<?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="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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">D</font> being    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b1">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font> holds <br/><font color="Olive" title="b1">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1 is    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">D</font> be    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <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"> (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font> implies <font color="Maroon" title="c2">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1 is    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:17"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</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" title="c2">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1 is    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font></span><br/>

<a NAME="E2:17"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <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="c2">D</font>
 
;<br/>
<a NAME="E3:17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A2</font></span>: 
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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="c2">D</font>
 
<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="E4:17"/>
1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">D</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="17_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:17_1"/>
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">D</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="E2:17_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D2" onmouseover="rs('integra1/D2')" onmouseout="rh()">INTEGRA1:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:17_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E18">A1</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>
<a NAME="E5:17"/><span class="kw">then </span>
1 <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="c2">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/>
<a NAME="E6:17"/><span class="kw">then </span>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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="c2">D</font>
 
<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="E7:17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A5</font></span>: 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/>
<a NAME="E8:17"/><span class="lab"><font color="Green" title="E21">A6</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="rfinseq.html#D1" onmouseover="rs('rfinseq/D1')" onmouseout="rh()">RFINSEQ:def 1</a></span>;<br/>
<a NAME="E9:17"/><span class="lab"><font color="Green" title="E22">A8</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">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:17"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="ref" href="rfinseq.html#D1" onmouseover="rs('rfinseq/D1')" onmouseout="rh()">RFINSEQ:def 1</a>, <a class="txt" href="#E3:17"><span class="lab"><font color="Green" title="E19">A2</font></span></a></span>;<br/>
<a NAME="E10:17"/><span class="kw">then </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">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" onmouseover="rs('finseq_1/T3')" onmouseout="rh()">FINSEQ_1:3</a></span>;<br/>
<a NAME="E11:17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A9</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">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E12:17"/>
 for <font color="Olive" title="b1">e1</font>, <font color="Olive" title="b2">e2</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">e1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span> &amp; <font color="Olive" title="b2">e2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span> &amp; <font color="Olive" title="b1">e1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">e2</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">e2</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="17_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">e1</font>, <font color="Maroon" title="c4">e2</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"> ( <font color="Maroon" title="c3">e1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span> &amp; <font color="Maroon" title="c4">e2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span> &amp; <font color="Maroon" title="c3">e1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">e2</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">e1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">e2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:17_2"/><span class="lab"><font color="Green" title="E24">A10</font></span>: 
( <font color="Maroon" title="c3">e1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span> &amp; <font color="Maroon" title="c4">e2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span> &amp; <font color="Maroon" title="c3">e1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">e2</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">e1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">e2</font></span><br/>

<a NAME="E2:17_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A11</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">e1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">e2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="rfinseq.html#D1" onmouseover="rs('rfinseq/D1')" onmouseout="rh()">RFINSEQ:def 1</a></span>;<br/>
<a NAME="E3:17_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">e1</font> &amp; <font color="Maroon" title="c3">e1</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_2"><span class="lab"><font color="Green" title="E24">A10</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E4:17_2"/><span class="kw">then </span>
( 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">e1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c3">e1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="E5:17_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A14</font></span>: 
<font color="Maroon" title="c3">e1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:17"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E6:17_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">e2</font> &amp; <font color="Maroon" title="c4">e2</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_2"><span class="lab"><font color="Green" title="E24">A10</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E7:17_2"/><span class="kw">then </span>
( 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c4">e2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c4">e2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="E8:17_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A15</font></span>: 
<font color="Maroon" title="c4">e2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a>, <a class="txt" href="#E8:17"><span class="lab"><font color="Green" title="E21">A6</font></span></a></span>;<br/>
<a NAME="E9:17_2"/>
<font color="Maroon" title="c3">e1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">e2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_2"><span class="lab"><font color="Green" title="E24">A10</font></span></a>, <a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:17_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">e1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_2"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="txt" href="#E5:17_2"><span class="lab"><font color="Green" title="E26">A14</font></span></a>, <a class="txt" href="#E8:17_2"><span class="lab"><font color="Green" title="E27">A15</font></span></a>, <a class="ref" href="valued_0.html#D13" onmouseover="rs('valued_0/D13')" onmouseout="rh()">VALUED_0:def 13</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="E13:17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A16</font></span>: 
<font color="Maroon" title="c2">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1 is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a> 
 
;<br/>
<a NAME="E14:17"/><span class="lab"><font color="Green" title="E25">A17</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">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="rfinseq.html#D1" onmouseover="rs('rfinseq/D1')" onmouseout="rh()">RFINSEQ:def 1</a></span>;<br/>
<a NAME="E15:17"/><span class="lab"><font color="Green" title="E26">B23</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">D</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="17_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span> implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">D</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:17_3"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</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"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">D</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:17_3"/><span class="lab"><font color="Green" title="E27">A18</font></span>: 
<font color="Maroon" title="c4">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:17_3"/><span class="lab"><font color="Green" title="E28">A19</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T10" onmouseover="rs('finseq_2/T10')" onmouseout="rh()">FINSEQ_2:10</a></span>;<br/>
<a NAME="E5:17_3"/><span class="lab"><font color="Green" title="E29">B20</font></span>: 
<font color="Maroon" title="c4">m</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_3"><span class="lab"><font color="Green" title="E27">A18</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E6:17_3"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E7:17_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A22</font></span>: 
<font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:17_3"><span class="lab"><font color="Green" title="E29">B20</font></span></a>, <a class="txt" href="#E14:17"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E8:17_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E3:17_3"><span class="lab"><font color="Green" title="E27">A18</font></span></a>, <a class="ref" href="rfinseq.html#D1" onmouseover="rs('rfinseq/D1')" onmouseout="rh()">RFINSEQ:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:17_3"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17_3"><span class="lab"><font color="Green" title="E28">A19</font></span></a>, <a class="txt" href="#E7:17_3"><span class="lab"><font color="Green" title="E30">A22</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</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="E16:17"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D2" onmouseover="rs('integra1/D2')" onmouseout="rh()">INTEGRA1:def 2</a></span>;<br/>
<a NAME="E17:17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A24</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:17"><span class="lab"><font color="Green" title="E26">B23</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E28">A25</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span></span>)</span> = 
<font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E11:17"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="ref" href="rfinseq.html#D1" onmouseover="rs('rfinseq/D1')" onmouseout="rh()">RFINSEQ:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <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="txt" href="#E3:17"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="rfinseq.html#D1" onmouseover="rs('rfinseq/D1')" onmouseout="rh()">RFINSEQ:def 1</a></span>
<br/>.= 
 <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <font color="Maroon" title="c1">A</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D2" onmouseover="rs('integra1/D2')" onmouseout="rh()">INTEGRA1:def 2</a></span>
;<br/>
<a NAME="E19:17"/>
<font color="Maroon" title="c2">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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="#E9:17"><span class="lab"><font color="Green" title="E22">A8</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E20:17"/>
<font color="Maroon" title="c2">D</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1 is    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:17"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E17:17"><span class="lab"><font color="Green" title="E27">A24</font></span></a>, <a class="txt" href="#E18:17"><span class="lab"><font color="Green" title="E28">A25</font></span></a>, <a class="ref" href="integra1.html#D2" onmouseover="rs('integra1/D2')" onmouseout="rh()">INTEGRA1:def 2</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/>


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