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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">F</font>, <font color="Maroon" title="c2">G</font>, <font color="Maroon" title="c3">H</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"> ( <font color="Maroon" title="c2">G</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>  &amp; <font color="Maroon" title="c3">H</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>  &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" title="c1">F</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</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="c3">H</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font></span>)</span> ) 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">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> <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">G</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</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="c3">H</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font></span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:3"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
<font color="Maroon" title="c2">G</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> 
 <span class="kw">and </span><br/><a NAME="E2:3"/><span class="lab"><font color="Green" title="E5">A2</font></span>: 
<font color="Maroon" title="c3">H</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <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>  st  not <font color="Maroon" title="c1">F</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</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="c3">H</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font></span>)</span> or  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">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> <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">G</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</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="c3">H</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font></span>)</span> )</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">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> $1 <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">G</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> $1</span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</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="c3">H</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> $1</span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E3:3"/><span class="lab"><font color="Green" title="E6">A3</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">F</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</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="c3">H</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font></span>)</span>
 ; <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">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> <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">G</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</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="c3">H</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font></span>)</span></span><br/>

<a NAME="E4:3"/><span class="lab"><font color="Green" title="E7">A4</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="3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">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="c4">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:3_1"/><span class="lab"><font color="Green" title="E8">A5</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">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">k</font> <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">G</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</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="c3">H</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">k</font></span>)</span>
 ; <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="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">n</font> = <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/>
<a NAME="E2:3_1"/><span class="lab"><font color="Green" title="E9">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="c2">G</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="c4">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">G</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">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">G</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c3">H</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="c4">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="c3">H</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">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="c3">H</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">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="ref" href="supinf_2.html#T44" title="SUPINF_2:th.44">SUPINF_2:44</a></span>;<br/>
<a NAME="E3:3_1"/><span class="lab"><font color="Green" title="E10">A7</font></span>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">H</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">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="#E2:3"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="ref" href="supinf_2.html#T39" title="SUPINF_2:th.39">SUPINF_2:39</a></span>;<br/>
<a NAME="E4:3_1"/><span class="lab"><font color="Green" title="E11">A8</font></span>: 
( <font color="Maroon" title="c1">F</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">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"><font color="Maroon" title="c2">G</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp;  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">G</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">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="#E1:3"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E3:3"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="ref" href="supinf_2.html#T39" title="SUPINF_2:th.39">SUPINF_2:39</a></span>;<br/>
<a NAME="E5:3_1"/><span class="lab"><font color="Green" title="E12">A9</font></span>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <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">G</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="supinf_2.html#T40" title="SUPINF_2:th.40">SUPINF_2:40</a></span>;<br/>
<a NAME="E6:3_1"/><span class="lab"><font color="Green" title="E13">A10</font></span>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <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">G</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="c4">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="#E1:3"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="supinf_2.html#T40" title="SUPINF_2:th.40">SUPINF_2:40</a></span>;<br/>
<a NAME="E7:3_1"/><span class="lab"><font color="Green" title="E14">A11</font></span>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <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="c3">H</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="ref" href="supinf_2.html#T40" title="SUPINF_2:th.40">SUPINF_2:40</a></span>;<br/>
<a NAME="E8:3_1"/>
(  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">G</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">H</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">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="#E1:3"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E2:3"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="ref" href="supinf_2.html#T39" title="SUPINF_2:th.39">SUPINF_2:39</a></span>;<br/>
<a NAME="E9:3_1"/><span class="kw">then </span>
<a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">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="ref" href="xxreal_3.html#T36" title="XXREAL_3:th.36">XXREAL_3:36</a></span>;<br/>
<a NAME="E10:3_1"/><span class="kw">then </span>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">F</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">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="#E3:3"><span class="lab"><font color="Green" title="E6">A3</font></span></a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">k</font></span>)</span></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">F</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</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="c3">H</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:3_1"><span class="lab"><font color="Green" title="E12">A9</font></span></a>, <a class="txt" href="#E7:3_1"><span class="lab"><font color="Green" title="E14">A11</font></span></a>, <a class="ref" href="xxreal_3.html#T44" title="XXREAL_3:th.44">XXREAL_3:44</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</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="c3">H</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:3_1"><span class="lab"><font color="Green" title="E12">A9</font></span></a>, <a class="txt" href="#E3:3_1"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="txt" href="#E4:3_1"><span class="lab"><font color="Green" title="E11">A8</font></span></a>, <a class="ref" href="xxreal_3.html#T44" title="XXREAL_3:th.44">XXREAL_3:44</a></span>
<br/>.= 
<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">G</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</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="c3">H</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">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="#E2:3_1"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="txt" href="#E7:3_1"><span class="lab"><font color="Green" title="E14">A11</font></span></a>, <a class="txt" href="#E6:3_1"><span class="lab"><font color="Green" title="E13">A10</font></span></a>, <a class="txt" href="#E3:3_1"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="ref" href="xxreal_3.html#T44" title="XXREAL_3:th.44">XXREAL_3:44</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E12:3_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">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="#E1:3_1"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="ref" href="supinf_2.html#T44" title="SUPINF_2:th.44">SUPINF_2:44</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="E5:3"/><span class="lab"><font color="Green" title="E8">A12</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="c3">H</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="c3">H</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#T44" title="SUPINF_2:th.44">SUPINF_2:44</a></span>;<br/>
<a NAME="E6:3"/>
( <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c1">F</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">F</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> &amp; <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c2">G</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">G</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#T44" title="SUPINF_2:th.44">SUPINF_2:44</a></span>;<br/>
<a NAME="E7:3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E9">A13</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="#E3:3"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="txt" href="#E5:3"><span class="lab"><font color="Green" title="E8">A12</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E8:3"/>
 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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</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="#E7:3"><span class="lab"><font color="Green" title="E9">A13</font></span></a>, <a class="txt" href="#E4:3"><span class="lab"><font color="Green" title="E7">A4</font></span></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>
