<?xml version="1.0"?>
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<span class="kw">let </span><font color="Maroon" title="c1">F1</font>, <font color="Maroon" title="c2">F2</font> be   <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#K7" title="NUMBERS:func.7">ExtREAL</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  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" title="c1">F1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">F2</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> ) implies  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  <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c1">F1</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c2">F2</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c3">N2</font> = <font color="Maroon" title="c2">F2</font> as    <a href="supinf_2.html#M1" title="SUPINF_2:mode.1">Num</a> of  <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c2">F2</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#D10" target="_self" title="SUPINF_2:def.10">Def16</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <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> ] <span class="kw">means </span><span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c1">F1</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c2">F2</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> $1;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">N1</font> = <font color="Maroon" title="c1">F1</font> as    <a href="supinf_2.html#M1" title="SUPINF_2:mode.1">Num</a> of  <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c1">F1</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#D10" target="_self" title="SUPINF_2:def.10">Def16</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:63"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
 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" title="c1">F1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">F2</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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  <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c1">F1</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c2">F2</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c1">F1</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c2">F2</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">n</font></span><br/>

<a NAME="E4:63"/><span class="lab"><font color="Green" title="E40">A2</font></span>: 
 <a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c2">F2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="supinf_2.html#K15" title="SUPINF_2:func.15">Ser</a> (<span class="p1">(<span class="default"><a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c2">F2</font></span>)</span>,<font color="Maroon" title="c3">N2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#D15" target="_self" title="SUPINF_2:def.15">Def21</a></span>;<br/>
<a NAME="E5:63"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A3</font></span>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c2">F2</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F2</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#D11" target="_self" title="SUPINF_2:def.11">Def17</a></span>;<br/>
<a NAME="E6:63"/><span class="lab"><font color="Green" title="E42">A4</font></span>: 
 <a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c1">F1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="supinf_2.html#K15" title="SUPINF_2:func.15">Ser</a> (<span class="p1">(<span class="default"><a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c1">F1</font></span>)</span>,<font color="Maroon" title="c4">N1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#D15" target="_self" title="SUPINF_2:def.15">Def21</a></span>;<br/>
<a NAME="E7:63"/><span class="lab"><font color="Green" title="E43">A5</font></span>: 
 for <font color="Olive" title="b1">k</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="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">k</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:63_1"/><span class="lab"><font color="Green" title="E44">A6</font></span>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c1">F1</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c2">F2</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<a NAME="E2:63_1"/><span class="lab"><font color="Green" title="E45">A7</font></span>: 
<font color="Maroon" title="c1">F1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">F2</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63"><span class="lab"><font color="Green" title="E39">A1</font></span></a></span>;<br/>
<a NAME="E3:63_1"/><span class="lab"><font color="Green" title="E46">A8</font></span>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c2">F2</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c2">F2</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F2</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="ref" href="supinf_2.html#D11" target="_self" title="SUPINF_2:def.11">Def17</a></span>;<br/>
<a NAME="E4:63_1"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c1">F1</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c1">F1</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:63"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="ref" href="supinf_2.html#D11" target="_self" title="SUPINF_2:def.11">Def17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:63_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_1"><span class="lab"><font color="Green" title="E44">A6</font></span></a>, <a class="txt" href="#E3:63_1"><span class="lab"><font color="Green" title="E46">A8</font></span></a>, <a class="txt" href="#E2:63_1"><span class="lab"><font color="Green" title="E45">A7</font></span></a>, <a class="ref" href="xxreal_3.html#T36" title="XXREAL_3:th.36">XXREAL_3:36</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:63"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c1">F1</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:63"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="ref" href="supinf_2.html#D11" target="_self" title="SUPINF_2:def.11">Def17</a></span>;<br/>
<a NAME="E9:63"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A9</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E5:63"><span class="lab"><font color="Green" title="E41">A3</font></span></a></span>;<br/>
<a NAME="E10:63"/>
 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>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E9:63"><span class="lab"><font color="Green" title="E44">A9</font></span></a>, <a class="txt" href="#E7:63"><span class="lab"><font color="Green" title="E43">A5</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E11:63"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c1">F1</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c2">F2</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
