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

<a NAME="E1:52_1_1"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_i00.html#T11" target="_self" title="AOFA_I00:th.11">Th11</a></span>;<br/>
<a NAME="E2:52_1_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_i00.html#T6" target="_self" title="AOFA_I00:th.6">Th6</a></span>;<br/>
<a NAME="E3:52_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A2</font></span>: 
<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_1_1"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="numbers.html#T17" title="NUMBERS:th.17">NUMBERS:17</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">x</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="aofa_i00.html#K19" title="AOFA_I00:func.19">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">a</font> = <span class="p1">(<span class="default"><a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="aofa_i00.html#K19" title="AOFA_I00:func.19">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="aofa_i00.html#K19" title="AOFA_I00:func.19">"</a> </span>)</span>;<br/>
<a NAME="E4:52_1_1"/><span class="lab"><font color="Green" title="E36">A3</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="aofa_i00.html#K19" title="AOFA_I00:func.19">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></span>;<br/>
<a NAME="E5:52_1_1"/><span class="lab"><font color="Green" title="E37">A4</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="aofa_i00.html#K19" title="AOFA_I00:func.19">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="aofa_i00.html#K19" title="AOFA_I00:func.19">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T14" title="FUNCOP_1:th.14">FUNCOP_1:14</a></span>;<br/>
<a NAME="E6:52_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="aofa_i00.html#K19" title="AOFA_I00:func.19">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a>, <a class="ref" href="ordinal3.html#T10" title="ORDINAL3:th.10">ORDINAL3:10</a></span>;<br/>
<a NAME="E7:52_1_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="aofa_i00.html#K19" title="AOFA_I00:func.19">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></span>;<br/>
<a NAME="E8:52_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A5</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="aofa_i00.html#K19" title="AOFA_I00:func.19">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></span>;<br/>
<a NAME="E9:52_1_1"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="aofa_i00.html#K19" title="AOFA_I00:func.19">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_i00.html#T7" target="_self" title="AOFA_I00:th.7">Th7</a></span>;<br/>
<a NAME="E10:52_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A6</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="aofa_i00.html#K19" title="AOFA_I00:func.19">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="aofa_i00.html#K19" title="AOFA_I00:func.19">"</a> </span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:52_1_1"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="txt" href="#E8:52_1_1"><span class="lab"><font color="Green" title="E38">A5</font></span></a>, <a class="ref" href="funct_4.html#T18" title="FUNCT_4:th.18">FUNCT_4:18</a></span>;<br/>
<a NAME="E11:52_1_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="aofa_i00.html#K19" title="AOFA_I00:func.19">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_i00.html#T7" target="_self" title="AOFA_I00:th.7">Th7</a></span>;<br/>
<a NAME="E12:52_1_1"/><span class="kw">then </span>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="aofa_i00.html#K19" title="AOFA_I00:func.19">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="aofa_i00.html#K19" title="AOFA_I00:func.19">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:52_1_1"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="txt" href="#E3:52_1_1"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">a</font> = <span class="p1">(<span class="default"><a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="aofa_i00.html#K19" title="AOFA_I00:func.19">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="aofa_i00.html#K19" title="AOFA_I00:func.19">"</a> </span>)</span> as   <a href="aofa_i00.html#NM6" title="AOFA_I00:NM.6">INT-Array</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:52_1_1"><span class="lab"><font color="Green" title="E39">A6</font></span></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c7">f</font> be    <a href="aofa_i00.html#M6" title="AOFA_I00:mode.6">INT-Exec</a> of <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c2">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">f</font> is  <a href="aofa_i00.html#V1" title="AOFA_I00:attr.1">Euclidean</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c8">S</font> =  <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">G</font> =  <a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">A</font> =  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ;<br/>
<span class="kw">thus </span><a NAME="E14:52_1_1"/>
for <font color="Olive" title="b1">v</font> being   <a href="aofa_i00.html#M2" title="AOFA_I00:mode.2">INT-Variable</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font><br/> for <font color="Olive" title="b2">t</font> being   <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> holds  <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">t</font> <a href="aofa_i00.html#R2" title="AOFA_I00:pred.2">form_assignment_wrt</a> <font color="Maroon" title="c7">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_i00.html#T19" target="_self" title="AOFA_I00:th.19">Th19</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="aofa_i00.html#D22" target="_self" title="AOFA_I00:def.22">AOFA_I00:def 22</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">i</font> being  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds   <a href="aofa_i00.html#K25" title="AOFA_I00:func.25">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c1">X</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> ) &amp; ( for <font color="Olive" title="b1">v</font> being   <a href="aofa_i00.html#M2" title="AOFA_I00:mode.2">INT-Variable</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> holds   <a href="aofa_i00.html#K23" title="AOFA_I00:func.23">.</a> <font color="Olive" title="b1">v</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> ) &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds   <a href="aofa_i00.html#K24" title="AOFA_I00:func.24">^</a> <font color="Olive" title="b1">x</font> is    <a href="aofa_i00.html#M2" title="AOFA_I00:mode.2">INT-Variable</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> ) &amp;  ex <font color="Olive" title="b1">a</font> being  <a href="aofa_i00.html#NM6" title="AOFA_I00:NM.6">INT-Array</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">X</font></span>)</span> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; ( for <font color="Olive" title="b2">t</font> being   <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> holds  <font color="Olive" title="b1">a</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">t</font> is    <a href="aofa_i00.html#M2" title="AOFA_I00:mode.2">INT-Variable</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> ) ) &amp; ( for <font color="Olive" title="b1">t</font> being   <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> holds   <a href="valued_1.html#K34" title="VALUED_1:func.34">-</a> <font color="Olive" title="b1">t</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> ) &amp; ( for <font color="Olive" title="b1">t1</font>, <font color="Olive" title="b2">t2</font> being   <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> holds <br/> ( <font color="Olive" title="b1">t1</font> <a href="valued_1.html#K22" title="VALUED_1:func.22">(#)</a> <font color="Olive" title="b2">t2</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> &amp; <font color="Olive" title="b1">t1</font> <a href="aofa_i00.html#K16" title="AOFA_I00:func.16">+</a> <font color="Olive" title="b2">t2</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> &amp; <font color="Olive" title="b1">t1</font> <a href="aofa_i00.html#K10" title="AOFA_I00:func.10">div</a> <font color="Olive" title="b2">t2</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> &amp; <font color="Olive" title="b1">t1</font> <a href="aofa_i00.html#K11" title="AOFA_I00:func.11">mod</a> <font color="Olive" title="b2">t2</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> &amp;  <a href="aofa_i00.html#K12" title="AOFA_I00:func.12">leq</a> <font color="Olive" title="b1">t1</font>,<font color="Olive" title="b2">t2</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> &amp;  <a href="aofa_i00.html#K13" title="AOFA_I00:func.13">gt</a> <font color="Olive" title="b1">t1</font>,<font color="Olive" title="b2">t2</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E15:52_1_1"/>
for <font color="Olive" title="b1">i</font> being  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds   <a href="aofa_i00.html#K25" title="AOFA_I00:func.25">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c1">X</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_i00.html#T21" target="_self" title="AOFA_I00:th.21">Th21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">v</font> being   <a href="aofa_i00.html#M2" title="AOFA_I00:mode.2">INT-Variable</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> holds   <a href="aofa_i00.html#K23" title="AOFA_I00:func.23">.</a> <font color="Olive" title="b1">v</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> ) &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds   <a href="aofa_i00.html#K24" title="AOFA_I00:func.24">^</a> <font color="Olive" title="b1">x</font> is    <a href="aofa_i00.html#M2" title="AOFA_I00:mode.2">INT-Variable</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> ) &amp;  ex <font color="Olive" title="b1">a</font> being  <a href="aofa_i00.html#NM6" title="AOFA_I00:NM.6">INT-Array</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">X</font></span>)</span> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; ( for <font color="Olive" title="b2">t</font> being   <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> holds  <font color="Olive" title="b1">a</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">t</font> is    <a href="aofa_i00.html#M2" title="AOFA_I00:mode.2">INT-Variable</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> ) ) &amp; ( for <font color="Olive" title="b1">t</font> being   <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> holds   <a href="valued_1.html#K34" title="VALUED_1:func.34">-</a> <font color="Olive" title="b1">t</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> ) &amp; ( for <font color="Olive" title="b1">t1</font>, <font color="Olive" title="b2">t2</font> being   <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> holds <br/> ( <font color="Olive" title="b1">t1</font> <a href="valued_1.html#K22" title="VALUED_1:func.22">(#)</a> <font color="Olive" title="b2">t2</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> &amp; <font color="Olive" title="b1">t1</font> <a href="aofa_i00.html#K16" title="AOFA_I00:func.16">+</a> <font color="Olive" title="b2">t2</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> &amp; <font color="Olive" title="b1">t1</font> <a href="aofa_i00.html#K10" title="AOFA_I00:func.10">div</a> <font color="Olive" title="b2">t2</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> &amp; <font color="Olive" title="b1">t1</font> <a href="aofa_i00.html#K11" title="AOFA_I00:func.11">mod</a> <font color="Olive" title="b2">t2</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> &amp;  <a href="aofa_i00.html#K12" title="AOFA_I00:func.12">leq</a> <font color="Olive" title="b1">t1</font>,<font color="Olive" title="b2">t2</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> &amp;  <a href="aofa_i00.html#K13" title="AOFA_I00:func.13">gt</a> <font color="Olive" title="b1">t1</font>,<font color="Olive" title="b2">t2</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E16:52_1_1"/>
for <font color="Olive" title="b1">v</font> being   <a href="aofa_i00.html#M2" title="AOFA_I00:mode.2">INT-Variable</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> holds   <a href="aofa_i00.html#K23" title="AOFA_I00:func.23">.</a> <font color="Olive" title="b1">v</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_i00.html#T21" target="_self" title="AOFA_I00:th.21">Th21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds   <a href="aofa_i00.html#K24" title="AOFA_I00:func.24">^</a> <font color="Olive" title="b1">x</font> is    <a href="aofa_i00.html#M2" title="AOFA_I00:mode.2">INT-Variable</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> ) &amp;  ex <font color="Olive" title="b1">a</font> being  <a href="aofa_i00.html#NM6" title="AOFA_I00:NM.6">INT-Array</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">X</font></span>)</span> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; ( for <font color="Olive" title="b2">t</font> being   <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> holds  <font color="Olive" title="b1">a</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">t</font> is    <a href="aofa_i00.html#M2" title="AOFA_I00:mode.2">INT-Variable</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> ) ) &amp; ( for <font color="Olive" title="b1">t</font> being   <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> holds   <a href="valued_1.html#K34" title="VALUED_1:func.34">-</a> <font color="Olive" title="b1">t</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> ) &amp; ( for <font color="Olive" title="b1">t1</font>, <font color="Olive" title="b2">t2</font> being   <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> holds <br/> ( <font color="Olive" title="b1">t1</font> <a href="valued_1.html#K22" title="VALUED_1:func.22">(#)</a> <font color="Olive" title="b2">t2</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> &amp; <font color="Olive" title="b1">t1</font> <a href="aofa_i00.html#K16" title="AOFA_I00:func.16">+</a> <font color="Olive" title="b2">t2</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> &amp; <font color="Olive" title="b1">t1</font> <a href="aofa_i00.html#K10" title="AOFA_I00:func.10">div</a> <font color="Olive" title="b2">t2</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> &amp; <font color="Olive" title="b1">t1</font> <a href="aofa_i00.html#K11" title="AOFA_I00:func.11">mod</a> <font color="Olive" title="b2">t2</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> &amp;  <a href="aofa_i00.html#K12" title="AOFA_I00:func.12">leq</a> <font color="Olive" title="b1">t1</font>,<font color="Olive" title="b2">t2</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> &amp;  <a href="aofa_i00.html#K13" title="AOFA_I00:func.13">gt</a> <font color="Olive" title="b1">t1</font>,<font color="Olive" title="b2">t2</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E17:52_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds   <a href="aofa_i00.html#K24" title="AOFA_I00:func.24">^</a> <font color="Olive" title="b1">x</font> is    <a href="aofa_i00.html#M2" title="AOFA_I00:mode.2">INT-Variable</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_i00.html#T20" target="_self" title="AOFA_I00:th.20">Th20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">a</font> being  <a href="aofa_i00.html#NM6" title="AOFA_I00:NM.6">INT-Array</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">X</font></span>)</span> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; ( for <font color="Olive" title="b2">t</font> being   <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> holds  <font color="Olive" title="b1">a</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">t</font> is    <a href="aofa_i00.html#M2" title="AOFA_I00:mode.2">INT-Variable</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> ) ) &amp; ( for <font color="Olive" title="b1">t</font> being   <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> holds   <a href="valued_1.html#K34" title="VALUED_1:func.34">-</a> <font color="Olive" title="b1">t</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> ) &amp; ( for <font color="Olive" title="b1">t1</font>, <font color="Olive" title="b2">t2</font> being   <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> holds <br/> ( <font color="Olive" title="b1">t1</font> <a href="valued_1.html#K22" title="VALUED_1:func.22">(#)</a> <font color="Olive" title="b2">t2</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> &amp; <font color="Olive" title="b1">t1</font> <a href="aofa_i00.html#K16" title="AOFA_I00:func.16">+</a> <font color="Olive" title="b2">t2</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> &amp; <font color="Olive" title="b1">t1</font> <a href="aofa_i00.html#K10" title="AOFA_I00:func.10">div</a> <font color="Olive" title="b2">t2</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> &amp; <font color="Olive" title="b1">t1</font> <a href="aofa_i00.html#K11" title="AOFA_I00:func.11">mod</a> <font color="Olive" title="b2">t2</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> &amp;  <a href="aofa_i00.html#K12" title="AOFA_I00:func.12">leq</a> <font color="Olive" title="b1">t1</font>,<font color="Olive" title="b2">t2</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> &amp;  <a href="aofa_i00.html#K13" title="AOFA_I00:func.13">gt</a> <font color="Olive" title="b1">t1</font>,<font color="Olive" title="b2">t2</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> ) ) )</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">t</font> being   <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> holds   <a href="valued_1.html#K34" title="VALUED_1:func.34">-</a> <font color="Olive" title="b1">t</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> ) &amp; ( for <font color="Olive" title="b1">t1</font>, <font color="Olive" title="b2">t2</font> being   <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> holds <br/> ( <font color="Olive" title="b1">t1</font> <a href="valued_1.html#K22" title="VALUED_1:func.22">(#)</a> <font color="Olive" title="b2">t2</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> &amp; <font color="Olive" title="b1">t1</font> <a href="aofa_i00.html#K16" title="AOFA_I00:func.16">+</a> <font color="Olive" title="b2">t2</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> &amp; <font color="Olive" title="b1">t1</font> <a href="aofa_i00.html#K10" title="AOFA_I00:func.10">div</a> <font color="Olive" title="b2">t2</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> &amp; <font color="Olive" title="b1">t1</font> <a href="aofa_i00.html#K11" title="AOFA_I00:func.11">mod</a> <font color="Olive" title="b2">t2</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> &amp;  <a href="aofa_i00.html#K12" title="AOFA_I00:func.12">leq</a> <font color="Olive" title="b1">t1</font>,<font color="Olive" title="b2">t2</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> &amp;  <a href="aofa_i00.html#K13" title="AOFA_I00:func.13">gt</a> <font color="Olive" title="b1">t1</font>,<font color="Olive" title="b2">t2</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> ) ) )</span>
<div class="add"><span class="kw">take </span><font color="Maroon" title="c11">a</font> = <font color="Maroon" title="c6">a</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">a</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">X</font></span>)</span> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; ( for <font color="Olive" title="b1">t</font> being   <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> holds  <font color="Maroon" title="c11">a</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">t</font> is    <a href="aofa_i00.html#M2" title="AOFA_I00:mode.2">INT-Variable</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> ) )</span><br/><a NAME="E1:52_1_1_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="aofa_i00.html#K19" title="AOFA_I00:func.19">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_i00.html#T7" target="_self" title="AOFA_I00:th.7">Th7</a></span>;<br/><span class="kw">hence </span><a NAME="E2:52_1_1_1"/>
<font color="Maroon" title="c11">a</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">X</font></span>)</span> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T24" title="FUNCT_4:th.24">FUNCT_4:24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">t</font> being   <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> holds  <font color="Maroon" title="c11">a</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">t</font> is    <a href="aofa_i00.html#M2" title="AOFA_I00:mode.2">INT-Variable</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font></span><br/><span class="kw">thus </span><a NAME="E3:52_1_1_1"/>
for <font color="Olive" title="b1">t</font> being   <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> holds  <font color="Maroon" title="c11">a</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">t</font> is    <a href="aofa_i00.html#M2" title="AOFA_I00:mode.2">INT-Variable</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_i00.html#T20" target="_self" title="AOFA_I00:th.20">Th20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E19:52_1_1"/>
( ( for <font color="Olive" title="b1">t</font> being   <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> holds   <a href="valued_1.html#K34" title="VALUED_1:func.34">-</a> <font color="Olive" title="b1">t</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> ) &amp; ( for <font color="Olive" title="b1">t1</font>, <font color="Olive" title="b2">t2</font> being   <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> holds <br/> ( <font color="Olive" title="b1">t1</font> <a href="valued_1.html#K22" title="VALUED_1:func.22">(#)</a> <font color="Olive" title="b2">t2</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> &amp; <font color="Olive" title="b1">t1</font> <a href="aofa_i00.html#K16" title="AOFA_I00:func.16">+</a> <font color="Olive" title="b2">t2</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> &amp; <font color="Olive" title="b1">t1</font> <a href="aofa_i00.html#K10" title="AOFA_I00:func.10">div</a> <font color="Olive" title="b2">t2</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> &amp; <font color="Olive" title="b1">t1</font> <a href="aofa_i00.html#K11" title="AOFA_I00:func.11">mod</a> <font color="Olive" title="b2">t2</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> &amp;  <a href="aofa_i00.html#K12" title="AOFA_I00:func.12">leq</a> <font color="Olive" title="b1">t1</font>,<font color="Olive" title="b2">t2</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> &amp;  <a href="aofa_i00.html#K13" title="AOFA_I00:func.13">gt</a> <font color="Olive" title="b1">t1</font>,<font color="Olive" title="b2">t2</font> is    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ,<font color="Maroon" title="c7">f</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_i00.html#T21" target="_self" title="AOFA_I00:th.21">Th21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
