<?xml version="1.0"?>
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<b>let </b><font color="Maroon" title="c1">D</font> be   <a href="supinf_2.html#NM1" title="SUPINF_2:NM.1">Denum_Set_of_R_EAL</a>;<br/><b>let </b><font color="Maroon" title="c2">N</font> be    <a href="supinf_2.html#M1" title="SUPINF_2:mode.1">Num</a> of <font color="Maroon" title="c1">D</font>;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><br/> for <font color="Olive" title="b3">s</font> being   <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>   st <font color="Olive" title="b3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $3 holds <br/><font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">s</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>;<br/>
<a NAME="E1:79"/><i><font color="Green" title="E58">A1</font></i>: 
for <font color="Olive" title="b1">n</font> being   <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> <br/> for <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>   ex <font color="Olive" title="b3">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">n</font> be    <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> ;<br/><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> ;<br/>



<b>reconsider </b><font color="Maroon" title="c5">y</font> = <font color="Maroon" title="c4">x</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> ;<br/>
<b>take </b>
<font color="Maroon" title="c5">y</font>
;<br/>

<b>thus </b><a NAME="E2:79_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>]
 ;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c3">A</font> = <font color="Maroon" title="c2">N</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> 0 as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> ;<br/>
<b>consider </b><font color="Maroon" title="c4">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> <b> such that </b><br/><a NAME="E4:79"/><i><font color="Green" title="E59">A2</font></i>: 
( <font color="Maroon" title="c4">F</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> &amp; ( for <font color="Olive" title="b1">n</font> being   <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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c4">F</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c4">F</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>] ) )
 <b>from </b><i><a class="ref" href="recdef_1.html#S2" title="RECDEF_1:sch.2">RECDEF_1:sch 2</a>(<a class="txt" href="supinf_2.html#E1:79"><i><font color="Green" title="E58">A1</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c4">F</font>
;<br/>

<b>thus </b><a NAME="E5:79"/>
( <font color="Maroon" title="c4">F</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">N</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> 0 &amp; ( for <font color="Olive" title="b1">n</font> being   <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> <br/> for <font color="Olive" title="b2">y</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>  st <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b1">n</font> holds <br/><font color="Maroon" title="c4">F</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="supinf_2.html#E4:79"><i><font color="Green" title="E59">A2</font></i></a></i>;<br/>


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