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

<b>let </b><font color="Maroon" title="c8">b</font>, <font color="Maroon" title="c9">a</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>



<b>assume </b><a NAME="E1:10"/><i><font color="Green" title="E13">A1</font></i>: 
<font color="Maroon" title="c8">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ;<br/>

<b>let </b><font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c11">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="taxonom2.html#D5" title="TAXONOM2:def.5">TAXONOM2:def 5</a><br/>

<b>assume </b><a NAME="E2:10"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="toprealb.html#K1" title="TOPREALB:func.1">IntIntervals</a> <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c8">b</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">nx</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <b> such that </b><br/><a NAME="E4:10"/><i><font color="Green" title="E14">A2</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">nx</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">nx</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 ;<br/>
<b>assume </b><a NAME="E5:10"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="toprealb.html#K1" title="TOPREALB:func.1">IntIntervals</a> <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c8">b</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c13">ny</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <b> such that </b><br/><a NAME="E7:10"/><i><font color="Green" title="E15">A3</font></i>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c13">ny</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c13">ny</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 ;<br/>
<b>assume </b><a NAME="E8:10"/><i><font color="Green" title="E16">A4</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">y</font>
 ;<br/>

<b>assume </b><a NAME="E9:10"/>
<font color="Maroon" title="c10">x</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c11">y</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c14">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:10"/><i><font color="Green" title="E17">A5</font></i>: 
<font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">x</font>
 <b>and </b><br/><a NAME="E12:10"/><i><font color="Green" title="E18">A6</font></i>: 
<font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">y</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">z</font> = <font color="Maroon" title="c14">z</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="toprealb.html#E4:10"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="toprealb.html#E11:10"><i><font color="Green" title="E17">A5</font></i></a></i>;<br/>
<a NAME="E14:10"/><i><font color="Green" title="E19">A7</font></i>: 
( <font color="Maroon" title="c9">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">nx</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">z</font> &amp; <font color="Maroon" title="c15">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">nx</font> )
 
<b>by </b><i><a class="txt" href="toprealb.html#E4:10"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="toprealb.html#E11:10"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/>
<a NAME="E15:10"/><i><font color="Green" title="E20">A8</font></i>: 
( <font color="Maroon" title="c9">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c13">ny</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">z</font> &amp; <font color="Maroon" title="c15">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c13">ny</font> )
 
<b>by </b><i><a class="txt" href="toprealb.html#E7:10"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="toprealb.html#E12:10"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/>
<div><i><font color="Green" title="E21">A9</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><a NAME="E1:10_1"/><i><font color="Green" title="E22">A10</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">a</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E1:10"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E2:10_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c16">k</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E3:10_1"/>
<font color="Maroon" title="c9">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">b</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E1:10_1"><i><font color="Green" title="E22">A10</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_2"/>
( <font color="Maroon" title="c12">nx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">ny</font> or <font color="Maroon" title="c12">nx</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">ny</font> or <font color="Maroon" title="c12">nx</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">ny</font> )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_1"><b>suppose </b></a><a NAME="E1:10_2_1"/>
<font color="Maroon" title="c12">nx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">ny</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:10_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="toprealb.html#E4:10"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="toprealb.html#E7:10"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="toprealb.html#E8:10"><i><font color="Green" title="E16">A4</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_2"><b>suppose </b></a><a NAME="E1:10_2_2"/>
<font color="Maroon" title="c12">nx</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">ny</font>
 ;<br/><div class="add"><a NAME="E2:10_2_2"/><b>then </b>
<font color="Maroon" title="c12">nx</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">ny</font>
 <b>by </b><i><a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c16">k</font> = <font color="Maroon" title="c13">ny</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">nx</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><a NAME="E4:10_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">nx</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c16">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">nx</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E14:10"><i><font color="Green" title="E19">A7</font></i></a>, <a class="txt" href="toprealb.html#E15:10"><i><font color="Green" title="E20">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E5:10_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">nx</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">nx</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">nx</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">nx</font>
 <b>by </b><i><a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><a NAME="E6:10_2_2"/><b>then </b>
<font color="Maroon" title="c9">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">b</font>
 ;<br/><b>hence </b><a NAME="E7:10_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="toprealb.html#E16:10"><i><font color="Green" title="E21">A9</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_3"><b>suppose </b></a><a NAME="E1:10_2_3"/>
<font color="Maroon" title="c12">nx</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">ny</font>
 ;<br/><div class="add"><a NAME="E2:10_2_3"/><b>then </b>
<font color="Maroon" title="c13">ny</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">nx</font>
 <b>by </b><i><a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c16">k</font> = <font color="Maroon" title="c12">nx</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ny</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><a NAME="E4:10_2_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c13">ny</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c16">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c13">ny</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E14:10"><i><font color="Green" title="E19">A7</font></i></a>, <a class="txt" href="toprealb.html#E15:10"><i><font color="Green" title="E20">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E5:10_2_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c13">ny</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">ny</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c13">ny</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">ny</font>
 <b>by </b><i><a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><a NAME="E6:10_2_3"/><b>then </b>
<font color="Maroon" title="c9">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">b</font>
 ;<br/><b>hence </b><a NAME="E7:10_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="toprealb.html#E16:10"><i><font color="Green" title="E21">A9</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
