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

<a NAME="E1:157_1_1"/><i><font color="Green">E46</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font> &amp; ( for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font> holds <br/>( ( <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> implies <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> ) &amp; ( <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> implies <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font> ) ) ) )
 
<b>by </b><i/>;<br/>
<a NAME="E2:157_1_1"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">H</font> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i><a class="ref" href="zf_lang1.html#T72" target="_self">Th72</a>, <a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/>
<a NAME="E4:157_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="157_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:157_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:157_1_1_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:157_1_1_1"/>
( ( <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> or <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> ) &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font> )
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T72" target="_self">Th72</a>, <a class="ref" href="zf_lang1.html#T74" target="_self">Th74</a>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:157_1_1_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> or ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> &amp; <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  ) )
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T72" target="_self">Th72</a>, <a class="ref" href="zf_lang1.html#T74" target="_self">Th74</a></i>;<br/>
<b>hence </b><a NAME="E6:157_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E6:157_1_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang.html#K9">WFF</a> 
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T71" target="_self">Th71</a></i>;<br/>
<b>hence </b><a NAME="E7:157_1_1"/>
<font color="Maroon">H</font> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> is   <a href="zf_lang.html#NM2">ZF-formula</a>
 <b>by </b><i><a class="ref" href="zf_lang.html#T14">ZF_LANG:14</a></i>;<br/>


</div>
