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

<b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="zfmisc_1.html#K1">bool</a> <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/>

<b>assume </b><a NAME="E1:17"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">n</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> holds <br/>( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> is <a href="henmodel.html#V1" target="_self">Consistent</a> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_1"/>
not  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span> is <a href="henmodel.html#V1" target="_self">Consistent</a>
 ;<br/><b>then consider </b><font color="Maroon">Z</font> being   <a href="subset_1.html#NM2">Subset</a> of <a href="cqc_lang.html#K7">CQC-WFF</a> <b> such that </b><br/><a NAME="E3:17_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">Z</font> is <a href="finset_1.html#V1">finite</a> &amp; not <font color="Maroon">Z</font> is <a href="henmodel.html#V1" target="_self">Consistent</a> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">p</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> <b> such that </b><br/><a NAME="E5:17_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">Z</font> <a href="henmodel.html#R1" target="_self">|-</a> <font color="Maroon">p</font> &amp; <font color="Maroon">Z</font> <a href="henmodel.html#R1" target="_self">|-</a>  <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E6:17_1"/>
for <font color="Olive">n</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> holds <br/><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:17_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> as   <a href="subset_1.html#NM2">Subset</a> of <a href="cqc_lang.html#K7">CQC-WFF</a>  ;<br/><b>consider </b><font color="Maroon">f1</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> <b> such that </b><br/><a NAME="E11:17_1"/><i><font color="Green">E51</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font> &amp;  <a href="calcul_1.html#R4">|-</a> <font color="Maroon">f1</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">f2</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> <b> such that </b><br/><a NAME="E13:17_1"/><i><font color="Green">E52</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font> &amp;  <a href="calcul_1.html#R4">|-</a> <font color="Maroon">f2</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i>, </i>;<br/><a NAME="E14:17_1"/><i><font color="Green">E53</font></i>: 
(  <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f2</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="finseq_1.html#K12">*&gt;</a></span> &amp;  <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f2</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i>, , , <a class="ref" href="calcul_2.html#T20">CALCUL_2:20</a></i>;<br/><a NAME="E15:17_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f2</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/><a NAME="E16:17_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><a NAME="E17:17_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E18:17_1"/><b>then </b>
( <font color="Maroon">Y</font> <a href="henmodel.html#R1" target="_self">|-</a> <font color="Maroon">p</font> &amp; <font color="Maroon">Y</font> <a href="henmodel.html#R1" target="_self">|-</a>  <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E19:17_1"/><b>then </b><i><font color="Green">E54</font></i>: 
not <font color="Maroon">Y</font> is <a href="henmodel.html#V1" target="_self">Consistent</a>
 <b>by </b><i/>;<br/><a NAME="E20:17_1"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> is <a href="henmodel.html#V1" target="_self">Consistent</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:17_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E2:17_1_1"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
;<br/>
<a NAME="E3:17_1_1"/><b>then </b>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E4:17_1_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> is <a href="henmodel.html#V1" target="_self">Consistent</a>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><a NAME="E21:17_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/><a NAME="E22:17_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E23:17_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:17"/>
 <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span> is <a href="henmodel.html#V1" target="_self">Consistent</a>
 ;<br/>


</div>
