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

<b>let </b><font color="Maroon" title="c1">S</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">i</font>, <font color="Maroon" title="c3">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>



<b>reconsider </b><font color="Maroon" title="c4">i'</font> = <font color="Maroon" title="c2">i</font>, <font color="Maroon" title="c5">j'</font> = <font color="Maroon" title="c3">j</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="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>assume </b><a NAME="E2:32"/><i><font color="Green" title="E18">A1</font></i>: 
( <font color="Maroon" title="c2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">S</font> &amp; <font color="Maroon" title="c3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">S</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">i</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c6">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>consider </b><font color="Maroon" title="c7">A0</font> being   <a href="unialg_1.html#NM2" title="UNIALG_1:NM.2">Universal_Algebra</a><b> such that </b><br/><a NAME="E4:32"/><i><font color="Green" title="E19">A2</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">A0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">X</font> &amp;  <a href="msualg_1.html#K6" title="MSUALG_1:func.6">signature</a> <font color="Maroon" title="c7">A0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">S</font> )
 <b>by </b><i><a class="ref" href="aofa_000.html#T13" target="_self" title="AOFA_000:th.13">Th13</a></i>;<br/>
<b>let </b><font color="Maroon" title="c8">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">j</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c6">X</font>;<br/>

<a NAME="E5:32"/>
<font color="Maroon" title="c5">j'</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c6">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 
<b>by </b><i><a class="ref" href="catalg_1.html#T6" title="CATALG_1:th.6">CATALG_1:6</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">f0</font> = <font color="Maroon" title="c8">f</font> as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c6">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> ,<font color="Maroon" title="c6">X</font> <b>by </b><i><a class="ref" href="relset_1.html#T15" title="RELSET_1:th.15">RELSET_1:15</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">z</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">j</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c6">X</font>;<br/>
<a NAME="E8:32"/><i><font color="Green" title="E20">A3</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">f0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">j</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c6">X</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:32"/>
<font color="Maroon" title="c9">f0</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="32_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c12">y</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="unialg_1.html#D1" title="UNIALG_1:def.1">UNIALG_1:def 1</a><br/>

<b>assume </b><a NAME="E1:32_1"/>
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">f0</font> &amp; <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">f0</font> )
 ;<br/>

<a NAME="E2:32_1"/><b>then </b>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">j</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">j</font> )
 
<b>by </b><i><a class="txt" href="aofa_000.html#E8:32"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/>
<b>hence </b><a NAME="E3:32_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">y</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c11">f0</font> = <font color="Maroon" title="c9">f0</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a> <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c6">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> ,<font color="Maroon" title="c6">X</font> ;<br/>
<a NAME="E11:32"/><i><font color="Green" title="E21">A4</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">j</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E12:32"/><b>then </b><i><font color="Green" title="E22">A5</font></i>: 
 <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c11">f0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">j</font>
 
<b>by </b><i><a class="txt" href="aofa_000.html#E8:32"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="unialg_1.html#D10" title="UNIALG_1:def.10">UNIALG_1:def 10</a></i>;<br/>
<b>set </b><font color="Maroon" title="c12">ch</font> = the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c7">A0</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c11">f0</font>;<br/>
<a NAME="E13:32"/>
<font color="Maroon" title="c11">f0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<font color="Maroon" title="c6">X</font>
 
<b>by </b><i><a class="ref" href="partfun1.html#T119" title="PARTFUN1:th.119">PARTFUN1:119</a></i>;<br/>
<a NAME="E14:32"/><b>then </b>
( <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">f0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<font color="Maroon" title="c6">X</font> &amp;  <a href="pua2mss1.html#K1" title="PUA2MSS1:func.1">rng</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c7">A0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<font color="Maroon" title="c6">X</font> )
 
<b>by </b><i><a class="txt" href="aofa_000.html#E4:32"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E15:32"/><b>then </b>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c7">A0</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c11">f0</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K1" title="PUA2MSS1:func.1">rng</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c7">A0</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">f0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <span class="p1">(<span class="default"><a href="pua2mss1.html#K1" title="PUA2MSS1:func.1">rng</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c7">A0</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">f0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<font color="Maroon" title="c6">X</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a>, <a class="ref" href="polynom1.html#T6" title="POLYNOM1:th.6">POLYNOM1:6</a></i>;<br/>
<a NAME="E16:32"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c7">A0</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c11">f0</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<font color="Maroon" title="c6">X</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">ch</font> = the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c7">A0</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c11">f0</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<font color="Maroon" title="c6">X</font> <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E18:32"/><i><font color="Green" title="E23">A6</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">ch</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c7">A0</font>
 
<b>by </b><i><a class="ref" href="funct_7.html#T32" title="FUNCT_7:th.32">FUNCT_7:32</a></i>;<br/>
<b>set </b><font color="Maroon" title="c14">A</font> =  <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c13">ch</font> #);<br/>
<a NAME="E19:32"/><i><font color="Green" title="E24">A7</font></i>: 
 <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c13">ch</font> #) is <a href="unialg_1.html#V7" title="UNIALG_1:attr.7">quasi_total</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="unialg_1.html#D5" title="UNIALG_1:def.5">UNIALG_1:def 5</a>,<a class="ref" href="unialg_1.html#D8" title="UNIALG_1:def.8">UNIALG_1:def 8</a><br/><b>let </b><font color="Maroon" title="c16">h</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c13">ch</font> #) <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c13">ch</font> #);<br/>



<b>assume </b><a NAME="E1:32_2"/><i><font color="Green" title="E25">A8</font></i>: 
( <font color="Maroon" title="c15">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>(# <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c13">ch</font> #) &amp; <font color="Maroon" title="c16">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>(# <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c13">ch</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font> )
 ;<br/>

<a NAME="E2:32_2"/><b>then </b>
( ( <font color="Maroon" title="c15">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> implies <font color="Maroon" title="c16">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f0</font> ) &amp; ( <font color="Maroon" title="c15">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">i</font> implies <font color="Maroon" title="c16">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 <font color="Maroon" title="c7">A0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font> ) &amp; the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c7">A0</font> is <a href="unialg_1.html#V5" title="UNIALG_1:attr.5">quasi_total</a> )
 
<b>by </b><i><a class="txt" href="aofa_000.html#E18:32"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/>
<b>hence </b><a NAME="E3:32_2"/>
<font color="Maroon" title="c16">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="aofa_000.html#E4:32"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="aofa_000.html#E12:32"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="aofa_000.html#E8:32"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="comput_1.html#T25" title="COMPUT_1:th.25">COMPUT_1:25</a>, <a class="txt" href="aofa_000.html#E1:32_2"><i><font color="Green" title="E25">A8</font></i></a>, <a class="txt" href="aofa_000.html#E18:32"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="unialg_1.html#D5" title="UNIALG_1:def.5">UNIALG_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:32"/><i><font color="Green" title="E25">A9</font></i>: 
 <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c13">ch</font> #) is <a href="unialg_1.html#V8" title="UNIALG_1:attr.8">non-empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:32_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>(# <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c13">ch</font> #) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="aofa_000.html#E18:32"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="unialg_1.html#D9" title="UNIALG_1:def.9">UNIALG_1:def 9</a><br/>

<b>assume </b><a NAME="E2:32_3"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pua2mss1.html#K1" title="PUA2MSS1:func.1">rng</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>(# <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c13">ch</font> #)
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_1.html#D9" title="RELAT_1:def.9">RELAT_1:def 9</a><br/>

<b>then consider </b><font color="Maroon" title="c15">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:32_3"/><i><font color="Green" title="E26">A10</font></i>: 
( <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">ch</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">ch</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</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/>
<b>reconsider </b><font color="Maroon" title="c16">a</font> = <font color="Maroon" title="c15">a</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="aofa_000.html#E4:32_3"><i><font color="Green" title="E26">A10</font></i></a></i>;<br/>
<a NAME="E6:32_3"/>
( ( <font color="Maroon" title="c16">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> implies  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f0</font> ) &amp; ( <font color="Maroon" title="c16">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">i</font> implies  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <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 <font color="Maroon" title="c7">A0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">a</font> ) &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a>  <a href="pua2mss1.html#K1" title="PUA2MSS1:func.1">rng</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c7">A0</font> )
 
<b>by </b><i><a class="txt" href="aofa_000.html#E4:32_3"><i><font color="Green" title="E26">A10</font></i></a>, <a class="txt" href="aofa_000.html#E18:32"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a>, <a class="ref" href="relat_1.html#D9" title="RELAT_1:def.9">RELAT_1:def 9</a></i>;<br/>
<b>hence </b><a NAME="E7:32_3"/>
contradiction
 <b>by </b><i><a class="txt" href="aofa_000.html#E18:32"><i><font color="Green" title="E23">A6</font></i></a>, <a class="txt" href="aofa_000.html#E4:32_3"><i><font color="Green" title="E26">A10</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:32"/>
 <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c13">ch</font> #) is <a href="unialg_1.html#V6" title="UNIALG_1:attr.6">partial</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="unialg_1.html#D4" title="UNIALG_1:def.4">UNIALG_1:def 4</a>,<a class="ref" href="unialg_1.html#D7" title="UNIALG_1:def.7">UNIALG_1:def 7</a><br/><b>let </b><font color="Maroon" title="c16">h</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c13">ch</font> #) <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c13">ch</font> #);<br/>



<b>assume </b><a NAME="E1:32_4"/>
( <font color="Maroon" title="c15">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>(# <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c13">ch</font> #) &amp; <font color="Maroon" title="c16">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>(# <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c13">ch</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font> )
 ;<br/>

<a NAME="E2:32_4"/><b>then </b>
( ( <font color="Maroon" title="c15">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> implies <font color="Maroon" title="c16">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f0</font> ) &amp; ( <font color="Maroon" title="c15">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">i</font> implies <font color="Maroon" title="c16">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 <font color="Maroon" title="c7">A0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font> ) &amp; the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c7">A0</font> is <a href="unialg_1.html#V4" title="UNIALG_1:attr.4">homogeneous</a> )
 
<b>by </b><i><a class="txt" href="aofa_000.html#E18:32"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/>
<b>hence </b><a NAME="E3:32_4"/>
<font color="Maroon" title="c16">h</font> is <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c15">A</font> =  <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c13">ch</font> #) as   <a href="unialg_1.html#NM2" title="UNIALG_1:NM.2">Universal_Algebra</a> <b>by </b><i><a class="txt" href="aofa_000.html#E19:32"><i><font color="Green" title="E24">A7</font></i></a>, <a class="txt" href="aofa_000.html#E20:32"><i><font color="Green" title="E25">A9</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c15">A</font>
;<br/>

<b>thus </b><a NAME="E23:32"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c15">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">X</font>
 ;<br/>

<a NAME="E24:32"/><i><font color="Green" title="E26">A11</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">S</font> <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="c7">A0</font>
 
<b>by </b><i><a class="txt" href="aofa_000.html#E4:32"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="unialg_1.html#D11" title="UNIALG_1:def.11">UNIALG_1:def 11</a></i>;<br/>
<a NAME="E25:32"/><b>then </b><i><font color="Green" title="E27">A12</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c7">A0</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E26:32"/><i><font color="Green" title="E28">A13</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">S</font> <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="c15">A</font>
 
<b>by </b><i><a class="txt" href="aofa_000.html#E24:32"><i><font color="Green" title="E26">A11</font></i></a>, <a class="txt" href="aofa_000.html#E18:32"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:32_5"/><i><font color="Green" title="E29">A14</font></i>: 
<font color="Maroon" title="c16">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> <font color="Maroon" title="c1">S</font>
 ;<br/><b>let </b><font color="Maroon" title="c17">h</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a> <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c15">A</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c15">A</font>;<br/><b>assume </b><a NAME="E2:32_5"/>
<font color="Maroon" title="c17">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 <font color="Maroon" title="c15">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">n</font>
 ;<br/><a NAME="E3:32_5"/><b>then </b>
( ( <font color="Maroon" title="c16">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> implies <font color="Maroon" title="c17">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f0</font> ) &amp; ( <font color="Maroon" title="c16">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">i</font> implies <font color="Maroon" title="c17">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 <font color="Maroon" title="c7">A0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">n</font> ) )
 <b>by </b><i><a class="txt" href="aofa_000.html#E25:32"><i><font color="Green" title="E27">A12</font></i></a>, <a class="txt" href="aofa_000.html#E1:32_5"><i><font color="Green" title="E29">A14</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><b>hence </b><a NAME="E4:32_5"/>
<font color="Maroon" title="c1">S</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c17">h</font>
 <b>by </b><i><a class="txt" href="aofa_000.html#E4:32"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="aofa_000.html#E2:32"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="aofa_000.html#E1:32_5"><i><font color="Green" title="E29">A14</font></i></a>, <a class="txt" href="aofa_000.html#E11:32"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="aofa_000.html#E8:32"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="unialg_1.html#D10" title="UNIALG_1:def.10">UNIALG_1:def 10</a>, <a class="ref" href="unialg_1.html#D11" title="UNIALG_1:def.11">UNIALG_1:def 11</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E28:32"/>
 <a href="msualg_1.html#K6" title="MSUALG_1:func.6">signature</a> <font color="Maroon" title="c15">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">S</font>
 <b>by </b><i><a class="txt" href="aofa_000.html#E26:32"><i><font color="Green" title="E28">A13</font></i></a>, <a class="ref" href="unialg_1.html#D11" title="UNIALG_1:def.11">UNIALG_1:def 11</a></i>;<br/>

<b>thus </b><a NAME="E29:32"/>
the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c15">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font>
 <b>by </b><i><a class="txt" href="aofa_000.html#E2:32"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="aofa_000.html#E25:32"><i><font color="Green" title="E27">A12</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/>


</div>
