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

<b>set </b><font color="Maroon">X</font> = <a href="numbers.html#K1">REAL</a>  <a href="finseq_2.html#K3">*</a> ;<br/>
<b>set </b><font color="Maroon">mi</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/>
<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> ] <b>means </b>for <font color="Olive">f</font> being   <a href="finseq_2.html#M2">Element</a> of <a href="numbers.html#K1">REAL</a>  <a href="finseq_2.html#K3">*</a>   st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graphsp.html#K2" target="_self">:=</a> <span class="p1">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p2">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Olive">f</font>,<font color="Maroon">n</font></span>)</span>,<font color="Olive">f</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>;<br/>
<div><i><font color="Green">E34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">xx</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:55_1_1_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">xx</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K1">REAL</a>  <a href="finseq_2.html#K3">*</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">xx</font>,<font color="Maroon">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">xx</font>,<font color="Maroon">y2</font>] )
 ;<br/><b>then reconsider </b><font color="Maroon">h</font> = <font color="Maroon">xx</font> as    <a href="finseq_2.html#M2">Element</a> of <a href="numbers.html#K1">REAL</a>  <a href="finseq_2.html#K3">*</a>  ;<br/><b>thus </b><font color="Maroon">y1</font> = 
<font color="Maroon">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graphsp.html#K2" target="_self">:=</a> <span class="p1">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p2">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">h</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">h</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">y2</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<div><i><font color="Green">E53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">xx</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:55_1_1_2"/>
<font color="Maroon">xx</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K1">REAL</a>  <a href="finseq_2.html#K3">*</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">h</font> = <font color="Maroon">xx</font> as    <a href="finseq_2.html#M2">Element</a> of <a href="numbers.html#K1">REAL</a>  <a href="finseq_2.html#K3">*</a>  ;<br/><b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graphsp.html#K2" target="_self">:=</a> <span class="p1">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p2">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">h</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">h</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">y</font> = <font color="Maroon">y</font>;<br/><b>thus </b><a NAME="E4:55_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">xx</font>,<font color="Maroon">y</font>]
 ;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">F</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:55_1_1"/><i><font color="Green">E54</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <a href="numbers.html#K1">REAL</a>  <a href="finseq_2.html#K3">*</a>  &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K1">REAL</a>  <a href="finseq_2.html#K3">*</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(, );<br/></i>
<a NAME="E5:55_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <a href="numbers.html#K1">REAL</a>  <a href="finseq_2.html#K3">*</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:55_1_1_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon">xx</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:55_1_1_3"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">xx</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">xx</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">h</font> = <font color="Maroon">xx</font> as    <a href="finseq_2.html#M2">Element</a> of <a href="numbers.html#K1">REAL</a>  <a href="finseq_2.html#K3">*</a>  <b>by </b><i>, </i>;<br/>
<a NAME="E5:55_1_1_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graphsp.html#K2" target="_self">:=</a> <span class="p1">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p2">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">h</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">h</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:55_1_1_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K1">REAL</a>  <a href="finseq_2.html#K3">*</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="numbers.html#K1">REAL</a>  <a href="finseq_2.html#K3">*</a> </span>)</span>,<span class="p1">(<span class="default"><a href="numbers.html#K1">REAL</a>  <a href="finseq_2.html#K3">*</a> </span>)</span> <b>by </b><i>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<b>thus </b><a NAME="E7:55_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <a href="numbers.html#K1">REAL</a>  <a href="finseq_2.html#K3">*</a> 
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">f</font> be    <a href="finseq_2.html#M2">Element</a> of <a href="numbers.html#K1">REAL</a>  <a href="finseq_2.html#K3">*</a> ;<br/>

<b>thus </b><a NAME="E8:55_1_1"/>
<font color="Maroon">F</font> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graphsp.html#K2" target="_self">:=</a> <span class="p1">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p2">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <font color="Maroon">f</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">f</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
 <b>by </b><i/>;<br/>


</div>
