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

<b>thus </b><a NAME="E1:5"/><i><font color="Green" title="E5">A1</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>

<b>thus </b><a NAME="E2:5"/><i><font color="Green" title="E6">A2</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c1">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:5_1_1_1"/>
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> 
 ;<br/><b>then reconsider </b><font color="Maroon" title="c2">p'</font> = <font color="Maroon" title="c1">p</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>  ;<br/><b>take </b><font color="Maroon" title="c3">q</font> =  <a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c2">p'</font>;<br/><b>thus </b><a NAME="E3:5_1_1_1"/>
<font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a> 
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:5"><i><font color="Green" title="E5">A1</font></i></a></i>;<br/><b>thus </b><a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">q</font> = 
 <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c2">p'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="zf_fund1.html#D2" target="_self" title="ZF_FUND1:def.2">Def2</a></i>
<br/>.= 
 <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K4" title="ZF_FUND1:func.4">x".</a> <font color="Maroon" title="c2">p'</font></span>)</span>
<b>by </b><i><a class="ref" href="card_1.html#T66" title="CARD_1:th.66">CARD_1:66</a>, <a class="ref" href="finseq_1.html#T78" title="FINSEQ_1:th.78">FINSEQ_1:78</a></i>
<br/>.= 
<font color="Maroon" title="c1">p</font>
<b>by </b><i><a class="ref" href="zf_fund1.html#D3" target="_self" title="ZF_FUND1:def.3">Def3</a></i>
;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:5_1_1"/>
( <font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>  iff  ex <font color="Olive" title="b1">q</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a>  &amp; <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">q</font> ) )
 <b>by </b><i><a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:5_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> 
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:5"/><i><font color="Green" title="E7">A3</font></i>: 
 <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a>  is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:5_2_1"/><i><font color="Green" title="E8">A4</font></i>: 
( <font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a>  &amp; <font color="Maroon" title="c2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a>  &amp; <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">q</font> )
 ;<br/><b>then reconsider </b><font color="Maroon" title="c3">p'</font> = <font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c4">q'</font> = <font color="Maroon" title="c2">q</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  ;<br/> <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">p'</font></span>)</span> = 
<a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">q</font>
<b>by </b><i><a class="txt" href="zf_fund1.html#E1:5_2_1"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="zf_fund1.html#D2" target="_self" title="ZF_FUND1:def.2">Def2</a></i>
<br/>.= 
 <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">q'</font></span>)</span>
<b>by </b><i><a class="ref" href="zf_fund1.html#D2" target="_self" title="ZF_FUND1:def.2">Def2</a></i>
;<br/><b>then </b><i><font color="Green" title="E9">A5</font></i>: 5 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">p'</font></span>)</span> = 
 <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">q'</font></span>)</span>
<b>by </b><i><a class="ref" href="zf_lang.html#D2" title="ZF_LANG:def.2">ZF_LANG:def 2</a></i>
<br/>.= 
5 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">q'</font></span>)</span>
<b>by </b><i><a class="ref" href="zf_lang.html#D2" title="ZF_LANG:def.2">ZF_LANG:def 2</a></i>
;<br/><b>consider </b><font color="Maroon" title="c5">k</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> <b> such that </b><br/><a NAME="E6:5_2_1"/><i><font color="Green" title="E10">A6</font></i>: 
<font color="Maroon" title="c3">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font>
 ;<br/><b>consider </b><font color="Maroon" title="c6">k1</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> <b> such that </b><br/><a NAME="E8:5_2_1"/><i><font color="Green" title="E11">A7</font></i>: 
<font color="Maroon" title="c4">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k1</font>
 ;<br/><font color="Maroon" title="c5">k</font> = 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">p'</font>
<b>by </b><i><a class="txt" href="zf_fund1.html#E6:5_2_1"><i><font color="Green" title="E10">A6</font></i></a>, <a class="ref" href="card_1.html#T66" title="CARD_1:th.66">CARD_1:66</a>, <a class="ref" href="finseq_1.html#T78" title="FINSEQ_1:th.78">FINSEQ_1:78</a></i>
<br/>.= 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">k1</font>
<b>by </b><i><a class="txt" href="zf_fund1.html#E4:5_2_1"><i><font color="Green" title="E9">A5</font></i></a>, <a class="txt" href="zf_fund1.html#E8:5_2_1"><i><font color="Green" title="E11">A7</font></i></a>, <a class="ref" href="xcmplx_1.html#T2" title="XCMPLX_1:th.2">XCMPLX_1:2</a></i>
<br/>.= 
<font color="Maroon" title="c6">k1</font>
<b>by </b><i><a class="ref" href="card_1.html#T66" title="CARD_1:th.66">CARD_1:66</a>, <a class="ref" href="finseq_1.html#T78" title="FINSEQ_1:th.78">FINSEQ_1:78</a></i>
;<br/><b>hence </b><a NAME="E10:5_2_1"/>
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">q</font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E6:5_2_1"><i><font color="Green" title="E10">A6</font></i></a>, <a class="txt" href="zf_fund1.html#E8:5_2_1"><i><font color="Green" title="E11">A7</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:5_2"/>
 <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a>  is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>


</div><b>end;</b></div>

<b>hence </b><a NAME="E4:5"/>
<a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a>  <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#T62" title="FUNCT_1:th.62">FUNCT_1:62</a></i>;<br/>

<b>thus </b><a NAME="E5:5"/>
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a>  <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a>  <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  )
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:5"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="zf_fund1.html#E2:5"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="zf_fund1.html#E3:5"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></i>;<br/>


</div>
