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<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">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_1.html#K21" title="MODELC_1:func.21">EvalSet</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">Kai</font>,<font color="Olive" title="b1">n</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">X</font> =  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a> ,the <a href="modelc_1.html#U5" title="MODELC_1:sel.5">Assignations</a> of <font color="Maroon" title="c1">V</font></span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">IT</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E2:61_1_1"/><span class="lab"><font color="Green" title="E53">A1</font></span>: 
for <font color="Olive" title="b1">p</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">IT</font> iff ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a> ,the <a href="modelc_1.html#U5" title="MODELC_1:sel.5">Assignations</a> of <font color="Maroon" title="c1">V</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></span>
<a NAME="E3:61_1_1"/>
not <font color="Maroon" title="c4">IT</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c5">p</font> =  <a href="modelc_1.html#K21" title="MODELC_1:func.21">EvalSet</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">Kai</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>
<a NAME="E1:61_1_1_1"/>
 <a href="modelc_1.html#K21" title="MODELC_1:func.21">EvalSet</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">Kai</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a> ,the <a href="modelc_1.html#U5" title="MODELC_1:sel.5">Assignations</a> of <font color="Maroon" title="c1">V</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <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="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K21" title="MODELC_1:func.21">EvalSet</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">Kai</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a> ,the <a href="modelc_1.html#U5" title="MODELC_1:sel.5">Assignations</a> of <font color="Maroon" title="c1">V</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:61_1_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K21" title="MODELC_1:func.21">EvalSet</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">Kai</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</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="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a> ,the <a href="modelc_1.html#U5" title="MODELC_1:sel.5">Assignations</a> of <font color="Maroon" title="c1">V</font></span><br/>

<a NAME="E2:61_1_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">h</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a> ,the <a href="modelc_1.html#U5" title="MODELC_1:sel.5">Assignations</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> &amp; <font color="Olive" title="b1">h</font> <a href="modelc_1.html#R2" title="MODELC_1:pred.2">is-PreEvaluation-for</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c2">Kai</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:61_1_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a> ,the <a href="modelc_1.html#U5" title="MODELC_1:sel.5">Assignations</a> of <font color="Maroon" title="c1">V</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</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="E2:61_1_1_1"/>
not <font color="Maroon" title="c4">IT</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_1_1"><span class="lab"><font color="Green" title="E53">A1</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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">IT</font> = <font color="Maroon" title="c4">IT</font> as  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/>
<span class="kw">take </span>
<font color="Maroon" title="c5">IT</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">p</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">IT</font> iff ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a> ,the <a href="modelc_1.html#U5" title="MODELC_1:sel.5">Assignations</a> of <font color="Maroon" title="c1">V</font></span>)</span> &amp;  ex <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_1.html#K21" title="MODELC_1:func.21">EvalSet</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">Kai</font>,<font color="Olive" title="b2">n</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E5:61_1_1"/>
for <font color="Olive" title="b1">p</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">IT</font> iff ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a> ,the <a href="modelc_1.html#U5" title="MODELC_1:sel.5">Assignations</a> of <font color="Maroon" title="c1">V</font></span>)</span> &amp;  ex <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_1.html#K21" title="MODELC_1:func.21">EvalSet</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">Kai</font>,<font color="Olive" title="b2">n</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_1_1"><span class="lab"><font color="Green" title="E53">A1</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/>


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