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

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">a</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">n</font>, <font color="Maroon">b</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">n</font>;<br/>



<b>set </b><font color="Maroon">CUTA1</font> = 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">a</font>;<br/>
<b>set </b><font color="Maroon">CUTA2</font> = <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">a</font>;<br/>
<b>set </b><font color="Maroon">CUTB1</font> = 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">b</font>;<br/>
<b>set </b><font color="Maroon">CUTB2</font> = <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">b</font>;<br/>
<b>thus </b><a NAME="E1:11"/>
( <font color="Maroon">a</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b</font> implies ( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">a</font> <a href="pboole.html#R6">=</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">b</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">a</font> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">b</font> ) )
 ;<br/>

<b>assume </b><a NAME="E2:11"/><i><font color="Green">E30</font></i>: 
( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">a</font> <a href="pboole.html#R6">=</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">b</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">a</font> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">b</font> )
 ;<br/>

<div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:11_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E2:11_1"/>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 0</span>)</span> <a href="binarith.html#K3">-'</a> 0
 ;<br/><a NAME="E3:11_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 0
 <b>by </b><i>, <a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/><a NAME="E4:11_1"/><b>then </b>
<span class="p1">(<span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:11_1"/>
<font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E48</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:11_2"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> )
 ;<br/><b>set </b><font color="Maroon">k</font> = <font color="Maroon">x</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><a NAME="E2:11_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E3:11_2"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">x</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E4:11_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:11_2"/><b>then </b>
<font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E6:11_2"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E7:11_2"/>
<font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E8:11_2"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">x</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/><a NAME="E9:11_2"/><b>then </b><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E10:11_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E11:11_2"/>
<font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , , <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x'</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:11_3"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E2:11_3"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="axioms.html#T30">AXIOMS:30</a></i>;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:11_3"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x'</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:11_3_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 or  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_3_1_1"><b>suppose </b></a><a NAME="E1:11_3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:11_3_1_1"/>
<font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_3_1_2"><b>suppose </b></a><a NAME="E1:11_3_1_2"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:11_3_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> )
 <b>by </b><i>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/><b>hence </b><a NAME="E3:11_3_1_2"/>
<font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:11"/>
<font color="Maroon">a</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


</div>
