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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">o</font> being  <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">U0</font>  st <font color="Olive">o</font> <a href="hidden.html#R1">=</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">o</font> <a href="unialg_2.html#K2" target="_self">/.</a> <font color="Maroon">A</font>;<br/>
<a NAME="E1:15_1_1"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U0</font></span>)</span> holds <br/> ex <font color="Olive">x</font> being   <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">A</font> <a href="finseq_2.html#K3">*</a> </span>)</span>,<font color="Maroon">A</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:15_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U0</font></span>)</span>
 ;<br/>

<a NAME="E2:15_1_1_1"/><b>then </b>
<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 <font color="Maroon">U0</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">o1</font> = the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> as   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">U0</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">o1</font> <a href="unialg_2.html#K2" target="_self">/.</a> <font color="Maroon">A</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">A</font> <a href="finseq_2.html#K3">*</a> </span>)</span>,<font color="Maroon">A</font> <b>by </b><i><a class="ref" href="partfun1.html#T119">PARTFUN1:119</a></i>;<br/>
<b>take </b>
<font color="Maroon">x</font>
;<br/>

<b>let </b><font color="Maroon">o</font> be   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">U0</font>;<br/>

<b>assume </b><a NAME="E5:15_1_1_1"/>
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 ;<br/>

<b>hence </b><a NAME="E6:15_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> <a href="unialg_2.html#K2" target="_self">/.</a> <font color="Maroon">A</font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">p</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="partfun1.html#K4">PFuncs</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="finseq_2.html#K3">*</a> </span>)</span>,<font color="Maroon">A</font><b> such that </b><br/><a NAME="E3:15_1_1"/><i><font color="Green">E26</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U0</font></span>)</span>
 <b>and </b><br/><a NAME="E4:15_1_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U0</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="finseq_1.html#S5">FINSEQ_1:sch 5</a>();<br/></i>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as   <a href="unialg_1.html#NM1">PFuncFinSequence</a> of <font color="Maroon">A</font> ;<br/>
<b>take </b>
<font color="Maroon">p</font>
;<br/>

<b>thus </b><a NAME="E6:15_1_1"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U0</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>

<b>let </b><font color="Maroon">n</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">o</font> be   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">U0</font>;<br/>



<b>assume </b><a NAME="E7:15_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 ;<br/>

<b>hence </b><a NAME="E8:15_1_1"/>
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> <a href="unialg_2.html#K2" target="_self">/.</a> <font color="Maroon">A</font>
 <b>by </b><i>, </i>;<br/>


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