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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">g</font> being  <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a>  st <br/>( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="diff_1.html#K7" target="_self">cD</a> <font color="Olive">g</font>,<font color="Maroon">h</font> );<br/>
<a NAME="E1:31_1_1"/><i><font color="Green">E40</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> <br/> for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <a href="numbers.html#K1">REAL</a> ,<a href="numbers.html#K1">REAL</a>   ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <a href="numbers.html#K1">REAL</a> ,<a href="numbers.html#K1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_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>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <a href="numbers.html#K1">REAL</a> ,<a href="numbers.html#K1">REAL</a> ;<br/>



<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="partfun1.html#T120">PARTFUN1:120</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> =  <a href="diff_1.html#K7" target="_self">cD</a> <font color="Maroon">x'</font>,<font color="Maroon">h</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <a href="numbers.html#K1">REAL</a> ,<a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="partfun1.html#T119">PARTFUN1:119</a></i>;<br/>
<a NAME="E3:31_1_1_1"/>
 ex <font color="Olive">w</font> being  <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">w</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="diff_1.html#K7" target="_self">cD</a> <font color="Olive">w</font>,<font color="Maroon">h</font> )
 
;<br/>
<b>hence </b><a NAME="E4:31_1_1_1"/>
 ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <a href="numbers.html#K1">REAL</a> ,<a href="numbers.html#K1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">x</font>,<font color="Olive">y</font>]
 ;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">fZ</font> = <font color="Maroon">f</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <a href="numbers.html#K1">REAL</a> ,<a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="partfun1.html#T119">PARTFUN1:119</a></i>;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="partfun1.html#K4">PFuncs</a> <a href="numbers.html#K1">REAL</a> ,<a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E4:31_1_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">fZ</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>,<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>] ) )
 <b>from </b><i><a class="ref" href="recdef_1.html#S2">RECDEF_1:sch 2</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> is   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/>
<a NAME="E5:31_1_1"/><i><font color="Green">E44</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:31_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i/>;<br/>
<a NAME="E7:31_1_1"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:31_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<b>consider </b><font color="Maroon">w</font> being   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:31_1_1_2"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="diff_1.html#K7" target="_self">cD</a> <font color="Maroon">w</font>,<font color="Maroon">h</font> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">gk</font> = <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> as   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a>  <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E5:31_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E8:31_1_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</font> as    <a href="seqfunc.html#M1">Functional_Sequence</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a>  <b>by </b><i>, <a class="ref" href="seqfunc.html#T1">SEQFUNC:1</a></i>;<br/>
<b>take </b>
<font color="Maroon">g</font>
;<br/>

<b>thus </b><a NAME="E10:31_1_1"/>
<font color="Maroon">g</font> <a href="seqfunc.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">i</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<a NAME="E11:31_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E12:31_1_1"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>,<font color="Maroon">g</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>]
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E13:31_1_1"/>
( <font color="Maroon">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  implies <font color="Maroon">g</font> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="diff_1.html#K7" target="_self">cD</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">h</font> )
 ;<br/>


</div>
