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

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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">A</font>, <font color="Olive" title="b2">B</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">B</font> holds <br/>(<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font>) <a href="funct_7.html#K12" title="FUNCT_7:func.12">followed_by</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> is  <a href="prob_2.html#V1" title="PROB_2:attr.1">disjoint_valued</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</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">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">B</font> implies (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>) <a href="funct_7.html#K12" title="FUNCT_7:func.12">followed_by</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> is  <a href="prob_2.html#V1" title="PROB_2:attr.1">disjoint_valued</a>  )</span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c4">A1</font> = (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>) <a href="funct_7.html#K13" title="FUNCT_7:func.13">followed_by</a> <span class="p1">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c1">X</font></span>)</span> as   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<span class="kw">assume </span><a NAME="E2:8"/><span class="lab"><font color="Green" title="E5">A1</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">B</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">A</font>,<font color="Maroon" title="c3">B</font>) <a href="funct_7.html#K12" title="FUNCT_7:func.12">followed_by</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> is  <a href="prob_2.html#V1" title="PROB_2:attr.1">disjoint_valued</a> </span><br/>

<a NAME="E3:8"/>
<font color="Maroon" title="c4">A1</font> is <a href="prob_2.html#V1" title="PROB_2:attr.1">V128</a>()
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="8_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:8_1"/><span class="lab"><font color="Green" title="E6">A2</font></span>: 
<font color="Maroon" title="c4">A1</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T123" onmouseover="rs('funct_7/T123')" onmouseout="rh()">FUNCT_7:123</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">m</font>, <font color="Maroon" title="c6">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="prob_3.html#D4" onmouseover="rs('prob_3/D4')" onmouseout="rh()">PROB_3:def 4</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="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> or <font color="Maroon" title="c4">A1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c5">m</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">A1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c6">n</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:8_1"/><span class="lab"><font color="Green" title="E7">A3</font></span>: 
<font color="Maroon" title="c5">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">n</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="c4">A1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c5">m</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">A1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c6">n</font></span><br/>

<a NAME="E3:8_1"/><span class="lab"><font color="Green" title="E8">A4</font></span>: 
<font color="Maroon" title="c4">A1</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T122" onmouseover="rs('funct_7/T122')" onmouseout="rh()">FUNCT_7:122</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_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="E1:8_1_1"/>
( <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or ( <font color="Maroon" title="c5">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c5">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c5">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:8_1_1_1"/><span class="lab"><font color="Green" title="E9">A5</font></span>: 
<font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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="c4">A1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c5">m</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">A1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c6">n</font></span><br/><div class="add"><a NAME="E2:8_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_1"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="ref" href="nat_1.html#T14" onmouseover="rs('nat_1/T14')" onmouseout="rh()">NAT_1:14</a></span>;<br/><a NAME="E3:8_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A6</font></span>: 
<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <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/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_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="E1:8_1_1_1_1"/>
( <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 or <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:8_1_1_1"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="ref" href="nat_1.html#T22" onmouseover="rs('nat_1/T22')" onmouseout="rh()">NAT_1:22</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:8_1_1_1_1_1"/>
<font color="Maroon" title="c6">n</font> <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"> <font color="Maroon" title="c4">A1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c5">m</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">A1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c6">n</font></span><br/><div class="add"><a NAME="E2:8_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">A1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c6">n</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="ref" href="funct_7.html#T124" onmouseover="rs('funct_7/T124')" onmouseout="rh()">FUNCT_7:124</a></span>;<br/><span class="kw">hence </span><a NAME="E3:8_1_1_1_1_1"/>
<font color="Maroon" title="c4">A1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c5">m</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">A1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c6">n</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:8_1_1_1_1_2"/>
<font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"> <font color="Maroon" title="c4">A1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c5">m</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">A1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c6">n</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:8_1_1_1_1_2"/>
<font color="Maroon" title="c4">A1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c5">m</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">A1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c6">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E3:8_1"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="txt" href="#E1:8_1_1_1"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="ref" href="funct_7.html#T123" onmouseover="rs('funct_7/T123')" onmouseout="rh()">FUNCT_7:123</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:8_1_1_2"/><span class="lab"><font color="Green" title="E9">A7</font></span>: 
( <font color="Maroon" title="c5">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"> <font color="Maroon" title="c4">A1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c5">m</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">A1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c6">n</font></span><br/><div class="add"><a NAME="E2:8_1_1_2"/>
( <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 or <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ;<br/><a NAME="E3:8_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A8</font></span>: 
( <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 or <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <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#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_2_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="E1:8_1_1_2_1"/>
( <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 or <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_1"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="txt" href="#E1:8_1_1_2"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="txt" href="#E3:8_1_1_2"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="ref" href="nat_1.html#T14" onmouseover="rs('nat_1/T14')" onmouseout="rh()">NAT_1:14</a>, <a class="ref" href="nat_1.html#T22" onmouseover="rs('nat_1/T22')" onmouseout="rh()">NAT_1:22</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:8_1_1_2_1_1"/>
<font color="Maroon" title="c6">n</font> <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"> <font color="Maroon" title="c4">A1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c5">m</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">A1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c6">n</font></span><br/><div class="add"><a NAME="E2:8_1_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">A1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c6">n</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="ref" href="funct_7.html#T124" onmouseover="rs('funct_7/T124')" onmouseout="rh()">FUNCT_7:124</a></span>;<br/><span class="kw">hence </span><a NAME="E3:8_1_1_2_1_1"/>
<font color="Maroon" title="c4">A1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c5">m</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">A1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c6">n</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:8_1_1_2_1_2"/>
<font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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="c4">A1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c5">m</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">A1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c6">n</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:8_1_1_2_1_2"/>
<font color="Maroon" title="c4">A1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c5">m</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">A1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c6">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E3:8_1"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="txt" href="#E1:8_1"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="txt" href="#E1:8_1_1_2"><span class="lab"><font color="Green" title="E9">A7</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:8_1_1_3"/><span class="lab"><font color="Green" title="E9">A9</font></span>: 
( <font color="Maroon" title="c5">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c5">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&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"> <font color="Maroon" title="c4">A1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c5">m</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">A1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c6">n</font></span><br/><div class="add"><a NAME="E2:8_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" onmouseover="rs('nat_1/T14')" onmouseout="rh()">NAT_1:14</a></span>;<br/><a NAME="E3:8_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <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="E4:8_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c4">A1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c5">m</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="#E1:8_1_1_3"><span class="lab"><font color="Green" title="E9">A9</font></span></a>, <a class="ref" href="funct_7.html#T124" onmouseover="rs('funct_7/T124')" onmouseout="rh()">FUNCT_7:124</a>, <a class="ref" href="nat_1.html#T22" onmouseover="rs('nat_1/T22')" onmouseout="rh()">NAT_1:22</a></span>;<br/><span class="kw">hence </span><a NAME="E5:8_1_1_3"/>
<font color="Maroon" title="c4">A1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c5">m</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">A1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c6">n</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><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="E4:8"/>
(<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>) <a href="funct_7.html#K12" title="FUNCT_7:func.12">followed_by</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> is  <a href="prob_2.html#V1" title="PROB_2:attr.1">disjoint_valued</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"> verum</span><br/>


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