<?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">UU</font> =  <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <span class="p1">(<span class="default"><a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <font color="Maroon" title="c2">E</font></span>)</span>,<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> #);<br/>
<a NAME="E1:31"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">h</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K9" title="EQREL_1:func.9">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="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <font color="Maroon" title="c2">E</font>  st <font color="Olive" title="b1">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="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> holds <br/><font color="Olive" title="b2">h</font> is <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">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" title="c4">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c5">h</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K9" title="EQREL_1:func.9">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="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <font color="Maroon" title="c2">E</font>;<br/>



<b>assume </b><a NAME="E1:31_1"/><i><font color="Green" title="E26">A1</font></i>: 
( <font color="Maroon" title="c4">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="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> )
 ;<br/>

<a NAME="E2:31_1"/><b>then </b>
<font color="Maroon" title="c4">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="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></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="E3:31_1"/><b>then </b>
<font color="Maroon" title="c4">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="alg_1.html#D13" target="_self" title="ALG_1:def.13">Def13</a></i>;<br/>
<a NAME="E4:31_1"/><b>then </b>
<font color="Maroon" title="c4">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>
 
<b>by </b><i><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="c6">o</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="c4">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="E6:31_1"/>
<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="c4">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="c6">o</font>,<font color="Maroon" title="c2">E</font>
 
<b>by </b><i><a class="txt" href="alg_1.html#E1:31_1"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="alg_1.html#D13" target="_self" title="ALG_1:def.13">Def13</a></i>;<br/>
<b>hence </b><a NAME="E7:31_1"/>
<font color="Maroon" title="c5">h</font> is <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a>
 <b>by </b><i><a class="txt" href="alg_1.html#E1:31_1"><i><font color="Green" title="E26">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:31"/><b>then </b><i><font color="Green" title="E26">A2</font></i>: 
the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <span class="p1">(<span class="default"><a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <font color="Maroon" title="c2">E</font></span>)</span>,<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> #) is <a href="unialg_1.html#V4" title="UNIALG_1:attr.4">homogeneous</a>
 
<b>by </b><i><a class="ref" href="unialg_1.html#D4" title="UNIALG_1:def.4">UNIALG_1:def 4</a></i>;<br/>
<a NAME="E3:31"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">h</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K9" title="EQREL_1:func.9">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="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <font color="Maroon" title="c2">E</font>  st <font color="Olive" title="b1">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="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> holds <br/><font color="Olive" title="b2">h</font> is <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">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" title="c4">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c5">h</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K9" title="EQREL_1:func.9">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="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <font color="Maroon" title="c2">E</font>;<br/>



<b>assume </b><a NAME="E1:31_2"/><i><font color="Green" title="E27">A3</font></i>: 
( <font color="Maroon" title="c4">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="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> )
 ;<br/>

<a NAME="E2:31_2"/><b>then </b>
<font color="Maroon" title="c4">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="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></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="E3:31_2"/><b>then </b>
<font color="Maroon" title="c4">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="alg_1.html#D13" target="_self" title="ALG_1:def.13">Def13</a></i>;<br/>
<a NAME="E4:31_2"/><b>then </b>
<font color="Maroon" title="c4">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>
 
<b>by </b><i><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="c6">o</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="c4">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="E6:31_2"/>
<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="c4">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="c6">o</font>,<font color="Maroon" title="c2">E</font>
 
<b>by </b><i><a class="txt" href="alg_1.html#E1:31_2"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="alg_1.html#D13" target="_self" title="ALG_1:def.13">Def13</a></i>;<br/>
<b>hence </b><a NAME="E7:31_2"/>
<font color="Maroon" title="c5">h</font> is <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>
 <b>by </b><i><a class="txt" href="alg_1.html#E1:31_2"><i><font color="Green" title="E27">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:31"/><b>then </b><i><font color="Green" title="E27">A4</font></i>: 
the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <span class="p1">(<span class="default"><a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <font color="Maroon" title="c2">E</font></span>)</span>,<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> #) is <a href="unialg_1.html#V5" title="UNIALG_1:attr.5">quasi_total</a>
 
<b>by </b><i><a class="ref" href="unialg_1.html#D5" title="UNIALG_1:def.5">UNIALG_1:def 5</a></i>;<br/>
<a NAME="E5:31"/>
for <font color="Olive" title="b1">n</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">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> holds <br/> not <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="Olive" title="b1">n</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">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" title="c4">n</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:31_3"/><i><font color="Green" title="E28">A5</font></i>: 
<font color="Maroon" title="c4">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>
 ;<br/>

<a NAME="E2:31_3"/><b>then </b>
<font color="Maroon" title="c4">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="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></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="E3:31_3"/><b>then </b>
<font color="Maroon" title="c4">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="alg_1.html#D13" target="_self" title="ALG_1:def.13">Def13</a></i>;<br/>
<a NAME="E4:31_3"/><b>then </b><i><font color="Green" title="E29">A6</font></i>: 
<font color="Maroon" title="c4">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>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">n</font> = <font color="Maroon" title="c4">n</font> as    <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>  <b>by </b><i><a class="txt" href="alg_1.html#E1:31_3"><i><font color="Green" title="E28">A5</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">o</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="c5">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="txt" href="alg_1.html#E4:31_3"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E7:31_3"/>
<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="c5">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="c6">o</font>,<font color="Maroon" title="c2">E</font>
 
<b>by </b><i><a class="txt" href="alg_1.html#E1:31_3"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="alg_1.html#D13" target="_self" title="ALG_1:def.13">Def13</a></i>;<br/>
<b>hence </b><a NAME="E8:31_3"/>
not <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="c4">n</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:31"/><b>then </b><i><font color="Green" title="E28">A7</font></i>: 
the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <span class="p1">(<span class="default"><a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <font color="Maroon" title="c2">E</font></span>)</span>,<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> #) is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>
 
<b>by </b><i><a class="ref" href="funct_1.html#D15" title="FUNCT_1:def.15">FUNCT_1:def 15</a></i>;<br/>
<a NAME="E7:31"/>
the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <span class="p1">(<span class="default"><a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <font color="Maroon" title="c2">E</font></span>)</span>,<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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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" title="UNIALG_1:sel.1">charact</a> of <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <span class="p1">(<span class="default"><a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <font color="Maroon" title="c2">E</font></span>)</span>,<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="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<a NAME="E2:31_4"/><b>then </b><i><font color="Green" title="E29">A8</font></i>: 
 <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 <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <span class="p1">(<span class="default"><a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <font color="Maroon" title="c2">E</font></span>)</span>,<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> 0
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E3:31_4"/>
 <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 <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <span class="p1">(<span class="default"><a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <font color="Maroon" title="c2">E</font></span>)</span>,<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>
 
<b>by </b><i><a class="ref" href="alg_1.html#D13" target="_self" title="ALG_1:def.13">Def13</a></i>;<br/>
<b>hence </b><a NAME="E4:31_4"/>
contradiction
 <b>by </b><i><a class="txt" href="alg_1.html#E2:31_4"><i><font color="Green" title="E29">A8</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:31"/>
 <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <span class="p1">(<span class="default"><a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <font color="Maroon" title="c2">E</font></span>)</span>,<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> #) is  <a href="unialg_1.html#V3" title="UNIALG_1:attr.3">strict</a> <a href="unialg_1.html#NM2" title="UNIALG_1:NM.2">Universal_Algebra</a>
 <b>by </b><i><a class="txt" href="alg_1.html#E2:31"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="alg_1.html#E4:31"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="alg_1.html#E6:31"><i><font color="Green" title="E28">A7</font></i></a>, <a class="ref" href="unialg_1.html#D7" title="UNIALG_1:def.7">UNIALG_1:def 7</a>, <a class="ref" href="unialg_1.html#D8" title="UNIALG_1:def.8">UNIALG_1:def 8</a>, <a class="ref" href="unialg_1.html#D9" title="UNIALG_1:def.9">UNIALG_1:def 9</a></i>;<br/>


</div>
