maroon" title="c7">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K54" title="MODELC_1:func.54">Tau</a> <font color="Maroon" title="c4">T</font>,<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="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">T</font></span><br/>

<a NAME="E2:153_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">t</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="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> &amp; <span class="p1">(<span class="default"><a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K54" title="MODELC_1:func.54">Tau</a> <font color="Maroon" title="c4">T</font>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font></span>)</span>,<font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:153"><span class="lab"><font color="Green" title="E115">A1</font></span></a></span>;<br/>
<a NAME="E3:153_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c4">T</font>,<font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#D64" target="_self" title="MODELC_1:def.64">Def64</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:153_1_1"/>
<font color="Maroon" title="c7">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#T42" title="FUNCT_3:th.42">FUNCT_3:42</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">assume </span><a NAME="E2:153_1"/><span class="lab"><font color="Green" title="E116">A2</font></span>: 
<font color="Maroon" title="c7">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">T</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">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K54" title="MODELC_1:func.54">Tau</a> <font color="Maroon" title="c4">T</font>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font></span>)</span></span><br/>

<span class="kw">then </span><span class="p1">(<span class="default"><a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K54" title="MODELC_1:func.54">Tau</a> <font color="Maroon" title="c4">T</font>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font></span>)</span>,<font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">s</font> = 
<span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c4">T</font>,<font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">s</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#D64" target="_self" title="MODELC_1:def.64">Def64</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:153_1"><span class="lab"><font color="Green" title="E116">A2</font></span></a>, <a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">FUNCT_3:def 3</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:153_1"/>
<font color="Maroon" title="c7">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K54" title="MODELC_1:func.54">Tau</a> <font color="Maroon" title="c4">T</font>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:153"><span class="lab"><font color="Green" title="E115">A1</font></span></a>, <a class="txt" href="#E2:153_1"><span class="lab"><font color="Green" title="E116">A2</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="E3:153"/>
 <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K54" title="MODELC_1:func.54">Tau</a> <font color="Maroon" title="c4">T</font>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</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>
