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<b>let </b><font color="Maroon" title="c1">f</font> be    <a href="aofa_i00.html#M4" title="AOFA_I00:mode.4">INT-Exec</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">f</font> is <a href="aofa_i00.html#V1" title="AOFA_I00:attr.1">Euclidean</a></span><br/>

<b>set </b><font color="Maroon" title="c2">X</font> =  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>
<b>set </b><font color="Maroon" title="c3">S</font> =  <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ;<br/>
<b>set </b><font color="Maroon" title="c4">G</font> =  <a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ;<br/>
<b>set </b><font color="Maroon" title="c5">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/>
<b>set </b><font color="Maroon" title="c6">T</font> = <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>
<b>thus </b><a NAME="E1:48_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="c1">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="c1">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="c1">f</font>
 <b>by </b><i><a class="ref" href="aofa_i00.html#T16" target="_self" title="AOFA_I00:th.16">II1</a></i>; <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></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>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  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="c1">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="c1">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="c1">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  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  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="c1">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  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  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> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </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="c1">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="c1">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="c1">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="c1">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="c1">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="c1">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="c1">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="c1">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="c1">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="c1">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="c1">f</font> ) ) )</span><br/>

<b>thus </b><a NAME="E2:48_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>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  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="c1">f</font>
 <b>by </b><i><a class="ref" href="aofa_i00.html#T18" target="_self" title="AOFA_I00:th.18">II3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c1">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="c1">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  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  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="c1">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  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  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> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </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="c1">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="c1">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="c1">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="c1">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="c1">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="c1">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="c1">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="c1">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="c1">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="c1">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="c1">f</font> ) ) )</span><br/>

<b>thus </b><a NAME="E3:48_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="c1">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="c1">f</font>
 <b>by </b><i><a class="ref" href="aofa_i00.html#T18" target="_self" title="AOFA_I00:th.18">II3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  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="c1">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  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  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> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </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="c1">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="c1">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="c1">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="c1">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="c1">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="c1">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="c1">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="c1">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="c1">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="c1">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="c1">f</font> ) ) )</span><br/>

<b>thus </b><a NAME="E4:48_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  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="c1">f</font>
 <b>by </b><i><a class="ref" href="aofa_i00.html#T17" target="_self" title="AOFA_I00:th.17">II2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  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> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </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="c1">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="c1">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="c1">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="c1">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="c1">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="c1">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="c1">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="c1">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="c1">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="c1">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="c1">f</font> ) ) )</span><br/>

<b>set </b><font color="Maroon" title="c7">a</font> = <span class="p1">(<span class="default"><a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K1" title="PRALG_3:func.1">id</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span>;<br/>
<a NAME="E5:48_1_1"/>
<span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E6:48_1_1"/><b>then </b>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K1" title="PRALG_3:func.1">id</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K1" title="PRALG_3:func.1">id</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  &amp; <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="numbers.html#T17" title="NUMBERS:th.17">NUMBERS:17</a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a>, <a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a>, <a class="ref" href="funcop_1.html#T14" title="FUNCOP_1:th.14">FUNCOP_1:14</a>, <a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E7:48_1_1"/><b>then </b>
(  <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="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K1" title="PRALG_3:func.1">id</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  &amp;  <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="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K1" title="PRALG_3:func.1">id</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a>, <a class="ref" href="funct_4.html#T18" title="FUNCT_4:th.18">FUNCT_4:18</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">a</font> = <span class="p1">(<span class="default"><a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K1" title="PRALG_3:func.1">id</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> as   <a href="aofa_i00.html#NM6" title="AOFA_I00:NM.6">INT-Array</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/>
<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c1">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="c1">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="c1">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="c1">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="c1">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="c1">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="c1">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="c1">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="c1">f</font> ) ) )</span>
<div class="add"><b>take </b><font color="Maroon" title="c9">a</font> = <font color="Maroon" title="c8">a</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">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> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </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="c1">f</font> holds  <font color="Maroon" title="c9">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="c1">f</font> ) )</span><br/><a NAME="E1:48_1_1_1"/>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K1" title="PRALG_3:func.1">id</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 <b>by </b><i><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a>, <a class="ref" href="card_1.html#T84" title="CARD_1:th.84">CARD_1:84</a></i>;<br/><b>hence </b><a NAME="E2:48_1_1_1"/>
<font color="Maroon" title="c9">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> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> 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_4.html#T24" title="FUNCT_4:th.24">FUNCT_4:24</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c1">f</font> holds  <font color="Maroon" title="c9">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="c1">f</font></span><br/><b>thus </b><a NAME="E3:48_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="c1">f</font> holds  <font color="Maroon" title="c9">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="c1">f</font>
 <b>by </b><i><a class="ref" href="aofa_i00.html#T17" target="_self" title="AOFA_I00:th.17">II2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E10:48_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="c1">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="c1">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="c1">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="c1">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="c1">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="c1">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="c1">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="c1">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="c1">f</font> ) ) )
 <b>by </b><i><a class="ref" href="aofa_i00.html#T18" target="_self" title="AOFA_I00:th.18">II3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
