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

<b>let </b><font color="Maroon">U1</font> be   <a href="unialg_1.html#NM2">Universal_Algebra</a>;<br/><b>let </b><font color="Maroon">E</font> be    <a href="alg_1.html#M1" target="_self">Congruence</a> of <font color="Maroon">U1</font>;<br/>



<b>set </b><font color="Maroon">f</font> =  <a href="alg_1.html#K6" target="_self">Nat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">E</font>;<br/>
<b>set </b><font color="Maroon">u1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font>;<br/>
<b>set </b><font color="Maroon">qu</font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" target="_self">QuotUnivAlg</a> <font color="Maroon">U1</font>,<font color="Maroon">E</font></span>)</span>;<br/>
<a NAME="E1:34"/><i><font color="Green">E33</font></i>: 
( the <a href="unialg_1.html#U1">charact</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" target="_self">QuotUnivAlg</a> <font color="Maroon">U1</font>,<font color="Maroon">E</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="alg_1.html#K4" target="_self">QuotOpSeq</a> <font color="Maroon">U1</font>,<font color="Maroon">E</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="alg_1.html#K4" target="_self">QuotOpSeq</a> <font color="Maroon">U1</font>,<font color="Maroon">E</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U1</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K4">signature</a> <span class="p2">(<span class="default"><a href="alg_1.html#K5" target="_self">QuotUnivAlg</a> <font color="Maroon">U1</font>,<font color="Maroon">E</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" target="_self">QuotUnivAlg</a> <font color="Maroon">U1</font>,<font color="Maroon">E</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K4">signature</a> <font color="Maroon">U1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U1</font> )
 
<b>by </b><i>, <a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:34_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="unialg_1.html#K4">signature</a> <font color="Maroon">U1</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:34_1"/><b>then </b><i><font color="Green">E34</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K4">signature</a> <font color="Maroon">U1</font></span>)</span> &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U1</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="alg_1.html#K4" target="_self">QuotOpSeq</a> <font color="Maroon">U1</font>,<font color="Maroon">E</font></span>)</span> &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K4">signature</a> <span class="p2">(<span class="default"><a href="alg_1.html#K5" target="_self">QuotUnivAlg</a> <font color="Maroon">U1</font>,<font color="Maroon">E</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="alg_1.html#T1" target="_self">Th1</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">o1</font> = the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> as   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">U1</font> <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:34_1"/><i><font color="Green">E35</font></i>: 
<span class="p1">(<span class="default"><a href="unialg_1.html#K4">signature</a> <font color="Maroon">U1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o1</font>
 <b>by </b><i>, <a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>;<br/><a NAME="E5:34_1"/><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><a href="alg_1.html#K4" target="_self">QuotOpSeq</a> <font color="Maroon">U1</font>,<font color="Maroon">E</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="alg_1.html#K3" target="_self">QuotOp</a> <font color="Maroon">o1</font>,<font color="Maroon">E</font>
 <b>by </b><i>, </i>;<br/><a NAME="E6:34_1"/><i><font color="Green">E46</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="alg_1.html#K3" target="_self">QuotOp</a> <font color="Maroon">o1</font>,<font color="Maroon">E</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o1</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">E</font></span>)</span>
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">cl</font> =  <a href="alg_1.html#K3" target="_self">QuotOp</a> <font color="Maroon">o1</font>,<font color="Maroon">E</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="unialg_1.html#V1">homogeneous</a> <a href="unialg_1.html#V2">quasi_total</a> <a href="partfun1.html#NM1">PartFunc</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" target="_self">QuotUnivAlg</a> <font color="Maroon">U1</font>,<font color="Maroon">E</font></span>)</span> <a href="finseq_2.html#K3">*</a> ,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" target="_self">QuotUnivAlg</a> <font color="Maroon">U1</font>,<font color="Maroon">E</font></span>)</span> ;<br/><a NAME="E8:34_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">cl</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="unialg_1.html#T1">UNIALG_1:1</a></i>;<br/><b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:34_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">cl</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" target="_self">QuotUnivAlg</a> <font color="Maroon">U1</font>,<font color="Maroon">E</font></span>)</span> <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="txt" href="alg_1.html#E1:34"><i><font color="Green">E33</font></i></a></i>;<br/><a NAME="E12:34_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">E</font></span>)</span> <a href="finseq_2.html#K3">*</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o1</font> </span>}</span> 
 <b>by </b><i>, <a class="txt" href="alg_1.html#E1:34"><i><font color="Green">E33</font></i></a>, <a class="ref" href="finseq_2.html#D4">FINSEQ_2:def 4</a></i>;<br/><b>then consider </b><font color="Maroon">w</font> being    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">E</font></span>)</span> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E14:34_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o1</font> )
 ;<br/><a NAME="E15:34_1"/>
 <a href="unialg_1.html#K3">arity</a> <font color="Maroon">cl</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o1</font>
 <b>by </b><i><a class="txt" href="alg_1.html#E1:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="alg_1.html#E2:34_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="unialg_1.html#D10">UNIALG_1:def 10</a></i>;<br/><b>hence </b><a NAME="E16:34_1"/>
<span class="p1">(<span class="default"><a href="unialg_1.html#K4">signature</a> <font color="Maroon">U1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K4">signature</a> <span class="p2">(<span class="default"><a href="alg_1.html#K5" target="_self">QuotUnivAlg</a> <font color="Maroon">U1</font>,<font color="Maroon">E</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, , , <a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:34"/>
 <a href="unialg_1.html#K4">signature</a> <font color="Maroon">U1</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K4">signature</a> <span class="p1">(<span class="default"><a href="alg_1.html#K5" target="_self">QuotUnivAlg</a> <font color="Maroon">U1</font>,<font color="Maroon">E</font></span>)</span>
 <b>by </b><i><a class="ref" href="alg_1.html#T1" target="_self">Th1</a>, <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="unialg_2.html#D2">UNIALG_2:def 2</a>,<a class="ref" href="alg_1.html#D1" target="_self">ALG_1:def 1</a><br/>

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E4:34"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U1</font>
 ;<br/>

<a NAME="E5:34"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:34"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="alg_1.html#K4" target="_self">QuotOpSeq</a> <font color="Maroon">U1</font>,<font color="Maroon">E</font></span>)</span>
 
<b>by </b><i><a class="ref" href="alg_1.html#T1" target="_self">Th1</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>let </b><font color="Maroon">o1</font> be   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">U1</font>;<br/><b>let </b><font color="Maroon">o2</font> be   <a href="unialg_2.html#NM5">operation</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" target="_self">QuotUnivAlg</a> <font color="Maroon">U1</font>,<font color="Maroon">E</font></span>)</span>;<br/>



<b>assume </b><a NAME="E7:34"/><i><font color="Green">E50</font></i>: 
( the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o1</font> &amp; <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> the <a href="unialg_1.html#U1">charact</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" target="_self">QuotUnivAlg</a> <font color="Maroon">U1</font>,<font color="Maroon">E</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 ;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">U1</font>;<br/>

<b>assume </b><a NAME="E8:34"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">o1</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> as    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font> <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">o1</font> = 
<span class="p1">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o1</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font>
<b>by </b><i><a class="ref" href="unialg_2.html#T2">UNIALG_2:2</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font> <a href="finseq_2.html#K3">*</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o1</font> </span>}</span> 
<b>by </b><i><a class="ref" href="finseq_2.html#D4">FINSEQ_2:def 4</a></i>
;<br/>
<a NAME="E11:34"/><b>then </b><i><font color="Green">E63</font></i>: 
 ex <font color="Olive">p</font> being   <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font> <a href="finseq_2.html#K3">*</a>  st <br/>( <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o1</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E12:34"/>
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">o1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font> &amp; <font color="Maroon">o1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">o1</font> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">ox</font> = <font color="Maroon">o1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font> ;<br/>
<i><font color="Green">E71</font></i>: <span class="p1">(<span class="default"><a href="alg_1.html#K6" target="_self">Nat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">E</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> = 
 <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">E</font>,<font color="Maroon">ox</font>
<b>by </b><i/>
<br/>.= 
 <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">E</font>,<span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>

;<br/>
<a NAME="E15:34"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">o2</font> <a href="hidden.html#R1">=</a>  <a href="alg_1.html#K3" target="_self">QuotOp</a> <font color="Maroon">o1</font>,<font color="Maroon">E</font>
 
<b>by </b><i>, , </i>;<br/>
<b>reconsider </b><font color="Maroon">fx</font> = <span class="p1">(<span class="default"><a href="alg_1.html#K6" target="_self">Nat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">E</font></span>)</span> <a href="finseqop.html#K5">*</a> <font color="Maroon">x</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">E</font> ;<br/>
<b>reconsider </b><font color="Maroon">fx</font> = <font color="Maroon">fx</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">E</font></span>)</span> <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<i><font color="Green">E73</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="alg_1.html#K3" target="_self">QuotOp</a> <font color="Maroon">o1</font>,<font color="Maroon">E</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o1</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">E</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">E</font></span>)</span> <a href="finseq_2.html#K3">*</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o1</font> </span>}</span> 
<b>by </b><i><a class="ref" href="finseq_2.html#D4">FINSEQ_2:def 4</a></i>
;<br/>
<a NAME="E19:34"/><i><font color="Green">E74</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="alg_1.html#K6" target="_self">Nat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">E</font></span>)</span> <a href="finseqop.html#K5">*</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<a NAME="E20:34"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">fx</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="alg_1.html#K3" target="_self">QuotOp</a> <font color="Maroon">o1</font>,<font color="Maroon">E</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:34_5"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E2:34_5"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="alg_1.html#K6" target="_self">Nat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">E</font></span>)</span> <a href="finseqop.html#K5">*</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:34_5"/>
( <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">x</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">xm</font> = <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font> ;<br/><b>thus </b> <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">E</font>,<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="alg_1.html#K6" target="_self">Nat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">E</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">xm</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">fx</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div>
<a NAME="E22:34"/><b>then </b>
<font color="Maroon">x</font> <a href="alg_1.html#R6" target="_self">is_representatives_FS</a> <font color="Maroon">fx</font>
 
<b>by </b><i><a class="ref" href="alg_1.html#T2" target="_self">Th2</a>, </i>;<br/>
<b>hence </b><a NAME="E23:34"/>
<span class="p1">(<span class="default"><a href="alg_1.html#K6" target="_self">Nat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">E</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">o2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="alg_1.html#K6" target="_self">Nat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">E</font></span>)</span> <a href="finseqop.html#K5">*</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="alg_1.html#E1:34"><i><font color="Green">E33</font></i></a>, , , </i>;<br/>


</div>
