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

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:6"/><i><font color="Green">E7</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 or ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1"><b>suppose </b></a><a NAME="E1:6_1_1"/><i><font color="Green">E8</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><div class="add"><b>take </b>
1
;<br/><b>thus </b><a NAME="E2:6_1_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="axioms.html#E1:6_1_1"><i><font color="Green">E8</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:6_1_2"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>and </b><br/><a NAME="E2:6_1_2"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E4:6_1_2"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="axioms.html#E1:6_1_2"><i><font color="Green">E8</font></i></a>, <a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E6:6_1_2"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="axioms.html#E2:6_1_2"><i><font color="Green">E9</font></i></a>, <a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/><a NAME="E7:6_1_2"/><i><font color="Green">E12</font></i>: 
 <a href="numbers.html#K1">REAL</a>  <a href="tarski.html#R1">c=</a> <a href="arytm_2.html#K2">REAL+</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="numbers.html#D1">NUMBERS:def 1</a></i>;<br/><a NAME="E8:6_1_2"/><b>then </b><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <a href="arytm_2.html#K2">REAL+</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E9:6_1_2"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <a href="arytm_2.html#K2">REAL+</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="axioms.html#E7:6_1_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>thus </b><a NAME="E10:6_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/>( <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_1_2_1_1"/>
( ( <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> ) or <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> or <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="arytm_2.html#K2">REAL+</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_2_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:6_1_2_1_1_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="arytm_2.html#K2">REAL+</a> 
 <b>and </b><br/><a NAME="E2:6_1_2_1_1_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><div class="add"><a NAME="E3:6_1_2_1_1_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="arytm_2.html#K2">REAL+</a> 
 <b>by </b><i><a class="txt" href="axioms.html#E9:6_1_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="axioms.html#E2:6_1_2_1_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xboole_1.html#T73">XBOOLE_1:73</a></i>;<br/><b>take </b><font color="Maroon">c<sub>6</sub></font> = 0;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E4:6_1_2_1_1_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E5:6_1_2_1_1_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E6:6_1_2_1_1_1"/>
( not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> &amp; not <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  )
 <b>by </b><i><a class="txt" href="axioms.html#E1:6_1_2_1_1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="axioms.html#E4:6_1_2_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="arytm_0.html#T5">ARYTM_0:5</a>, <a class="ref" href="arytm_2.html#T21">ARYTM_2:21</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E7:6_1_2_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="axioms.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a>  <b>by </b><i><a class="txt" href="axioms.html#E3:6_1_2_1_1_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="axioms.html#E5:6_1_2_1_1_1"><i><font color="Green">E19</font></i></a></i>;<br/><a NAME="E9:6_1_2_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="arytm_1.html#T6">ARYTM_1:6</a></i>;<br/><b>hence </b><a NAME="E10:6_1_2_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="axioms.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:6_1_2_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:6_1_2_1_1_2"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E4:6_1_2_1_1_2"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:6_1_2_1_1_2"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 <b>and </b><br/><a NAME="E7:6_1_2_1_1_2"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 <b>and </b><br/><a NAME="E8:6_1_2_1_1_2"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="axioms.html#E4:6_1_2_1_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="zfmisc_1.html#T103">ZFMISC_1:103</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a>  <b>by </b><i><a class="txt" href="axioms.html#E7:6_1_2_1_1_2"><i><font color="Green">E18</font></i></a></i>;<br/><a NAME="E10:6_1_2_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="txt" href="axioms.html#E3:6_1_2_1_1_2"><i><font color="Green">E15</font></i></a></i>;<br/><a NAME="E11:6_1_2_1_1_2"/><b>then </b><i><font color="Green">E20</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="txt" href="axioms.html#E6:6_1_2_1_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="axioms.html#E8:6_1_2_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K4">]</a></span> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><a NAME="E13:6_1_2_1_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a>  : <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="arytm_2.html#K2">REAL+</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:6_1_2_1_1_2_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a>  : <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> 
 ;<br/>

<a NAME="E2:6_1_2_1_1_2_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a>  st <br/>( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E3:6_1_2_1_1_2_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a>  : <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_2.html#K2">REAL+</a>  ;<br/><a NAME="E15:6_1_2_1_1_2"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="axioms.html#E26"><i><font color="Green">Lemma6</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E16:6_1_2_1_1_2"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="axioms.html#E3:6_1_2_1_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="axioms.html#E6:6_1_2_1_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="axioms.html#E8:6_1_2_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E17:6_1_2_1_1_2"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><a NAME="E18:6_1_2_1_1_2"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:6_1_2_1_1_2_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_1_1_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:6_1_2_1_1_2_2_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 ;<br/><a NAME="E2:6_1_2_1_1_2_2_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="axioms.html#E1:6"><i><font color="Green">E7</font></i></a>, <a class="txt" href="axioms.html#E16:6_1_2_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="axioms.html#E1:6_1_2_1_1_2_2"><i><font color="Green">E25</font></i></a></i>;<br/><a NAME="E3:6_1_2_1_1_2_2_1"/>
( not <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; not <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="txt" href="axioms.html#E15:6_1_2_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="axioms.html#E1:6_1_2_1_1_2_2_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="arytm_0.html#T5">ARYTM_0:5</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E4:6_1_2_1_1_2_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="axioms.html#E2:6_1_2_1_1_2_2_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="axioms.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:6_1_2_1_1_2_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="axioms.html#E8:6_1_2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="axioms.html#E1:6_1_2_1_1_2_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E20:6_1_2_1_1_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 <b>and </b><br/><a NAME="E21:6_1_2_1_1_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 <b>and </b><br/><a NAME="E22:6_1_2_1_1_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="axioms.html#E4:6_1_2"><i><font color="Green">E10</font></i></a>, <a class="ref" href="zfmisc_1.html#T103">ZFMISC_1:103</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a>  <b>by </b><i><a class="txt" href="axioms.html#E21:6_1_2_1_1_2"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E24:6_1_2_1_1_2"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="axioms.html#E20:6_1_2_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="axioms.html#E22:6_1_2_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E25:6_1_2_1_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a>  : <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="arytm_2.html#K2">REAL+</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_1_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:6_1_2_1_1_2_3"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a>  : <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> 
 ;<br/>

<a NAME="E2:6_1_2_1_1_2_3"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a>  st <br/>( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E3:6_1_2_1_1_2_3"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a>  : <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_2.html#K2">REAL+</a>  ;<br/><a NAME="E27:6_1_2_1_1_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="axioms.html#E4:6_1_2"><i><font color="Green">E10</font></i></a>, <a class="txt" href="axioms.html#E24:6_1_2_1_1_2"><i><font color="Green">E28</font></i></a></i>;<br/><a NAME="E28:6_1_2_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Olive">b<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_1_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> ;<br/>

<b>assume </b><a NAME="E1:6_1_2_1_1_2_4"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>18</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E3:6_1_2_1_1_2_4"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font>
 <b>and </b><br/><a NAME="E4:6_1_2_1_1_2_4"/><i><font color="Green">E31</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>18</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>
<b>assume </b><a NAME="E5:6_1_2_1_1_2_4"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>19</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E7:6_1_2_1_1_2_4"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font>
 <b>and </b><br/><a NAME="E8:6_1_2_1_1_2_4"/><i><font color="Green">E33</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>19</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>17</sub></font></span><a href="tarski.html#K4">]</a></span>, <font color="Maroon">c<sub>21</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  <b>by </b><i><a class="txt" href="axioms.html#E3:6_1_2_1_1_2_4"><i><font color="Green">E30</font></i></a>, <a class="txt" href="axioms.html#E4:6_1_2_1_1_2_4"><i><font color="Green">E31</font></i></a>, <a class="txt" href="axioms.html#E7:6_1_2_1_1_2_4"><i><font color="Green">E32</font></i></a>, <a class="txt" href="axioms.html#E8:6_1_2_1_1_2_4"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E10:6_1_2_1_1_2_4"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="txt" href="axioms.html#E26"><i><font color="Green">Lemma6</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E11:6_1_2_1_1_2_4"/>
<font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="txt" href="axioms.html#E1:6"><i><font color="Green">E7</font></i></a>, <a class="txt" href="axioms.html#E3:6_1_2_1_1_2_4"><i><font color="Green">E30</font></i></a>, <a class="txt" href="axioms.html#E4:6_1_2_1_1_2_4"><i><font color="Green">E31</font></i></a>, <a class="txt" href="axioms.html#E7:6_1_2_1_1_2_4"><i><font color="Green">E32</font></i></a>, <a class="txt" href="axioms.html#E8:6_1_2_1_1_2_4"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>22</sub></font>, <font color="Maroon">c<sub>23</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E13:6_1_2_1_1_2_4"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>22</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E14:6_1_2_1_1_2_4"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>23</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E15:6_1_2_1_1_2_4"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="axioms.html#E10:6_1_2_1_1_2_4"><i><font color="Green">E34</font></i></a>, <a class="txt" href="axioms.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/>
<a NAME="E16:6_1_2_1_1_2_4"/>
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> )
 
<b>by </b><i><a class="txt" href="axioms.html#E13:6_1_2_1_1_2_4"><i><font color="Green">E35</font></i></a>, <a class="txt" href="axioms.html#E14:6_1_2_1_1_2_4"><i><font color="Green">E36</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E17:6_1_2_1_1_2_4"/>
<font color="Maroon">c<sub>16</sub></font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="axioms.html#E15:6_1_2_1_1_2_4"><i><font color="Green">E37</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>16</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E30:6_1_2_1_1_2"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="axioms.html#E17:6_1_2_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="axioms.html#E27:6_1_2_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="arytm_2.html#T9">ARYTM_2:9</a></i>;<br/><a NAME="E31:6_1_2_1_1_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="axioms.html#E11:6_1_2_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="arytm_0.html#T3">ARYTM_0:3</a></i>;<br/><a NAME="E32:6_1_2_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="axioms.html#E17:6_1_2_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="axioms.html#E27:6_1_2_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="axioms.html#E30:6_1_2_1_1_2"><i><font color="Green">E30</font></i></a></i>;<br/><a NAME="E33:6_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="axioms.html#E31:6_1_2_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="arytm_1.html#T5">ARYTM_1:5</a></i>;<br/><a NAME="E34:6_1_2_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="arytm_0.html#T2">ARYTM_0:2</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>17</sub></font>
;<br/><a NAME="E36:6_1_2_1_1_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="axioms.html#E26"><i><font color="Green">Lemma6</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>let </b><font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E37:6_1_2_1_1_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E38:6_1_2_1_1_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>20</sub></font>, <font color="Maroon">c<sub>21</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E40:6_1_2_1_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 <b>and </b><br/><a NAME="E41:6_1_2_1_1_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 <b>and </b><br/><a NAME="E42:6_1_2_1_1_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="axioms.html#E18:6_1_2_1_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="axioms.html#E37:6_1_2_1_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="zfmisc_1.html#T103">ZFMISC_1:103</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>21</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a>  <b>by </b><i><a class="txt" href="axioms.html#E41:6_1_2_1_1_2"><i><font color="Green">E36</font></i></a></i>;<br/><a NAME="E44:6_1_2_1_1_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>22</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="axioms.html#E40:6_1_2_1_1_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="axioms.html#E42:6_1_2_1_1_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E45:6_1_2_1_1_2"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="axioms.html#E37:6_1_2_1_1_2"><i><font color="Green">E33</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_1_1_2_5"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_2_1_1_2_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_1_2_1_1_2_5_1"/>
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> )
 </span><b>by </b><i><a class="txt" href="axioms.html#E9:6_1_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="axioms.html#E38:6_1_2_1_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_2_1_1_2_5_1_1"><b>suppose </b></a><a NAME="E1:6_1_2_1_1_2_5_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 ;<br/><div class="add"><a NAME="E2:6_1_2_1_1_2_5_1_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="txt" href="axioms.html#E17:6_1_2_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="axioms.html#E30:6_1_2_1_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="axioms.html#E45:6_1_2_1_1_2"><i><font color="Green">E39</font></i></a></i>;<br/><b>hence </b><a NAME="E3:6_1_2_1_1_2_5_1_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="axioms.html#E18:6_1_2_1_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="axioms.html#E36:6_1_2_1_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="axioms.html#E37:6_1_2_1_1_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="axioms.html#E44:6_1_2_1_1_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="axioms.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/><a NAME="E4:6_1_2_1_1_2_5_1_1"/>
( not <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; not <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="txt" href="axioms.html#E36:6_1_2_1_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="axioms.html#E1:6_1_2_1_1_2_5_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="arytm_0.html#T5">ARYTM_0:5</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E5:6_1_2_1_1_2_5_1_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="axioms.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_2_1_1_2_5_1_2"><b>suppose </b></a><a NAME="E1:6_1_2_1_1_2_5_1_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>23</sub></font>, <font color="Maroon">c<sub>24</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:6_1_2_1_1_2_5_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 <b>and </b><br/><a NAME="E4:6_1_2_1_1_2_5_1_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 <b>and </b><br/><a NAME="E5:6_1_2_1_1_2_5_1_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T103">ZFMISC_1:103</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>24</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a>  <b>by </b><i><a class="txt" href="axioms.html#E4:6_1_2_1_1_2_5_1_2"><i><font color="Green">E42</font></i></a></i>;<br/><a NAME="E7:6_1_2_1_1_2_5_1_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>25</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="axioms.html#E3:6_1_2_1_1_2_5_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="axioms.html#E5:6_1_2_1_1_2_5_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E8:6_1_2_1_1_2_5_1_2"/><b>then </b>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="axioms.html#E38:6_1_2_1_1_2"><i><font color="Green">E34</font></i></a></i>;<br/><a NAME="E9:6_1_2_1_1_2_5_1_2"/><b>then </b>
( <font color="Maroon">c<sub>25</sub></font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="txt" href="axioms.html#E30:6_1_2_1_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="axioms.html#E45:6_1_2_1_1_2"><i><font color="Green">E39</font></i></a></i>;<br/><b>hence </b><a NAME="E10:6_1_2_1_1_2_5_1_2"/>
( <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="txt" href="axioms.html#E18:6_1_2_1_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="axioms.html#E36:6_1_2_1_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="axioms.html#E37:6_1_2_1_1_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="axioms.html#E44:6_1_2_1_1_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="axioms.html#E1:6_1_2_1_1_2_5_1_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="axioms.html#E7:6_1_2_1_1_2_5_1_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="axioms.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E47:6_1_2_1_1_2"/>
( <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>19</sub></font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:6_1_2_1_1_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="arytm_2.html#K2">REAL+</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:6_1_2_1_1_3"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E4:6_1_2_1_1_3"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a>  <b>by </b><i><a class="txt" href="axioms.html#E4:6_1_2_1_1_3"><i><font color="Green">E16</font></i></a></i>;<br/><a NAME="E6:6_1_2_1_1_3"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  <b>by </b><i><a class="ref" href="arytm_0.html#T1">ARYTM_0:1</a>, <a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K8">/\</a> <a href="arytm_2.html#K2">REAL+</a>  as   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_2.html#K2">REAL+</a>  <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E9:6_1_2_1_1_3"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="axioms.html#E3:6_1_2_1_1_3"><i><font color="Green">E15</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E10:6_1_2_1_1_3"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="arytm_2.html#K2">REAL+</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:6_1_2_1_1_3_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_1_1_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:6_1_2_1_1_3_1_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><a NAME="E2:6_1_2_1_1_3_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="axioms.html#E1:6"><i><font color="Green">E7</font></i></a>, <a class="txt" href="axioms.html#E3:6_1_2_1_1_3"><i><font color="Green">E15</font></i></a>, <a class="txt" href="axioms.html#E1:6_1_2_1_1_3_1"><i><font color="Green">E19</font></i></a></i>;<br/><a NAME="E3:6_1_2_1_1_3_1_1"/>
( not <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="txt" href="axioms.html#E1:6_1_2_1_1_3_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="arytm_0.html#T5">ARYTM_0:5</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E4:6_1_2_1_1_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="axioms.html#E2:6_1_2_1_1_3_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="axioms.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:6_1_2_1_1_3_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 <b>by </b><i><a class="txt" href="axioms.html#E9:6_1_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="axioms.html#E1:6_1_2_1_1_3_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_2.html#K2">REAL+</a>  ;<br/><a NAME="E12:6_1_2_1_1_3"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Olive">b<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_1_1_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> ;<br/>

<b>assume </b><a NAME="E1:6_1_2_1_1_3_2"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> )
 ;<br/>

<a NAME="E2:6_1_2_1_1_3_2"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  )
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>12</sub></font> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  <b>by </b><i><a class="ref" href="arytm_0.html#T1">ARYTM_0:1</a>, <a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E4:6_1_2_1_1_3_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E5:6_1_2_1_1_3_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="axioms.html#E1:6"><i><font color="Green">E7</font></i></a>, <a class="txt" href="axioms.html#E1:6_1_2_1_1_3_2"><i><font color="Green">E19</font></i></a></i>;<br/>
<a NAME="E6:6_1_2_1_1_3_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a>  st <br/>( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Olive">b<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="axioms.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:6_1_2_1_1_3_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E14:6_1_2_1_1_3"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="axioms.html#E6:6_1_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="axioms.html#E9:6_1_2_1_1_3"><i><font color="Green">E17</font></i></a>, <a class="ref" href="arytm_2.html#T9">ARYTM_2:9</a></i>;<br/><a NAME="E15:6_1_2_1_1_3"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  <b>by </b><i><a class="ref" href="arytm_0.html#T1">ARYTM_0:1</a>, <a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>12</sub></font>
;<br/><b>let </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E17:6_1_2_1_1_3"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E18:6_1_2_1_1_3"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a>  <b>by </b><i><a class="txt" href="axioms.html#E10:6_1_2_1_1_3"><i><font color="Green">E18</font></i></a>, <a class="txt" href="axioms.html#E18:6_1_2_1_1_3"><i><font color="Green">E21</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_1_1_3_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_2_1_1_3_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_1_2_1_1_3_3_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> )
 </span><b>by </b><i><a class="txt" href="axioms.html#E8:6_1_2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="axioms.html#E17:6_1_2_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_2_1_1_3_3_1_1"><b>suppose </b></a><a NAME="E1:6_1_2_1_1_3_3_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>13</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a>  ;<br/><a NAME="E3:6_1_2_1_1_3_3_1_1"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="axioms.html#E17:6_1_2_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="axioms.html#E18:6_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:6_1_2_1_1_3_3_1_1"/><b>then </b>
( <font color="Maroon">c<sub>16</sub></font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="txt" href="axioms.html#E14:6_1_2_1_1_3"><i><font color="Green">E19</font></i></a></i>;<br/><b>hence </b><a NAME="E5:6_1_2_1_1_3_3_1_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="axioms.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_2_1_1_3_3_1_2"><b>suppose </b></a><a NAME="E1:6_1_2_1_1_3_3_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><div class="add"><a NAME="E2:6_1_2_1_1_3_3_1_2"/><b>then </b>
( not <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; not <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="arytm_0.html#T5">ARYTM_0:5</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E3:6_1_2_1_1_3_3_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="axioms.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/><a NAME="E4:6_1_2_1_1_3_3_1_2"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="txt" href="axioms.html#E9:6_1_2_1_1_3"><i><font color="Green">E17</font></i></a>, <a class="txt" href="axioms.html#E14:6_1_2_1_1_3"><i><font color="Green">E19</font></i></a>, <a class="txt" href="axioms.html#E18:6_1_2_1_1_3"><i><font color="Green">E21</font></i></a></i>;<br/><b>hence </b><a NAME="E5:6_1_2_1_1_3_3_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="axioms.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E21:6_1_2_1_1_3"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div>
