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<b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon">f</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon">g</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/>





<b>assume </b><a NAME="E1:35"/><i><font color="Green">E52</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp;  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">g</font> &amp;  <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="cqc_lang.html#K10">'not'</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">g</font> &amp;  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font> <a href="calcul_1.html#R10" target="_self">|=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">f</font> &amp;  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">g</font> <a href="calcul_1.html#R10" target="_self">|=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">g</font> )
 ;<br/>

<a NAME="E2:35"/><b>then </b><i><font color="Green">E53</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="calcul_1.html#D15" target="_self">CALCUL_1:def 15</a><br/><b>let </b><font color="Maroon">J</font> be    <a href="valuat_1.html#M1">interpretation</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">v</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2">Valuations_in</a> <font color="Maroon">A</font>;<br/>





<b>assume </b><a NAME="E3:35"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="calcul_1.html#R9" target="_self">|=</a>  <a href="calcul_1.html#K1" target="_self">Ant</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>

<div><i><font color="Green">E55</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">A</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">J</font> being    <a href="valuat_1.html#M1">interpretation</a> of <font color="Maroon">A</font>, <font color="Maroon">v</font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2">Valuations_in</a> <font color="Maroon">A</font><b> such that </b><a NAME="E1:35_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="calcul_1.html#R9" target="_self">|=</a>  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font>
 ;<br/><a NAME="E2:35_1"/>
( <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="valuat_1.html#R1">|=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">f</font> &amp; <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="valuat_1.html#R1">|=</a>  <a href="cqc_lang.html#K10">'not'</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">f</font></span>)</span> )
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E3:35_1"/>
contradiction
 <b>by </b><i><a class="ref" href="valuat_1.html#T28">VALUAT_1:28</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_2"/>
<font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="valuat_1.html#R1">|=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font></span>)</span>
 ;<br/><a NAME="E2:35_2"/><b>then </b>
<font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="calcul_1.html#R9" target="_self">|=</a>  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E3:35_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:35"/>
<font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="valuat_1.html#T28">VALUAT_1:28</a></i>;<br/>


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