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

<span class="kw">consider </span><font color="Maroon" title="c3">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">J</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">ua</font> =  <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K21" title="PRALG_1:func.21">ProdOpSeq</a> <font color="Maroon" title="c2">A</font></span>)</span> #);<br/>
<span class="kw">set </span><font color="Maroon" title="c5">pr</font> =  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span>;<br/>
<a NAME="E2:55_1_1"/>
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"><span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>  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> 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="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K21" title="PRALG_1:func.21">ProdOpSeq</a> <font color="Maroon" title="c2">A</font></span>)</span> #) &amp; <font color="Olive" title="b2">h</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 <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K21" title="PRALG_1:func.21">ProdOpSeq</a> <font color="Maroon" title="c2">A</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="55_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">h</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>  st <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 <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K21" title="PRALG_1:func.21">ProdOpSeq</a> <font color="Maroon" title="c2">A</font></span>)</span> #) &amp; <font color="Olive" title="b1">h</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 <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K21" title="PRALG_1:func.21">ProdOpSeq</a> <font color="Maroon" title="c2">A</font></span>)</span> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> holds <br/><font color="Olive" title="b1">h</font> is  <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c7">h</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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 <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K21" title="PRALG_1:func.21">ProdOpSeq</a> <font color="Maroon" title="c2">A</font></span>)</span> #) &amp; <font color="Maroon" title="c7">h</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 <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K21" title="PRALG_1:func.21">ProdOpSeq</a> <font color="Maroon" title="c2">A</font></span>)</span> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> implies <font color="Maroon" title="c7">h</font> is  <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>  )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:55_1_1_1"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
<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 <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K21" title="PRALG_1:func.21">ProdOpSeq</a> <font color="Maroon" title="c2">A</font></span>)</span> #)
 <span class="kw">and </span><br/><a NAME="E2:55_1_1_1"/><span class="lab"><font color="Green" title="E27">A2</font></span>: 
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="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K21" title="PRALG_1:func.21">ProdOpSeq</a> <font color="Maroon" title="c2">A</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> <font color="Maroon" title="c7">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">h</font> is  <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a> </span><br/>

<a NAME="E3:55_1_1_1"/>
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="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K21" title="PRALG_1:func.21">ProdOpSeq</a> <font color="Maroon" title="c2">A</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"><a href="pralg_1.html#K19" title="PRALG_1:func.19">[[:</a><span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K20" title="PRALG_1:func.20">ProdOp</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c6">n</font></span>)</span></span><a href="pralg_1.html#K19" title="PRALG_1:func.19">:]]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55_1_1_1"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="pralg_1.html#D23" target="_self" title="PRALG_1:def.23">Def23</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:55_1_1_1"/>
<font color="Maroon" title="c7">h</font> is  <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55_1_1_1"><span class="lab"><font color="Green" title="E27">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:55_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A3</font></span>: 
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="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K21" title="PRALG_1:func.21">ProdOpSeq</a> <font color="Maroon" title="c2">A</font></span>)</span> #) is  <a href="unialg_1.html#V5" title="UNIALG_1:attr.5">quasi_total</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_1.html#D5" title="UNIALG_1:def.5">UNIALG_1:def 5</a></span>;<br/>
<a NAME="E4:55_1_1"/>
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"><span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>  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> 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="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K21" title="PRALG_1:func.21">ProdOpSeq</a> <font color="Maroon" title="c2">A</font></span>)</span> #) &amp; <font color="Olive" title="b2">h</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 <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K21" title="PRALG_1:func.21">ProdOpSeq</a> <font color="Maroon" title="c2">A</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="55_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">h</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>  st <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 <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K21" title="PRALG_1:func.21">ProdOpSeq</a> <font color="Maroon" title="c2">A</font></span>)</span> #) &amp; <font color="Olive" title="b1">h</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 <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K21" title="PRALG_1:func.21">ProdOpSeq</a> <font color="Maroon" title="c2">A</font></span>)</span> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> holds <br/><font color="Olive" title="b1">h</font> is  <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c7">h</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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 <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K21" title="PRALG_1:func.21">ProdOpSeq</a> <font color="Maroon" title="c2">A</font></span>)</span> #) &amp; <font color="Maroon" title="c7">h</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 <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K21" title="PRALG_1:func.21">ProdOpSeq</a> <font color="Maroon" title="c2">A</font></span>)</span> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> implies <font color="Maroon" title="c7">h</font> is  <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a>  )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:55_1_1_2"/><span class="lab"><font color="Green" title="E27">A4</font></span>: 
<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 <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K21" title="PRALG_1:func.21">ProdOpSeq</a> <font color="Maroon" title="c2">A</font></span>)</span> #)
 <span class="kw">and </span><br/><a NAME="E2:55_1_1_2"/><span class="lab"><font color="Green" title="E28">A5</font></span>: 
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="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K21" title="PRALG_1:func.21">ProdOpSeq</a> <font color="Maroon" title="c2">A</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> <font color="Maroon" title="c7">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">h</font> is  <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a> </span><br/>

<a NAME="E3:55_1_1_2"/>
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="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K21" title="PRALG_1:func.21">ProdOpSeq</a> <font color="Maroon" title="c2">A</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"><a href="pralg_1.html#K19" title="PRALG_1:func.19">[[:</a><span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K20" title="PRALG_1:func.20">ProdOp</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c6">n</font></span>)</span></span><a href="pralg_1.html#K19" title="PRALG_1:func.19">:]]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55_1_1_2"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="ref" href="pralg_1.html#D23" target="_self" title="PRALG_1:def.23">Def23</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:55_1_1_2"/>
<font color="Maroon" title="c7">h</font> is  <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55_1_1_2"><span class="lab"><font color="Green" title="E28">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:55_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A6</font></span>: 
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="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K21" title="PRALG_1:func.21">ProdOpSeq</a> <font color="Maroon" title="c2">A</font></span>)</span> #) is  <a href="unialg_1.html#V4" title="UNIALG_1:attr.4">homogeneous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_1.html#D4" title="UNIALG_1:def.4">UNIALG_1:def 4</a></span>;<br/>
<a NAME="E6:55_1_1"/>
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> 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="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K21" title="PRALG_1:func.21">ProdOpSeq</a> <font color="Maroon" title="c2">A</font></span>)</span> #) holds <br/> not 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="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K21" title="PRALG_1:func.21">ProdOpSeq</a> <font color="Maroon" title="c2">A</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="55_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">n</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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 <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K21" title="PRALG_1:func.21">ProdOpSeq</a> <font color="Maroon" title="c2">A</font></span>)</span> #) implies  not 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="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K21" title="PRALG_1:func.21">ProdOpSeq</a> <font color="Maroon" title="c2">A</font></span>)</span> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:55_1_1_3"/><span class="lab"><font color="Green" title="E28">A7</font></span>: 
<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 <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K21" title="PRALG_1:func.21">ProdOpSeq</a> <font color="Maroon" title="c2">A</font></span>)</span> #)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not 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="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K21" title="PRALG_1:func.21">ProdOpSeq</a> <font color="Maroon" title="c2">A</font></span>)</span> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> </span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">n9</font> = <font color="Maroon" title="c6">n</font> as    <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/>
<a NAME="E3:55_1_1_3"/>
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="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K21" title="PRALG_1:func.21">ProdOpSeq</a> <font color="Maroon" title="c2">A</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"><a href="pralg_1.html#K19" title="PRALG_1:func.19">[[:</a><span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K20" title="PRALG_1:func.20">ProdOp</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c7">n9</font></span>)</span></span><a href="pralg_1.html#K19" title="PRALG_1:func.19">:]]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55_1_1_3"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="pralg_1.html#D23" target="_self" title="PRALG_1:def.23">Def23</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:55_1_1_3"/>
not 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="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K21" title="PRALG_1:func.21">ProdOpSeq</a> <font color="Maroon" title="c2">A</font></span>)</span> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:55_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A8</font></span>: 
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="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K21" title="PRALG_1:func.21">ProdOpSeq</a> <font color="Maroon" title="c2">A</font></span>)</span> #) is  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D15" title="FUNCT_1:def.15">FUNCT_1:def 15</a></span>;<br/>
 <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="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K21" title="PRALG_1:func.21">ProdOpSeq</a> <font color="Maroon" title="c2">A</font></span>)</span> #) = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K13" title="PRALG_1:func.13">ComSign</a> <font color="Maroon" title="c2">A</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pralg_1.html#D23" target="_self" title="PRALG_1:def.23">Def23</a></span>
<br/>.= 
 <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"><font color="Maroon" title="c2">A</font> <a href="pralg_1.html#K11" title="PRALG_1:func.11">.</a> <font color="Maroon" title="c3">j</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pralg_1.html#D14" target="_self" title="PRALG_1:def.14">Def14</a></span>
<br/>.= 
 <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"><font color="Maroon" title="c2">A</font> <a href="pralg_1.html#K11" title="PRALG_1:func.11">.</a> <font color="Maroon" title="c3">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_1.html#D11" title="UNIALG_1:def.11">UNIALG_1:def 11</a></span>
;<br/>
<a NAME="E9:55_1_1"/><span class="kw">then </span>
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="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K21" title="PRALG_1:func.21">ProdOpSeq</a> <font color="Maroon" title="c2">A</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> 
 
;<br/>
<span class="kw">hence </span><a NAME="E10:55_1_1"/>
 <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K21" title="PRALG_1:func.21">ProdOpSeq</a> <font color="Maroon" title="c2">A</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:55_1_1"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="txt" href="#E5:55_1_1"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="txt" href="#E7:55_1_1"><span class="lab"><font color="Green" title="E28">A8</font></span></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></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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