<?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">UU</font> =  <a href="unialg_1.html#G1">UAStr</a>(# <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">E</font></span>)</span>,<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> #);<br/>
<a NAME="E1:31"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a><br/> for <font color="Olive">h</font> being  <a href="partfun1.html#NM1">PartFunc</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="eqrel_1.html#K9">Class</a> <font color="Maroon">E</font>  st <font color="Olive">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="Olive">h</font> <a href="hidden.html#R1">=</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="funct_1.html#K1">.</a> <font color="Olive">n</font> holds <br/><font color="Olive">h</font> is <a href="unialg_1.html#V1">homogeneous</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">h</font> be   <a href="partfun1.html#NM1">PartFunc</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="eqrel_1.html#K9">Class</a> <font color="Maroon">E</font>;<br/>



<b>assume </b><a NAME="E1:31_1"/><i><font color="Green">E33</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> &amp; <font color="Maroon">h</font> <a href="hidden.html#R1">=</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="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 ;<br/>

<a NAME="E2:31_1"/><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> <span class="p2">(<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></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:31_1"/><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/>;<br/>
<a NAME="E4:31_1"/><b>then </b>
<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>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">o</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="E6:31_1"/>
<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">o</font>,<font color="Maroon">E</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:31_1"/>
<font color="Maroon">h</font> is <a href="unialg_1.html#V1">homogeneous</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:31"/><b>then </b><i><font color="Green">E34</font></i>: 
the <a href="unialg_1.html#U1">charact</a> of <a href="unialg_1.html#G1">UAStr</a>(# <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">E</font></span>)</span>,<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> #) is <a href="unialg_1.html#V4">homogeneous</a>
 
<b>by </b><i><a class="ref" href="unialg_1.html#D4">UNIALG_1:def 4</a></i>;<br/>
<a NAME="E3:31"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a><br/> for <font color="Olive">h</font> being  <a href="partfun1.html#NM1">PartFunc</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="eqrel_1.html#K9">Class</a> <font color="Maroon">E</font>  st <font color="Olive">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="Olive">h</font> <a href="hidden.html#R1">=</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="funct_1.html#K1">.</a> <font color="Olive">n</font> holds <br/><font color="Olive">h</font> is <a href="unialg_1.html#V2">quasi_total</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">h</font> be   <a href="partfun1.html#NM1">PartFunc</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="eqrel_1.html#K9">Class</a> <font color="Maroon">E</font>;<br/>



<b>assume </b><a NAME="E1:31_2"/><i><font color="Green">E35</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> &amp; <font color="Maroon">h</font> <a href="hidden.html#R1">=</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="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 ;<br/>

<a NAME="E2:31_2"/><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> <span class="p2">(<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></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:31_2"/><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/>;<br/>
<a NAME="E4:31_2"/><b>then </b>
<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>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">o</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="E6:31_2"/>
<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">o</font>,<font color="Maroon">E</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:31_2"/>
<font color="Maroon">h</font> is <a href="unialg_1.html#V2">quasi_total</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E4:31"/><b>then </b><i><font color="Green">E41</font></i>: 
the <a href="unialg_1.html#U1">charact</a> of <a href="unialg_1.html#G1">UAStr</a>(# <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">E</font></span>)</span>,<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> #) is <a href="unialg_1.html#V5">quasi_total</a>
 
<b>by </b><i><a class="ref" href="unialg_1.html#D5">UNIALG_1:def 5</a></i>;<br/>
<a NAME="E5:31"/>
for <font color="Olive">n</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">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> holds <br/> not <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="Olive">n</font> is <a href="xboole_0.html#V1">empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:31_3"/><i><font color="Green">E46</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>
 ;<br/>

<a NAME="E2:31_3"/><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> <span class="p2">(<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></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:31_3"/><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/>;<br/>
<a NAME="E4:31_3"/><b>then </b><i><font color="Green">E47</font></i>: 
<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>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="alg_1.html#T1" target="_self">Th1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">o</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="E7:31_3"/>
<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">o</font>,<font color="Maroon">E</font>
 
<b>by </b><i><a class="ref" href="alg_1.html#T1" target="_self">Th1</a>, </i>;<br/>
<b>hence </b><a NAME="E8:31_3"/>
not <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> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:31"/><b>then </b><i><font color="Green">E48</font></i>: 
the <a href="unialg_1.html#U1">charact</a> of <a href="unialg_1.html#G1">UAStr</a>(# <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">E</font></span>)</span>,<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> #) is <a href="relat_1.html#V2">non-empty</a>
 
<b>by </b><i><a class="ref" href="funct_1.html#D15">FUNCT_1:def 15</a></i>;<br/>
<a NAME="E7:31"/>
the <a href="unialg_1.html#U1">charact</a> of <a href="unialg_1.html#G1">UAStr</a>(# <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">E</font></span>)</span>,<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#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_4"/>
the <a href="unialg_1.html#U1">charact</a> of <a href="unialg_1.html#G1">UAStr</a>(# <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">E</font></span>)</span>,<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="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:31_4"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <a href="unialg_1.html#G1">UAStr</a>(# <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">E</font></span>)</span>,<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> 0
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E3:31_4"/>
 <a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <a href="unialg_1.html#G1">UAStr</a>(# <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">E</font></span>)</span>,<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>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:31_4"/>
contradiction
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:31"/>
 <a href="unialg_1.html#G1">UAStr</a>(# <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">E</font></span>)</span>,<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> #) is  <a href="unialg_1.html#V3">strict</a> <a href="unialg_1.html#NM2">Universal_Algebra</a>
 <b>by </b><i>, , <a class="ref" href="alg_1.html#T1" target="_self">Th1</a>, <a class="ref" href="unialg_1.html#D7">UNIALG_1:def 7</a>, <a class="ref" href="unialg_1.html#D8">UNIALG_1:def 8</a>, <a class="ref" href="unialg_1.html#D9">UNIALG_1:def 9</a></i>;<br/>


</div>
