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

<b>consider </b><font color="Maroon" title="c2">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> =  <a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon" title="c1">X</font>;<br/>
<a NAME="E2:12_1_1"/>
(  ex <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font> st <br/>for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>) &amp; ( for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font>  st ( for <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Olive" title="b1">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</font>) ) &amp; ( for <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Olive" title="b2">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</font>) ) holds <br/><font color="Olive" title="b1">f1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">f2</font> ) )
 
<b>from </b><i><a class="ref" href="circcmb3.html#S7" title="CIRCCMB3:sch.7">CIRCCMB3:sch 7</a>();<br/></i>
<b>then consider </b><font color="Maroon" title="c3">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E4:12_1_1"/><i><font color="Green" title="E7">A1</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <font color="Olive" title="b1">a</font>,<font color="Maroon" title="c2">x</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font>
 ;<br/>
<a NAME="E5:12_1_1"/><i><font color="Green" title="E8">A2</font></i>: 
( 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">X</font> )
 
<b>by </b><i><a class="ref" href="catalg_1.html#T6" title="CATALG_1:th.6">CATALG_1:6</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">f</font> = <font color="Maroon" title="c3">f</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a> <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> ,<font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a>, <a class="ref" href="comput_1.html#T19" title="COMPUT_1:th.19">COMPUT_1:19</a></i>;<br/>
<a NAME="E7:12_1_1"/>
 <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<b>by </b><i><a class="txt" href="aofa_000.html#E5:12_1_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="comput_1.html#T27" title="COMPUT_1:th.27">COMPUT_1:27</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">f</font> = <font color="Maroon" title="c4">f</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a> <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a> <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> ,<font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="aofa_000.html#E5:12_1_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="comput_1.html#T25" title="COMPUT_1:th.25">COMPUT_1:25</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">f</font>
;<br/>

<b>thus </b><a NAME="E9:12_1_1"/>
 <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <b>by </b><i><a class="txt" href="aofa_000.html#E5:12_1_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="comput_1.html#T27" title="COMPUT_1:th.27">COMPUT_1:27</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="comput_1.html#D26" title="COMPUT_1:def.26">COMPUT_1:def 26</a><br/>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="aofa_000.html#D2" target="_self" title="AOFA_000:def.2">AOFA_000:def 2</a>
<div class="add"><b>let </b><font color="Maroon" title="c6">u</font>, <font color="Maroon" title="c7">y</font>, <font color="Maroon" title="c8">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:12_1_1_1"/>
( <span class="p1"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c6">u</font>,<font color="Maroon" title="c7">y</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">f</font> &amp; <span class="p1"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c8">z</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">f</font> )
 ;<br/><b>then reconsider </b><font color="Maroon" title="c9">u'</font> = <font color="Maroon" title="c6">u</font>, <font color="Maroon" title="c10">y'</font> = <font color="Maroon" title="c7">y</font>, <font color="Maroon" title="c11">z'</font> = <font color="Maroon" title="c8">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="aofa_000.html#E5:12_1_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="catalg_1.html#T11" title="CATALG_1:th.11">CATALG_1:11</a></i>;<br/><b>assume </b><a NAME="E3:12_1_1_1"/>
( <span class="p1"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c6">u</font>,<font color="Maroon" title="c7">y</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c8">z</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">f</font> &amp; <span class="p1"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c6">u</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c8">z</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span></span>)</span></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">f</font> )
 ;<br/><a NAME="E4:12_1_1_1"/><i><font color="Green" title="E9">A3</font></i>: 
( ( <font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> implies  <a href="funcop_1.html#K12" title="FUNCOP_1:func.12">IFEQ</a> <font color="Maroon" title="c6">u</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font> ) &amp; ( <font color="Maroon" title="c6">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font> implies  <a href="funcop_1.html#K12" title="FUNCOP_1:func.12">IFEQ</a> <font color="Maroon" title="c6">u</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">u</font> ) &amp; ( <font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> implies  <a href="funcop_1.html#K12" title="FUNCOP_1:func.12">IFEQ</a> <font color="Maroon" title="c6">u</font>,<font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><a href="funcop_1.html#K12" title="FUNCOP_1:func.12">IFEQ</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c8">z</font>,<font color="Maroon" title="c7">y</font></span>)</span>,<font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcop_1.html#K12" title="FUNCOP_1:func.12">IFEQ</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c8">z</font>,<font color="Maroon" title="c7">y</font> ) &amp; ( <font color="Maroon" title="c6">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font> implies  <a href="funcop_1.html#K12" title="FUNCOP_1:func.12">IFEQ</a> <font color="Maroon" title="c6">u</font>,<font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><a href="funcop_1.html#K12" title="FUNCOP_1:func.12">IFEQ</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c8">z</font>,<font color="Maroon" title="c7">y</font></span>)</span>,<font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">u</font> ) &amp; ( <font color="Maroon" title="c6">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font> implies  <a href="funcop_1.html#K12" title="FUNCOP_1:func.12">IFEQ</a> <font color="Maroon" title="c6">u</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c8">z</font>,<font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">u</font> ) )
 <b>by </b><i><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/><b>thus </b><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c6">u</font>,<font color="Maroon" title="c7">y</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c8">z</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span> = 
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <font color="Maroon" title="c9">u'</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c10">y'</font>,<font color="Maroon" title="c9">u'</font></span>)</span>,<font color="Maroon" title="c11">z'</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
<b>by </b><i><a class="txt" href="aofa_000.html#E4:12_1_1"><i><font color="Green" title="E7">A1</font></i></a></i>
<br/>.= 
 <a href="funcop_1.html#K12" title="FUNCOP_1:func.12">IFEQ</a> <span class="p1">(<span class="default"><a href="funcop_1.html#K12" title="FUNCOP_1:func.12">IFEQ</a> <font color="Maroon" title="c6">u</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">u</font></span>)</span>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c8">z</font>,<span class="p1">(<span class="default"><a href="funcop_1.html#K12" title="FUNCOP_1:func.12">IFEQ</a> <font color="Maroon" title="c6">u</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">u</font></span>)</span>
<b>by </b><i><a class="txt" href="aofa_000.html#E4:12_1_1"><i><font color="Green" title="E7">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c9">u'</font>,<span class="p2">(<span class="default"><a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <font color="Maroon" title="c10">y'</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c11">z'</font>,<font color="Maroon" title="c10">y'</font></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
<b>by </b><i><a class="txt" href="aofa_000.html#E4:12_1_1"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="aofa_000.html#E4:12_1_1_1"><i><font color="Green" title="E9">A3</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c6">u</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c8">z</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span></span>)</span></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span>
<b>by </b><i><a class="txt" href="aofa_000.html#E4:12_1_1"><i><font color="Green" title="E7">A1</font></i></a></i>
;<br/></div>
<b>end;</b></div>

<b>take </b>
<font color="Maroon" title="c2">x</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="aofa_000.html#D3" target="_self" title="AOFA_000:def.3">AOFA_000:def 3</a><br/>

<b>let </b><font color="Maroon" title="c6">y</font>, <font color="Maroon" title="c7">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="aofa_000.html#D1" target="_self" title="AOFA_000:def.1">AOFA_000:def 1</a><br/>

<b>assume </b><a NAME="E11:12_1_1"/>
( <span class="p1"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">z</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">f</font> or <span class="p1"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">y</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">f</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c8">y'</font> = <font color="Maroon" title="c6">y</font>, <font color="Maroon" title="c9">z'</font> = <font color="Maroon" title="c7">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="aofa_000.html#E5:12_1_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="catalg_1.html#T11" title="CATALG_1:th.11">CATALG_1:11</a></i>;<br/>
<a NAME="E13:12_1_1"/>
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c8">y'</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="catalg_1.html#T10" title="CATALG_1:th.10">CATALG_1:10</a></i>;<br/>
<b>hence </b><a NAME="E14:12_1_1"/>
<span class="p1"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">y</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">f</font>
 <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><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">y</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span> = 
 <a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c8">y'</font>,<font color="Maroon" title="c2">x</font>
<b>by </b><i><a class="txt" href="aofa_000.html#E4:12_1_1"><i><font color="Green" title="E7">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">y</font>
<b>by </b><i><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>
;<br/>

<a NAME="E16:12_1_1"/>
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c8">y'</font>,<font color="Maroon" title="c2">x</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="catalg_1.html#T10" title="CATALG_1:th.10">CATALG_1:10</a></i>;<br/>
<b>hence </b><a NAME="E17:12_1_1"/>
<span class="p1"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c2">x</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">f</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>

<a NAME="E18:12_1_1"/><i><font color="Green" title="E9">A4</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> or <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">y</font> )
 
;<br/>
<b>thus </b><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c2">x</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span> = 
 <a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <font color="Maroon" title="c8">y'</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c8">y'</font>
<b>by </b><i><a class="txt" href="aofa_000.html#E4:12_1_1"><i><font color="Green" title="E7">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">y</font>
<b>by </b><i><a class="txt" href="aofa_000.html#E18:12_1_1"><i><font color="Green" title="E9">A4</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>
;<br/>


</div>
