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

<b>set </b><font color="Maroon">ua</font> =  <a href="unialg_1.html#G1">UAStr</a>(# <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" target="_self">Carrier</a> <font color="Maroon">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K24" target="_self">ProdOpSeq</a> <font color="Maroon">A</font></span>)</span> #);<br/>
<b>set </b><font color="Maroon">pr</font> =  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" target="_self">Carrier</a> <font color="Maroon">A</font></span>)</span>;<br/>
<a NAME="E1:50_1_1"/>
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="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" target="_self">Carrier</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a> , <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" target="_self">Carrier</a> <font color="Maroon">A</font></span>)</span>  st <font color="Olive">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 <a href="unialg_1.html#G1">UAStr</a>(# <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" target="_self">Carrier</a> <font color="Maroon">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K24" target="_self">ProdOpSeq</a> <font color="Maroon">A</font></span>)</span> #) &amp; <font color="Olive">h</font> <a href="hidden.html#R1">=</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="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" target="_self">Carrier</a> <font color="Maroon">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K24" target="_self">ProdOpSeq</a> <font color="Maroon">A</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="50_1_1_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="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" target="_self">Carrier</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a> , <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" target="_self">Carrier</a> <font color="Maroon">A</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:50_1_1_1"/><i><font color="Green">E23</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 <a href="unialg_1.html#G1">UAStr</a>(# <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" target="_self">Carrier</a> <font color="Maroon">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K24" target="_self">ProdOpSeq</a> <font color="Maroon">A</font></span>)</span> #) &amp; 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="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" target="_self">Carrier</a> <font color="Maroon">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K24" target="_self">ProdOpSeq</a> <font color="Maroon">A</font></span>)</span> #) <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> )
 ;<br/>

<a NAME="E2:50_1_1_1"/><b>then </b>
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="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" target="_self">Carrier</a> <font color="Maroon">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K24" target="_self">ProdOpSeq</a> <font color="Maroon">A</font></span>)</span> #) <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="pralg_1.html#K22" target="_self">[[:</a><span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K23" target="_self">ProdOp</a> <font color="Maroon">A</font>,<font color="Maroon">n</font></span>)</span></span><a href="pralg_1.html#K22" target="_self">:]]</a></span>
 
<b>by </b><i><a class="txt" href="pralg_1.html#E2:50_1_1"><i><font color="Green">E26</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:50_1_1_1"/>
<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="E2:50_1_1"/><b>then </b><i><font color="Green">E26</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="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" target="_self">Carrier</a> <font color="Maroon">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K24" target="_self">ProdOpSeq</a> <font color="Maroon">A</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="E3:50_1_1"/>
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="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" target="_self">Carrier</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a> , <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" target="_self">Carrier</a> <font color="Maroon">A</font></span>)</span>  st <font color="Olive">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 <a href="unialg_1.html#G1">UAStr</a>(# <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" target="_self">Carrier</a> <font color="Maroon">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K24" target="_self">ProdOpSeq</a> <font color="Maroon">A</font></span>)</span> #) &amp; <font color="Olive">h</font> <a href="hidden.html#R1">=</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="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" target="_self">Carrier</a> <font color="Maroon">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K24" target="_self">ProdOpSeq</a> <font color="Maroon">A</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="50_1_1_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="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" target="_self">Carrier</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a> , <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" target="_self">Carrier</a> <font color="Maroon">A</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:50_1_1_2"/><i><font color="Green">E38</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 <a href="unialg_1.html#G1">UAStr</a>(# <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" target="_self">Carrier</a> <font color="Maroon">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K24" target="_self">ProdOpSeq</a> <font color="Maroon">A</font></span>)</span> #) &amp; 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="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" target="_self">Carrier</a> <font color="Maroon">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K24" target="_self">ProdOpSeq</a> <font color="Maroon">A</font></span>)</span> #) <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> )
 ;<br/>

<a NAME="E2:50_1_1_2"/><b>then </b>
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="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" target="_self">Carrier</a> <font color="Maroon">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K24" target="_self">ProdOpSeq</a> <font color="Maroon">A</font></span>)</span> #) <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="pralg_1.html#K22" target="_self">[[:</a><span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K23" target="_self">ProdOp</a> <font color="Maroon">A</font>,<font color="Maroon">n</font></span>)</span></span><a href="pralg_1.html#K22" target="_self">:]]</a></span>
 
<b>by </b><i><a class="txt" href="pralg_1.html#E2:50_1_1"><i><font color="Green">E26</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:50_1_1_2"/>
<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="E4:50_1_1"/><b>then </b><i><font color="Green">E39</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="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" target="_self">Carrier</a> <font color="Maroon">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K24" target="_self">ProdOpSeq</a> <font color="Maroon">A</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="E5:50_1_1"/>
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> 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="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" target="_self">Carrier</a> <font color="Maroon">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K24" target="_self">ProdOpSeq</a> <font color="Maroon">A</font></span>)</span> #) holds <br/> not 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="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" target="_self">Carrier</a> <font color="Maroon">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K24" target="_self">ProdOpSeq</a> <font color="Maroon">A</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="50_1_1_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="E1:50_1_1_3"/><i><font color="Green">E41</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 <a href="unialg_1.html#G1">UAStr</a>(# <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" target="_self">Carrier</a> <font color="Maroon">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K24" target="_self">ProdOpSeq</a> <font color="Maroon">A</font></span>)</span> #)
 ;<br/>

<b>then reconsider </b><font color="Maroon">n'</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E3:50_1_1_3"/>
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="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" target="_self">Carrier</a> <font color="Maroon">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K24" target="_self">ProdOpSeq</a> <font color="Maroon">A</font></span>)</span> #) <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="pralg_1.html#K22" target="_self">[[:</a><span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K23" target="_self">ProdOp</a> <font color="Maroon">A</font>,<font color="Maroon">n'</font></span>)</span></span><a href="pralg_1.html#K22" target="_self">:]]</a></span>
 
<b>by </b><i>, <a class="txt" href="pralg_1.html#E2:50_1_1"><i><font color="Green">E26</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:50_1_1_3"/>
not 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="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" target="_self">Carrier</a> <font color="Maroon">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K24" target="_self">ProdOpSeq</a> <font color="Maroon">A</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="E6:50_1_1"/><b>then </b><i><font color="Green">E42</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="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" target="_self">Carrier</a> <font color="Maroon">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K24" target="_self">ProdOpSeq</a> <font color="Maroon">A</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/>
<b>consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font>;<br/>
<a NAME="E8:50_1_1"/><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
 <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="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" target="_self">Carrier</a> <font color="Maroon">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K24" target="_self">ProdOpSeq</a> <font color="Maroon">A</font></span>)</span> #) = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K13" target="_self">ComSign</a> <font color="Maroon">A</font></span>)</span>
<b>by </b><i><a class="txt" href="pralg_1.html#E2:50_1_1"><i><font color="Green">E26</font></i></a></i>
<br/>.= 
 <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"><font color="Maroon">A</font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pralg_1.html#T1" target="_self">Th1</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>
;<br/>
<a NAME="E10:50_1_1"/><b>then </b>
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="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" target="_self">Carrier</a> <font color="Maroon">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K24" target="_self">ProdOpSeq</a> <font color="Maroon">A</font></span>)</span> #) <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="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<b>hence </b><a NAME="E11:50_1_1"/>
 <a href="unialg_1.html#G1">UAStr</a>(# <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" target="_self">Carrier</a> <font color="Maroon">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K24" target="_self">ProdOpSeq</a> <font color="Maroon">A</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="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/>


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