<?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">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b3">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="b4">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#K50" title="MODELC_1:func.50">CTLModel</a> <font color="Olive" title="b1">R</font>,<font color="Olive" title="b3">BASSIGN</font></span>)</span> holds <br/> ( <font color="Olive" title="b2">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K25" title="MODELC_1:func.25">'not'</a> <font color="Olive" title="b4">f</font> iff <font color="Olive" title="b2">s</font> <a href="modelc_1.html#NR4" title="MODELC_1:NR.4">|/=</a> <font color="Olive" title="b4">f</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">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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">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#K50" title="MODELC_1:func.50">CTLModel</a> <font color="Maroon" title="c2">R</font>,<font color="Olive" title="b2">BASSIGN</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K25" title="MODELC_1:func.25">'not'</a> <font color="Olive" title="b3">f</font> iff <font color="Olive" title="b1">s</font> <a href="modelc_1.html#NR4" title="MODELC_1:NR.4">|/=</a> <font color="Olive" title="b3">f</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">s</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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"> 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">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#K50" title="MODELC_1:func.50">CTLModel</a> <font color="Maroon" title="c2">R</font>,<font color="Olive" title="b1">BASSIGN</font></span>)</span> holds <br/> ( <font color="Maroon" title="c3">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K25" title="MODELC_1:func.25">'not'</a> <font color="Olive" title="b2">f</font> iff <font color="Maroon" title="c3">s</font> <a href="modelc_1.html#NR4" title="MODELC_1:NR.4">|/=</a> <font color="Olive" title="b2">f</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">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">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#K50" title="MODELC_1:func.50">CTLModel</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c4">BASSIGN</font></span>)</span> holds <br/> ( <font color="Maroon" title="c3">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K25" title="MODELC_1:func.25">'not'</a> <font color="Olive" title="b1">f</font> iff <font color="Maroon" title="c3">s</font> <a href="modelc_1.html#NR4" title="MODELC_1:NR.4">|/=</a> <font color="Olive" title="b1">f</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">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#K50" title="MODELC_1:func.50">CTLModel</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c4">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="c3">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K25" title="MODELC_1:func.25">'not'</a> <font color="Maroon" title="c5">f</font> iff <font color="Maroon" title="c3">s</font> <a href="modelc_1.html#NR4" title="MODELC_1:NR.4">|/=</a> <font color="Maroon" title="c5">f</font> )</span><br/>









<a NAME="E1:125"/><span class="lab"><font color="Green" title="E99">A1</font></span>: 
 <a href="modelc_1.html#K25" title="MODELC_1:func.25">'not'</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_1.html#K34" title="MODELC_1:func.34">Not_0</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c1">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#D43" target="_self" title="MODELC_1:def.43">Def43</a></span>;<br/>
<a NAME="E2:125"/><span class="lab"><font color="Green" title="E100">A2</font></span>: 
( <font color="Maroon" title="c3">s</font> <a href="modelc_1.html#NR4" title="MODELC_1:NR.4">|/=</a> <font color="Maroon" title="c5">f</font> implies <font color="Maroon" title="c3">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K25" title="MODELC_1:func.25">'not'</a> <font color="Maroon" title="c5">f</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:125_1"/>
<font color="Maroon" title="c3">s</font> <a href="modelc_1.html#NR4" title="MODELC_1:NR.4">|/=</a> <font color="Maroon" title="c5">f</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="c3">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K25" title="MODELC_1:func.25">'not'</a> <font color="Maroon" title="c5">f</font></span><br/>

<a NAME="E2:125_1"/><span class="kw">then </span>
not <span class="p1">(<span class="default"><a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">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#D59" target="_self" title="MODELC_1:def.59">Def59</a></span>;<br/>
<a NAME="E3:125_1"/><span class="kw">then </span>
not  <a href="modelc_1.html#K4" title="MODELC_1:func.4">Castboolean</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</font></span>)</span> <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#D4" target="_self" title="MODELC_1:def.4">Def4</a></span>;<br/>
<a NAME="E4:125_1"/><span class="kw">then </span>
 <a href="modelc_1.html#K4" title="MODELC_1:func.4">Castboolean</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboolean.html#D3" title="XBOOLEAN:def.3">XBOOLEAN:def 3</a></span>;<br/>
<a NAME="E5:125_1"/><span class="kw">then </span>
 <a href="xboolean.html#K3" title="XBOOLEAN:func.3">'not'</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K4" title="MODELC_1:func.4">Castboolean</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 
;<br/>
<a NAME="E6:125_1"/><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#K25" title="MODELC_1:func.25">'not'</a> <font color="Maroon" title="c5">f</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="Maroon" title="c3">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="txt" href="#E1:125"><span class="lab"><font color="Green" title="E99">A1</font></span></a>, <a class="ref" href="modelc_1.html#D42" target="_self" title="MODELC_1:def.42">Def42</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:125_1"/>
<font color="Maroon" title="c3">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K25" title="MODELC_1:func.25">'not'</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#D59" target="_self" title="MODELC_1:def.59">Def59</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>
<a NAME="E3:125"/>
( <font color="Maroon" title="c3">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K25" title="MODELC_1:func.25">'not'</a> <font color="Maroon" title="c5">f</font> implies <font color="Maroon" title="c3">s</font> <a href="modelc_1.html#NR4" title="MODELC_1:NR.4">|/=</a> <font color="Maroon" title="c5">f</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:125_2"/>
<font color="Maroon" title="c3">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K25" title="MODELC_1:func.25">'not'</a> <font color="Maroon" title="c5">f</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="c3">s</font> <a href="modelc_1.html#NR4" title="MODELC_1:NR.4">|/=</a> <font color="Maroon" title="c5">f</font></span><br/>

<a NAME="E2:125_2"/><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#K34" title="MODELC_1:func.34">Not_0</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c1">S</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="Maroon" title="c3">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="txt" href="#E1:125"><span class="lab"><font color="Green" title="E99">A1</font></span></a>, <a class="ref" href="modelc_1.html#D59" target="_self" title="MODELC_1:def.59">Def59</a></span>;<br/>
<a NAME="E3:125_2"/><span class="kw">then </span>
 <a href="xboolean.html#K3" title="XBOOLEAN:func.3">'not'</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K4" title="MODELC_1:func.4">Castboolean</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</font></span>)</span></span>)</span> <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#D42" target="_self" title="MODELC_1:def.42">Def42</a></span>;<br/>
<a NAME="E4:125_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#D4" target="_self" title="MODELC_1:def.4">Def4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:125_2"/>
<font color="Maroon" title="c3">s</font> <a href="modelc_1.html#NR4" title="MODELC_1:NR.4">|/=</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#D59" target="_self" title="MODELC_1:def.59">Def59</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="E4:125"/>
( <font color="Maroon" title="c3">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K25" title="MODELC_1:func.25">'not'</a> <font color="Maroon" title="c5">f</font> iff <font color="Maroon" title="c3">s</font> <a href="modelc_1.html#NR4" title="MODELC_1:NR.4">|/=</a> <font color="Maroon" title="c5">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:125"><span class="lab"><font color="Green" title="E100">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>
