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

<span class="kw">let </span><font color="Maroon" title="c1">a</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">h</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>   st <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binpack1.html#K6" title="BINPACK1:func.6">OnlinePackingHistory</a> (<font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><a href="binpack1.html#K8" title="BINPACK1:func.8">NextFit</a> <font color="Maroon" title="c1">a</font></span>)</span>) holds <br/> for <font color="Olive" title="b2">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">a</font> holds <br/> ex <font color="Olive" title="b3">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>(  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Olive" title="b3">k</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">k</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">h</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binpack1.html#K6" title="BINPACK1:func.6">OnlinePackingHistory</a> (<font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><a href="binpack1.html#K8" title="BINPACK1:func.8">NextFit</a> <font color="Maroon" title="c1">a</font></span>)</span>) implies  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">a</font> holds <br/> ex <font color="Olive" title="b2">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>(  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Olive" title="b2">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">k</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:43"/><span class="lab"><font color="Green" title="E36">HN00</font></span>: 
<font color="Maroon" title="c2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binpack1.html#K6" title="BINPACK1:func.6">OnlinePackingHistory</a> (<font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><a href="binpack1.html#K8" title="BINPACK1:func.8">NextFit</a> <font color="Maroon" title="c1">a</font></span>)</span>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">a</font> holds <br/> ex <font color="Olive" title="b2">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>(  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Olive" title="b2">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">k</font> )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span> ex <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>(  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Olive" title="b1">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> );<br/>
<a NAME="E2:43"/>
<span class="p1">(<span class="default"><a href="binpack1.html#K6" title="BINPACK1:func.6">OnlinePackingHistory</a> (<font color="Maroon" title="c1">a</font>,<span class="p2">(<span class="default"><a href="binpack1.html#K8" title="BINPACK1:func.8">NextFit</a> <font color="Maroon" title="c1">a</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="binpack1.html#D5" target="_self" onmouseover="rs('binpack1/D5')" onmouseout="rh()">defPackHistory</a></span>;<br/>
<a NAME="E3:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">L219</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E36">HN00</font></span></a>, <a class="ref" href="relat_1.html#T9" onmouseover="rs('relat_1/T9')" onmouseout="rh()">RELAT_1:9</a></span>;<br/>
<a NAME="E4:43"/><span class="lab"><font color="Green" title="E38">L289</font></span>: 
<font color="Maroon" title="c2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="binpack1.html#D5" target="_self" onmouseover="rs('binpack1/D5')" onmouseout="rh()">defPackHistory</a>, <a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E36">HN00</font></span></a></span>;<br/>
<a NAME="E5:43"/><span class="lab"><font color="Green" title="E39">L300</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E37">L219</font></span></a>, <a class="ref" href="finseq_1.html#T2" onmouseover="rs('finseq_1/T2')" onmouseout="rh()">FINSEQ_1:2</a>, <a class="txt" href="#E4:43"><span class="lab"><font color="Green" title="E38">L289</font></span></a></span>;<br/>
<a NAME="E6:43"/><span class="lab"><font color="Green" title="E40">L400</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">i0</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i0</font> &amp; <font color="Maroon" title="c3">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">i0</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:43_1"/><span class="lab"><font color="Green" title="E41">L410</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i0</font>
 <span class="kw">and </span><br/><a NAME="E2:43_1"/><span class="lab"><font color="Green" title="E42">L411</font></span>: 
<font color="Maroon" title="c3">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">a</font>
 <span class="kw">and </span><br/><a NAME="E3:43_1"/><span class="lab"><font color="Green" title="E43">L412</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">i0</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="c3">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">hi0</font> = <font color="Maroon" title="c2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i0</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E36">HN00</font></span></a>, <a class="txt" href="#E1:43_1"><span class="lab"><font color="Green" title="E41">L410</font></span></a>, <a class="txt" href="#E2:43_1"><span class="lab"><font color="Green" title="E42">L411</font></span></a>, <a class="ref" href="binpack1.html#T21" target="_self" onmouseover="rs('binpack1/T21')" onmouseout="rh()">NF505</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">i0p</font> = <font color="Maroon" title="c3">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E6:43_1"/><span class="lab"><font color="Green" title="E44">L074</font></span>: 
1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<a NAME="E7:43_1"/><span class="lab"><font color="Green" title="E45">L076</font></span>: 
<font color="Maroon" title="c5">i0p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_1"><span class="lab"><font color="Green" title="E42">L411</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">hi0p</font> = <font color="Maroon" title="c2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i0p</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E36">HN00</font></span></a>, <a class="txt" href="#E6:43_1"><span class="lab"><font color="Green" title="E44">L074</font></span></a>, <a class="txt" href="#E7:43_1"><span class="lab"><font color="Green" title="E45">L076</font></span></a>, <a class="ref" href="binpack1.html#T21" target="_self" onmouseover="rs('binpack1/T21')" onmouseout="rh()">NF505</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">k0</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E10:43_1"/><span class="lab"><font color="Green" title="E46">L427</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">hi0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">k0</font>
 <span class="kw">and </span><br/><a NAME="E11:43_1"/><span class="lab"><font color="Green" title="E47">L428</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i0</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1"><span class="lab"><font color="Green" title="E43">L412</font></span></a></span>;<br/>
<a NAME="E12:43_1"/><span class="lab"><font color="Green" title="E48">L090</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E36">HN00</font></span></a>, <a class="txt" href="#E1:43_1"><span class="lab"><font color="Green" title="E41">L410</font></span></a>, <a class="txt" href="#E2:43_1"><span class="lab"><font color="Green" title="E42">L411</font></span></a>, <a class="ref" href="binpack1.html#T22" target="_self" onmouseover="rs('binpack1/T22')" onmouseout="rh()">NF510</a></span>;<br/>
<a NAME="E13:43_1"/><span class="lab"><font color="Green" title="E49">L100</font></span>: 
( ( <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i0p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c4">hi0</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">hi0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i0</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 implies <span class="p1">(<span class="default"><a href="binpack1.html#K8" title="BINPACK1:func.8">NextFit</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i0p</font></span>)</span>,<font color="Maroon" title="c4">hi0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">hi0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i0</font> ) &amp; ( <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i0p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c4">hi0</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">hi0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i0</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 implies <span class="p1">(<span class="default"><a href="binpack1.html#K8" title="BINPACK1:func.8">NextFit</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i0p</font></span>)</span>,<font color="Maroon" title="c4">hi0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">hi0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:43_1"><span class="lab"><font color="Green" title="E48">L090</font></span></a>, <a class="ref" href="binpack1.html#D7" target="_self" onmouseover="rs('binpack1/D7')" onmouseout="rh()">defNextFit</a></span>;<br/>
<a NAME="E14:43_1"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>(  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i0p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Olive" title="b1">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i0p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i0p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:43_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i0p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c4">hi0</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">hi0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i0</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 or <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i0p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c4">hi0</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">hi0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i0</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:43_1_1_1_1"/><span class="lab"><font color="Green" title="E50">L500</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i0p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c4">hi0</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">hi0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i0</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>(  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i0p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Olive" title="b1">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i0p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i0p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> )</span><br/><div class="add"><a NAME="E2:43_1_1_1_1"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c6">hi0p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">k0</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">hi0p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i0p</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E36">HN00</font></span></a>, <a class="txt" href="#E1:43_1"><span class="lab"><font color="Green" title="E41">L410</font></span></a>, <a class="txt" href="#E2:43_1"><span class="lab"><font color="Green" title="E42">L411</font></span></a>, <a class="ref" href="binpack1.html#T24" target="_self" onmouseover="rs('binpack1/T24')" onmouseout="rh()">NF525</a>, <a class="txt" href="#E10:43_1"><span class="lab"><font color="Green" title="E46">L427</font></span></a></span>;<br/><span class="kw">then </span> <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c6">hi0p</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">k0</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">k0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E36">HN00</font></span></a>, <a class="txt" href="#E1:43_1"><span class="lab"><font color="Green" title="E41">L410</font></span></a>, <a class="txt" href="#E2:43_1"><span class="lab"><font color="Green" title="E42">L411</font></span></a>, <a class="ref" href="binpack1.html#T23" target="_self" onmouseover="rs('binpack1/T23')" onmouseout="rh()">NF520</a>, <a class="txt" href="#E13:43_1"><span class="lab"><font color="Green" title="E49">L100</font></span></a>, <a class="txt" href="#E1:43_1_1_1_1"><span class="lab"><font color="Green" title="E50">L500</font></span></a>, <a class="txt" href="#E11:43_1"><span class="lab"><font color="Green" title="E47">L428</font></span></a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T9" onmouseover="rs('finseq_1/T9')" onmouseout="rh()">FINSEQ_1:9</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:43_1_1_1_1"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>(  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i0p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Olive" title="b1">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i0p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i0p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E36">HN00</font></span></a>, <a class="txt" href="#E1:43_1"><span class="lab"><font color="Green" title="E41">L410</font></span></a>, <a class="txt" href="#E2:43_1"><span class="lab"><font color="Green" title="E42">L411</font></span></a>, <a class="ref" href="binpack1.html#T23" target="_self" onmouseover="rs('binpack1/T23')" onmouseout="rh()">NF520</a>, <a class="txt" href="#E13:43_1"><span class="lab"><font color="Green" title="E49">L100</font></span></a>, <a class="txt" href="#E1:43_1_1_1_1"><span class="lab"><font color="Green" title="E50">L500</font></span></a>, <a class="txt" href="#E11:43_1"><span class="lab"><font color="Green" title="E47">L428</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:43_1_1_1_2"/><span class="lab"><font color="Green" title="E50">L700</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i0p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c4">hi0</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">hi0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i0</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>(  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i0p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Olive" title="b1">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i0p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i0p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> )</span><br/><div class="add"><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_1_1_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k0</font></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:43_1_1_1_2_1"/>
 not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k0</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:43_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">k0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 ;<br/><a NAME="E3:43_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">k0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E4:43_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">hi0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:43_1"><span class="lab"><font color="Green" title="E46">L427</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:43_1_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E36">HN00</font></span></a>, <a class="txt" href="#E1:43_1"><span class="lab"><font color="Green" title="E41">L410</font></span></a>, <a class="txt" href="#E2:43_1"><span class="lab"><font color="Green" title="E42">L411</font></span></a>, <a class="ref" href="binpack1.html#T22" target="_self" onmouseover="rs('binpack1/T22')" onmouseout="rh()">NF510</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E3:43_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">L739</font></span>: 
<font color="Maroon" title="c7">k0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">k0</font>
 ;<br/> <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c6">hi0p</font> = 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">hi0</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">hi0p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i0p</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E36">HN00</font></span></a>, <a class="txt" href="#E1:43_1"><span class="lab"><font color="Green" title="E41">L410</font></span></a>, <a class="txt" href="#E2:43_1"><span class="lab"><font color="Green" title="E42">L411</font></span></a>, <a class="ref" href="binpack1.html#T24" target="_self" onmouseover="rs('binpack1/T24')" onmouseout="rh()">NF525</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">k0</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">k0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:43_1"><span class="lab"><font color="Green" title="E46">L427</font></span></a>, <a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E36">HN00</font></span></a>, <a class="txt" href="#E1:43_1"><span class="lab"><font color="Green" title="E41">L410</font></span></a>, <a class="txt" href="#E2:43_1"><span class="lab"><font color="Green" title="E42">L411</font></span></a>, <a class="ref" href="binpack1.html#T23" target="_self" onmouseover="rs('binpack1/T23')" onmouseout="rh()">NF520</a>, <a class="txt" href="#E13:43_1"><span class="lab"><font color="Green" title="E49">L100</font></span></a>, <a class="txt" href="#E1:43_1_1_1_2"><span class="lab"><font color="Green" title="E50">L700</font></span></a>, <a class="txt" href="#E11:43_1"><span class="lab"><font color="Green" title="E47">L428</font></span></a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">k0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1_1_1_2"><span class="lab"><font color="Green" title="E51">L739</font></span></a>, <a class="ref" href="zfmisc_1.html#T31" onmouseover="rs('zfmisc_1/T31')" onmouseout="rh()">ZFMISC_1:31</a>, <a class="ref" href="xboole_1.html#T12" onmouseover="rs('xboole_1/T12')" onmouseout="rh()">XBOOLE_1:12</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:43_1_1_1_2"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>(  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i0p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Olive" title="b1">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i0p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i0p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E36">HN00</font></span></a>, <a class="txt" href="#E1:43_1"><span class="lab"><font color="Green" title="E41">L410</font></span></a>, <a class="txt" href="#E2:43_1"><span class="lab"><font color="Green" title="E42">L411</font></span></a>, <a class="ref" href="binpack1.html#T23" target="_self" onmouseover="rs('binpack1/T23')" onmouseout="rh()">NF520</a>, <a class="txt" href="#E13:43_1"><span class="lab"><font color="Green" title="E49">L100</font></span></a>, <a class="txt" href="#E1:43_1_1_1_2"><span class="lab"><font color="Green" title="E50">L700</font></span></a>, <a class="txt" href="#E11:43_1"><span class="lab"><font color="Green" title="E47">L428</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E15:43_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E7:43"/><span class="lab"><font color="Green" title="E41">L800</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">a</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="int_1.html#S7" onmouseover="rs('int_1/S7')" onmouseout="rh()">INT_1:sch 7</a>(<a class="txt" href="#E5:43"><span class="lab"><font color="Green" title="E39">L300</font></span></a>, <a class="txt" href="#E6:43"><span class="lab"><font color="Green" title="E40">L400</font></span></a>);<br/></span>
<a NAME="E8:43"/>
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">a</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">i</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">a</font> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">i</font>] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:43_2"/><span class="lab"><font color="Green" title="E42">L810</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font>
 <span class="kw">and </span><br/><a NAME="E2:43_2"/><span class="lab"><font color="Green" title="E43">L811</font></span>: 
<font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">a</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="c3">i</font>]</span><br/>

<a NAME="E3:43_2"/>
<font color="Maroon" title="c3">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:43_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">i</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_2"><span class="lab"><font color="Green" title="E42">L810</font></span></a>, <a class="txt" href="#E2:43_2"><span class="lab"><font color="Green" title="E43">L811</font></span></a>, <a class="txt" href="#E7:43"><span class="lab"><font color="Green" title="E41">L800</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>
<span class="kw">hence </span><a NAME="E9:43"/>
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">a</font> holds <br/> ex <font color="Olive" title="b2">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>(  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Olive" title="b2">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">k</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
