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

<b>consider </b><font color="Maroon">c<sub>2</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K13">*</a> ,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="unialg_1.html#T2" target="_self">Th2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K13">*</a> </span>)</span>,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="partfun1.html#T119">PARTFUN1:119</a></i>;<br/>
<b>take </b>
<span class="p1"><a href="unialg_1.html#K1" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="unialg_1.html#K1" target="_self">*&gt;</a></span>
;<br/>

<b>thus </b><a NAME="E4:13_1_1"/>
<span class="p1"><a href="unialg_1.html#K1" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="unialg_1.html#K1" target="_self">*&gt;</a></span> is <a href="unialg_1.html#V4" target="_self">homogeneous</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="unialg_1.html#D4" target="_self">UNIALG_1:def 4</a><br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K13">*</a> ,<font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:13_1_1_1"/><i><font color="Green">E6</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="unialg_1.html#K1" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="unialg_1.html#K1" target="_self">*&gt;</a></span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="unialg_1.html#K1" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="unialg_1.html#K1" target="_self">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/>

<a NAME="E2:13_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/>
<a NAME="E3:13_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="unialg_1.html#K1" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="unialg_1.html#K1" target="_self">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1
 
<b>by </b><i><a class="txt" href="unialg_1.html#E1:13_1_1_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:13_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> is  <a href="unialg_1.html#V1" target="_self">homogeneous</a> <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K13">*</a> ,<font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="unialg_1.html#T2" target="_self">Th2</a>, <a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E5:13_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="unialg_1.html#V1" target="_self">homogeneous</a>
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E5:13_1_1"/>
<span class="p1"><a href="unialg_1.html#K1" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="unialg_1.html#K1" target="_self">*&gt;</a></span> is <a href="unialg_1.html#V5" target="_self">quasi_total</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="unialg_1.html#D5" target="_self">UNIALG_1:def 5</a><br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K13">*</a> ,<font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:13_1_1_2"/><i><font color="Green">E6</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="unialg_1.html#K1" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="unialg_1.html#K1" target="_self">*&gt;</a></span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="unialg_1.html#K1" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="unialg_1.html#K1" target="_self">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/>

<a NAME="E2:13_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/>
<a NAME="E3:13_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="unialg_1.html#K1" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="unialg_1.html#K1" target="_self">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1
 
<b>by </b><i><a class="txt" href="unialg_1.html#E1:13_1_1_2"><i><font color="Green">E6</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:13_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> is  <a href="unialg_1.html#V2" target="_self">quasi_total</a> <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K13">*</a> ,<font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="unialg_1.html#T2" target="_self">Th2</a>, <a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E5:13_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="unialg_1.html#V2" target="_self">quasi_total</a>
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E6:13_1_1"/>
<span class="p1"><a href="unialg_1.html#K1" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="unialg_1.html#K1" target="_self">*&gt;</a></span> is <a href="relat_1.html#V2">non-empty</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D15">FUNCT_1:def 15</a><br/>

<b>assume </b><a NAME="E1:13_1_1_3"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="unialg_1.html#K1" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="unialg_1.html#K1" target="_self">*&gt;</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<a NAME="E3:13_1_1_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="unialg_1.html#E1:13_1_1_3"><i><font color="Green">E6</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/>
<a NAME="E4:13_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:13_1_1_3"/><b>then </b>
<span class="p1"><a href="unialg_1.html#K1" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="unialg_1.html#K1" target="_self">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E6:13_1_1_3"/>
not <span class="p1"><a href="unialg_1.html#K1" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="unialg_1.html#K1" target="_self">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="unialg_1.html#T2" target="_self">Th2</a></i>;<br/>


</div><b>end;</b></div>


</div>
