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

<b>let </b><font color="Maroon" title="c1">U1</font> be   <a href="unialg_1.html#NM2" title="UNIALG_1:NM.2">Universal_Algebra</a>;<br/><b>let </b><font color="Maroon" title="c2">E</font> be    <a href="alg_1.html#M1" title="ALG_1:mode.1">Congruence</a> of <font color="Maroon" title="c1">U1</font>;<br/>



<b>set </b><font color="Maroon" title="c3">f</font> =  <a href="alg_1.html#K6" title="ALG_1:func.6">Nat_Hom</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>;<br/>
<b>set </b><font color="Maroon" title="c4">u1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U1</font>;<br/>
<b>set </b><font color="Maroon" title="c5">qu</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font></span>)</span>;<br/>
<a NAME="E1:34"/><i><font color="Green" title="E28">A1</font></i>: 
( the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="alg_1.html#K4" title="ALG_1:func.4">QuotOpSeq</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="alg_1.html#K4" title="ALG_1:func.4">QuotOpSeq</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">U1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <span class="p2">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <font color="Maroon" title="c1">U1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">U1</font> )
 
<b>by </b><i><a class="ref" href="alg_1.html#D13" target="_self" title="ALG_1:def.13">Def13</a>, <a class="ref" href="unialg_1.html#D11" title="UNIALG_1:def.11">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" title="c6">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:34_1"/>
<font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <font color="Maroon" title="c1">U1</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:34_1"/><b>then </b><i><font color="Green" title="E29">A2</font></i>: 
( <font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <font color="Maroon" title="c1">U1</font></span>)</span> &amp; <font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">U1</font> &amp; <font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="alg_1.html#K4" title="ALG_1:func.4">QuotOpSeq</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font></span>)</span> &amp; <font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <span class="p2">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="alg_1.html#E1:34"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c7">o1</font> = the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> as   <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <font color="Maroon" title="c1">U1</font> <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:34_1"/><i><font color="Green" title="E30">A3</font></i>: 
<span class="p1">(<span class="default"><a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <font color="Maroon" title="c1">U1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c7">o1</font>
 <b>by </b><i><a class="txt" href="alg_1.html#E2:34_1"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="unialg_1.html#D11" title="UNIALG_1:def.11">UNIALG_1:def 11</a></i>;<br/><a NAME="E5:34_1"/><i><font color="Green" title="E31">A4</font></i>: 
<span class="p1">(<span class="default"><a href="alg_1.html#K4" title="ALG_1:func.4">QuotOpSeq</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="alg_1.html#K3" title="ALG_1:func.3">QuotOp</a> <font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c2">E</font>
 <b>by </b><i><a class="txt" href="alg_1.html#E2:34_1"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="alg_1.html#D13" target="_self" title="ALG_1:def.13">Def13</a></i>;<br/><a NAME="E6:34_1"/><i><font color="Green" title="E32">A5</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="alg_1.html#K3" title="ALG_1:func.3">QuotOp</a> <font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c2">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c7">o1</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c2">E</font></span>)</span>
 <b>by </b><i><a class="ref" href="alg_1.html#D12" target="_self" title="ALG_1:def.12">Def12</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">cl</font> =  <a href="alg_1.html#K3" title="ALG_1:func.3">QuotOp</a> <font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c2">E</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a> <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a> <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font></span>)</span> ;<br/><b>consider </b><font color="Maroon" title="c9">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:34_1"/><i><font color="Green" title="E33">A6</font></i>: 
<font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">cl</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">b</font> = <font color="Maroon" title="c9">b</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <b>by </b><i><a class="txt" href="alg_1.html#E9:34_1"><i><font color="Green" title="E33">A6</font></i></a></i>;<br/><a NAME="E11:34_1"/>
<font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c2">E</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c7">o1</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="alg_1.html#E6:34_1"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="alg_1.html#E9:34_1"><i><font color="Green" title="E33">A6</font></i></a>, <a class="ref" href="finseq_2.html#D4" title="FINSEQ_2:def.4">FINSEQ_2:def 4</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">w</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c2">E</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> <b> such that </b><br/><a NAME="E13:34_1"/><i><font color="Green" title="E34">A7</font></i>: 
( <font color="Maroon" title="c11">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c7">o1</font> )
 ;<br/><a NAME="E14:34_1"/>
 <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c8">cl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c7">o1</font>
 <b>by </b><i><a class="txt" href="alg_1.html#E9:34_1"><i><font color="Green" title="E33">A6</font></i></a>, <a class="txt" href="alg_1.html#E13:34_1"><i><font color="Green" title="E34">A7</font></i></a>, <a class="ref" href="unialg_1.html#D10" title="UNIALG_1:def.10">UNIALG_1:def 10</a></i>;<br/><b>hence </b><a NAME="E15:34_1"/>
<span class="p1">(<span class="default"><a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <font color="Maroon" title="c1">U1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <span class="p2">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="txt" href="alg_1.html#E2:34_1"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="alg_1.html#E4:34_1"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="alg_1.html#E5:34_1"><i><font color="Green" title="E31">A4</font></i></a>, <a class="ref" href="unialg_1.html#D11" title="UNIALG_1:def.11">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" title="UNIALG_1:func.4">signature</a> <font color="Maroon" title="c1">U1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font></span>)</span>
 <b>by </b><i><a class="txt" href="alg_1.html#E1:34"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="unialg_2.html#D2" title="UNIALG_2:def.2">UNIALG_2:def 2</a>,<a class="ref" href="alg_1.html#D1" target="_self" title="ALG_1:def.1">ALG_1:def 1</a><br/>

<b>let </b><font color="Maroon" title="c6">n</font> be    <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> ;<br/>

<b>assume </b><a NAME="E4:34"/>
<font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">U1</font>
 ;<br/>

<a NAME="E5:34"/><b>then </b>
<font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">U1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:34"/><b>then </b><i><font color="Green" title="E29">A8</font></i>: 
<font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="alg_1.html#K4" title="ALG_1:func.4">QuotOpSeq</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font></span>)</span>
 
<b>by </b><i><a class="txt" href="alg_1.html#E1:34"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>let </b><font color="Maroon" title="c7">o1</font> be   <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <font color="Maroon" title="c1">U1</font>;<br/><b>let </b><font color="Maroon" title="c8">o2</font> be   <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font></span>)</span>;<br/>



<b>assume </b><a NAME="E7:34"/><i><font color="Green" title="E30">A9</font></i>: 
( the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">o1</font> &amp; <font color="Maroon" title="c8">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c9">x</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">U1</font>;<br/>

<b>assume </b><a NAME="E8:34"/><i><font color="Green" title="E31">A10</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">o1</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c10">x1</font> = <font color="Maroon" title="c9">x</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U1</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">o1</font> = 
<span class="p1">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c7">o1</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U1</font>
<b>by </b><i><a class="ref" href="unialg_2.html#T2" title="UNIALG_2:th.2">UNIALG_2:2</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U1</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c7">o1</font> </span>}</span> 
<b>by </b><i><a class="ref" href="finseq_2.html#D4" title="FINSEQ_2:def.4">FINSEQ_2:def 4</a></i>
;<br/>
<a NAME="E11:34"/><b>then </b><i><font color="Green" title="E32">A11</font></i>: 
 ex <font color="Olive" title="b1">p</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U1</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  st <br/>( <font color="Olive" title="b1">p</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c10">x1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c7">o1</font> )
 
<b>by </b><i><a class="txt" href="alg_1.html#E8:34"><i><font color="Green" title="E31">A10</font></i></a></i>;<br/>
<a NAME="E12:34"/>
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">o1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U1</font> &amp; <font color="Maroon" title="c7">o1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">o1</font> )
 
<b>by </b><i><a class="txt" href="alg_1.html#E8:34"><i><font color="Green" title="E31">A10</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">ox</font> = <font color="Maroon" title="c7">o1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U1</font> ;<br/>
<i><font color="Green" title="E33">A12</font></i>: <span class="p1">(<span class="default"><a href="alg_1.html#K6" title="ALG_1:func.6">Nat_Hom</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">o1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span> = 
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c11">ox</font>
<b>by </b><i><a class="ref" href="alg_1.html#D15" target="_self" title="ALG_1:def.15">Def15</a></i>
<br/>.= 
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c2">E</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">o1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span>

;<br/>
<a NAME="E15:34"/><i><font color="Green" title="E34">A13</font></i>: 
<font color="Maroon" title="c8">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="alg_1.html#K3" title="ALG_1:func.3">QuotOp</a> <font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c2">E</font>
 
<b>by </b><i><a class="txt" href="alg_1.html#E6:34"><i><font color="Green" title="E29">A8</font></i></a>, <a class="txt" href="alg_1.html#E7:34"><i><font color="Green" title="E30">A9</font></i></a>, <a class="ref" href="alg_1.html#D13" target="_self" title="ALG_1:def.13">Def13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">fx</font> = <span class="p1">(<span class="default"><a href="alg_1.html#K6" title="ALG_1:func.6">Nat_Hom</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c9">x</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c2">E</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c13">fx</font> = <font color="Maroon" title="c12">fx</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c2">E</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<i><font color="Green" title="E35">A14</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="alg_1.html#K3" title="ALG_1:func.3">QuotOp</a> <font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c2">E</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c7">o1</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c2">E</font></span>)</span>
<b>by </b><i><a class="ref" href="alg_1.html#D12" target="_self" title="ALG_1:def.12">Def12</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c2">E</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c7">o1</font> </span>}</span> 
<b>by </b><i><a class="ref" href="finseq_2.html#D4" title="FINSEQ_2:def.4">FINSEQ_2:def 4</a></i>
;<br/>
<a NAME="E19:34"/><i><font color="Green" title="E36">A15</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="alg_1.html#K6" title="ALG_1:func.6">Nat_Hom</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">x</font>
 
<b>by </b><i><a class="ref" href="alg_1.html#T1" target="_self" title="ALG_1:th.1">Th1</a></i>;<br/>
<a NAME="E20:34"/><b>then </b><i><font color="Green" title="E37">A16</font></i>: 
<font color="Maroon" title="c13">fx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="alg_1.html#K3" title="ALG_1:func.3">QuotOp</a> <font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c2">E</font></span>)</span>
 
<b>by </b><i><a class="txt" href="alg_1.html#E11:34"><i><font color="Green" title="E32">A11</font></i></a>, <a class="txt" href="alg_1.html#E18:34"><i><font color="Green" title="E35">A14</font></i></a></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" title="c14">m</font> be    <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> ;<br/><b>assume </b><a NAME="E1:34_5"/><i><font color="Green" title="E38">A17</font></i>: 
<font color="Maroon" title="c14">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">x</font>
 ;<br/><a NAME="E2:34_5"/><b>then </b><i><font color="Green" title="E39">A18</font></i>: 
<font color="Maroon" title="c14">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="alg_1.html#K6" title="ALG_1:func.6">Nat_Hom</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c9">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="alg_1.html#T1" target="_self" title="ALG_1:th.1">Th1</a></i>;<br/><a NAME="E3:34_5"/>
( <font color="Maroon" title="c9">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">x</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U1</font> )
 <b>by </b><i><a class="txt" href="alg_1.html#E1:34_5"><i><font color="Green" title="E38">A17</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c15">xm</font> = <font color="Maroon" title="c9">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U1</font> ;<br/><b>thus </b> <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c2">E</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="alg_1.html#K6" title="ALG_1:func.6">Nat_Hom</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">xm</font>
<b>by </b><i><a class="ref" href="alg_1.html#D15" target="_self" title="ALG_1:def.15">Def15</a></i>
<br/>.= 
<font color="Maroon" title="c13">fx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font>
<b>by </b><i><a class="txt" href="alg_1.html#E2:34_5"><i><font color="Green" title="E39">A18</font></i></a>, <a class="ref" href="alg_1.html#T1" target="_self" title="ALG_1:th.1">Th1</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E22:34"/><b>then </b>
<font color="Maroon" title="c9">x</font> <a href="alg_1.html#R6" title="ALG_1:pred.6">is_representatives_FS</a> <font color="Maroon" title="c13">fx</font>
 
<b>by </b><i><a class="txt" href="alg_1.html#E19:34"><i><font color="Green" title="E36">A15</font></i></a>, <a class="ref" href="alg_1.html#D11" target="_self" title="ALG_1:def.11">Def11</a></i>;<br/>
<b>hence </b><a NAME="E23:34"/>
<span class="p1">(<span class="default"><a href="alg_1.html#K6" title="ALG_1:func.6">Nat_Hom</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">o1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">o2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="alg_1.html#K6" title="ALG_1:func.6">Nat_Hom</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c9">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="alg_1.html#E14:34"><i><font color="Green" title="E33">A12</font></i></a>, <a class="txt" href="alg_1.html#E15:34"><i><font color="Green" title="E34">A13</font></i></a>, <a class="txt" href="alg_1.html#E20:34"><i><font color="Green" title="E37">A16</font></i></a>, <a class="ref" href="alg_1.html#D12" target="_self" title="ALG_1:def.12">Def12</a></i>;<br/>


</div>
