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<div class="add">

<b>let </b><font color="Maroon" title="c2">X</font>, <font color="Maroon" title="c3">Y</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:6"/><i><font color="Green" title="E5">A1</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> holds <br/><font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">y</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" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c3">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or ( <font color="Maroon" title="c2">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c3">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&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" title="E6">A2</font></i>: 
( <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c3">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 ;<br/><div class="add"><b>take </b>
1
;<br/><b>thus </b><a NAME="E2:6_1_1"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> holds <br/>( <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">y</font> )
 <b>by </b><i><a class="txt" href="axioms.html#E1:6_1_1"><i><font color="Green" title="E6">A2</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" title="E6">A3</font></i>: 
<font color="Maroon" title="c2">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>and </b><br/><a NAME="E2:6_1_2"/><i><font color="Green" title="E7">A4</font></i>: 
<font color="Maroon" title="c3">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c4">x1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E4:6_1_2"/><i><font color="Green" title="E8">A5</font></i>: 
<font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="axioms.html#E1:6_1_2"><i><font color="Green" title="E6">A3</font></i></a>, <a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></i>;<br/><b>consider </b><font color="Maroon" title="c5">x2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E6:6_1_2"/><i><font color="Green" title="E9">A6</font></i>: 
<font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 <b>by </b><i><a class="txt" href="axioms.html#E2:6_1_2"><i><font color="Green" title="E7">A4</font></i></a>, <a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></i>;<br/><a NAME="E7:6_1_2"/><i><font color="Green" title="E10">A7</font></i>: 
<font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="numbers.html#D1" title="NUMBERS:def.1">NUMBERS:def 1</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E8:6_1_2"/><i><font color="Green" title="E11">A8</font></i>: 
<font color="Maroon" title="c3">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="numbers.html#D1" title="NUMBERS:def.1">NUMBERS:def 1</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>thus </b><a NAME="E9:6_1_2"/>
 ex <font color="Olive" title="b1">z</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> holds <br/>( <font color="Olive" title="b2">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">z</font> &amp; <font color="Olive" title="b1">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">y</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" title="c2">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  &amp; <font color="Maroon" title="c3">Y</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ) or <font color="Maroon" title="c3">Y</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> or <font color="Maroon" title="c2">X</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">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" title="E12">A9</font></i>: 
<font color="Maroon" title="c2">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 <b>and </b><br/><a NAME="E2:6_1_2_1_1_1"/><i><font color="Green" title="E13">A10</font></i>: 
<font color="Maroon" title="c3">Y</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/><div class="add"><a NAME="E3:6_1_2_1_1_1"/><i><font color="Green" title="E14">A11</font></i>: 
<font color="Maroon" title="c3">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 <b>by </b><i><a class="txt" href="axioms.html#E8:6_1_2"><i><font color="Green" title="E11">A8</font></i></a>, <a class="txt" href="axioms.html#E2:6_1_2_1_1_1"><i><font color="Green" title="E13">A10</font></i></a>, <a class="ref" href="xboole_1.html#T73" title="XBOOLE_1:th.73">XBOOLE_1:73</a></i>;<br/><b>take </b><font color="Maroon" title="c6">z</font> = 0;<br/><b>let </b><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</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><b>that </b><br/><a NAME="E4:6_1_2_1_1_1"/><i><font color="Green" title="E15">A12</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <b>and </b><br/><a NAME="E5:6_1_2_1_1_1"/><i><font color="Green" title="E16">A13</font></i>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 ;<br/><a NAME="E6:6_1_2_1_1_1"/>
( not <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  )
 <b>by </b><i><a class="txt" href="axioms.html#E1:6_1_2_1_1_1"><i><font color="Green" title="E12">A9</font></i></a>, <a class="txt" href="axioms.html#E4:6_1_2_1_1_1"><i><font color="Green" title="E15">A12</font></i></a>, <a class="ref" href="arytm_0.html#T5" title="ARYTM_0:th.5">ARYTM_0:5</a>, <a class="ref" href="arytm_2.html#T21" title="ARYTM_2:th.21">ARYTM_2:21</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E7:6_1_2_1_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">z</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#D5" title="XXREAL_0:def.5">XXREAL_0:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">y'</font> = <font color="Maroon" title="c8">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  <b>by </b><i><a class="txt" href="axioms.html#E3:6_1_2_1_1_1"><i><font color="Green" title="E14">A11</font></i></a>, <a class="txt" href="axioms.html#E5:6_1_2_1_1_1"><i><font color="Green" title="E16">A13</font></i></a></i>;<br/><a NAME="E9:6_1_2_1_1_1"/>
<font color="Maroon" title="c1">o</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">y'</font>
 <b>by </b><i><a class="ref" href="arytm_1.html#T6" title="ARYTM_1:th.6">ARYTM_1:6</a></i>;<br/><b>hence </b><a NAME="E10:6_1_2_1_1_1"/>
<font color="Maroon" title="c6">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">y</font>
 <b>by </b><i><a class="txt" href="axioms.html#E1"><i><font color="Green" title="E1">Lm1</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" title="c3">Y</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c6">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:6_1_2_1_1_2"/><i><font color="Green" title="E12">A14</font></i>: 
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 <b>and </b><br/><a NAME="E4:6_1_2_1_1_2"/><i><font color="Green" title="E13">A15</font></i>: 
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">u</font>, <font color="Maroon" title="c8">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:6_1_2_1_1_2"/><i><font color="Green" title="E14">A16</font></i>: 
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>and </b><br/><a NAME="E7:6_1_2_1_1_2"/><i><font color="Green" title="E15">A17</font></i>: 
<font color="Maroon" title="c8">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 <b>and </b><br/><a NAME="E8:6_1_2_1_1_2"/><i><font color="Green" title="E16">A18</font></i>: 
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">u</font>,<font color="Maroon" title="c8">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="axioms.html#E4:6_1_2_1_1_2"><i><font color="Green" title="E13">A15</font></i></a>, <a class="ref" href="zfmisc_1.html#T103" title="ZFMISC_1:th.103">ZFMISC_1:103</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">y1</font> = <font color="Maroon" title="c8">y1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  <b>by </b><i><a class="txt" href="axioms.html#E7:6_1_2_1_1_2"><i><font color="Green" title="E15">A17</font></i></a></i>;<br/><a NAME="E10:6_1_2_1_1_2"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>by </b><i><a class="txt" href="axioms.html#E3:6_1_2_1_1_2"><i><font color="Green" title="E12">A14</font></i></a></i>;<br/><a NAME="E11:6_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E17">A19</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Maroon" title="c9">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>by </b><i><a class="txt" href="axioms.html#E6:6_1_2_1_1_2"><i><font color="Green" title="E14">A16</font></i></a>, <a class="txt" href="axioms.html#E8:6_1_2_1_1_2"><i><font color="Green" title="E16">A18</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c10">y0</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Maroon" title="c9">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">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" title="b1">r</font> where <font color="Olive" title="b1">r</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  : <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Olive" title="b1">r</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">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" title="c11">u</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="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:6_1_2_1_1_2_1"/>
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  : <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Olive" title="b1">r</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> </span>}</span> 
 ;<br/>

<a NAME="E2:6_1_2_1_1_2_1"/><b>then </b>
 ex <font color="Olive" title="b1">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  st <br/>( <font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Olive" title="b1">r</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> )
 
;<br/>
<b>hence </b><a NAME="E3:6_1_2_1_1_2_1"/>
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c11">Y'</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  : <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Olive" title="b1">r</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> </span>}</span>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  ;<br/><a NAME="E15:6_1_2_1_1_2"/><i><font color="Green" title="E18">A20</font></i>: 
<font color="Maroon" title="c10">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="axioms.html#E24"><i><font color="Green" title="E4">Lm4</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E16:6_1_2_1_1_2"/><i><font color="Green" title="E19">A21</font></i>: 
<font color="Maroon" title="c10">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 <b>by </b><i><a class="txt" href="axioms.html#E3:6_1_2_1_1_2"><i><font color="Green" title="E12">A14</font></i></a>, <a class="txt" href="axioms.html#E6:6_1_2_1_1_2"><i><font color="Green" title="E14">A16</font></i></a>, <a class="txt" href="axioms.html#E8:6_1_2_1_1_2"><i><font color="Green" title="E16">A18</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E17:6_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E20">A22</font></i>: 
<font color="Maroon" title="c9">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y'</font>
 ;<br/><a NAME="E18:6_1_2_1_1_2"/><i><font color="Green" title="E21">A23</font></i>: 
<font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</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" title="c12">u</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="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:6_1_2_1_1_2_2"/><i><font color="Green" title="E22">A24</font></i>: 
<font color="Maroon" title="c12">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c13">x</font> = <font color="Maroon" title="c12">u</font> as  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">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" title="E23">A25</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 ;<br/><a NAME="E2:6_1_2_1_1_2_2_1"/><i><font color="Green" title="E24">A26</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">y0</font>
 <b>by </b><i><a class="txt" href="axioms.html#E1:6"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="axioms.html#E16:6_1_2_1_1_2"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="axioms.html#E1:6_1_2_1_1_2_2"><i><font color="Green" title="E22">A24</font></i></a></i>;<br/><a NAME="E3:6_1_2_1_1_2_2_1"/>
( not <font color="Maroon" title="c10">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  &amp; not <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 <b>by </b><i><a class="txt" href="axioms.html#E15:6_1_2_1_1_2"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="axioms.html#E1:6_1_2_1_1_2_2_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="ref" href="arytm_0.html#T5" title="ARYTM_0:th.5">ARYTM_0:5</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">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" title="E24">A26</font></i></a>, <a class="ref" href="xxreal_0.html#D5" title="XXREAL_0:def.5">XXREAL_0:def 5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:6_1_2_1_1_2_2"/>
<font color="Maroon" title="c12">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="axioms.html#E7:6_1_2"><i><font color="Green" title="E10">A7</font></i></a>, <a class="txt" href="axioms.html#E1:6_1_2_1_1_2_2"><i><font color="Green" title="E22">A24</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c12">e</font>, <font color="Maroon" title="c13">x3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E20:6_1_2_1_1_2"/><i><font color="Green" title="E22">A27</font></i>: 
<font color="Maroon" title="c12">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>and </b><br/><a NAME="E21:6_1_2_1_1_2"/><i><font color="Green" title="E23">A28</font></i>: 
<font color="Maroon" title="c13">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 <b>and </b><br/><a NAME="E22:6_1_2_1_1_2"/><i><font color="Green" title="E24">A29</font></i>: 
<font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">e</font>,<font color="Maroon" title="c13">x3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="axioms.html#E4:6_1_2"><i><font color="Green" title="E8">A5</font></i></a>, <a class="ref" href="zfmisc_1.html#T103" title="ZFMISC_1:th.103">ZFMISC_1:103</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">x3</font> = <font color="Maroon" title="c13">x3</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  <b>by </b><i><a class="txt" href="axioms.html#E21:6_1_2_1_1_2"><i><font color="Green" title="E23">A28</font></i></a></i>;<br/><a NAME="E24:6_1_2_1_1_2"/><i><font color="Green" title="E25">A30</font></i>: 
<font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Maroon" title="c14">x3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="axioms.html#E20:6_1_2_1_1_2"><i><font color="Green" title="E22">A27</font></i></a>, <a class="txt" href="axioms.html#E22:6_1_2_1_1_2"><i><font color="Green" title="E24">A29</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E25:6_1_2_1_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  : <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Olive" title="b1">r</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">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" title="c15">u</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="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:6_1_2_1_1_2_3"/>
<font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  : <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Olive" title="b1">r</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> </span>}</span> 
 ;<br/>

<a NAME="E2:6_1_2_1_1_2_3"/><b>then </b>
 ex <font color="Olive" title="b1">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  st <br/>( <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Olive" title="b1">r</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> )
 
;<br/>
<b>hence </b><a NAME="E3:6_1_2_1_1_2_3"/>
<font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c15">X'</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  : <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Olive" title="b1">r</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> </span>}</span>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  ;<br/><a NAME="E27:6_1_2_1_1_2"/><i><font color="Green" title="E26">A31</font></i>: 
<font color="Maroon" title="c14">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">X'</font>
 <b>by </b><i><a class="txt" href="axioms.html#E4:6_1_2"><i><font color="Green" title="E8">A5</font></i></a>, <a class="txt" href="axioms.html#E24:6_1_2_1_1_2"><i><font color="Green" title="E25">A30</font></i></a></i>;<br/><a NAME="E28:6_1_2_1_1_2"/>
for <font color="Olive" title="b1">y'</font>, <font color="Olive" title="b2">x'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>   st <font color="Olive" title="b1">y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y'</font> &amp; <font color="Olive" title="b2">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">X'</font> holds <br/><font color="Olive" title="b1">y'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Olive" title="b2">x'</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" title="c16">y'</font>, <font color="Maroon" title="c17">x'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> ;<br/>

<b>assume </b><a NAME="E1:6_1_2_1_1_2_4"/>
<font color="Maroon" title="c16">y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y'</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c18">r1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> <b> such that </b><br/><a NAME="E3:6_1_2_1_1_2_4"/><i><font color="Green" title="E27">A32</font></i>: 
<font color="Maroon" title="c16">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">r1</font>
 <b>and </b><br/><a NAME="E4:6_1_2_1_1_2_4"/><i><font color="Green" title="E28">A33</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Maroon" title="c18">r1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 ;<br/>
<b>assume </b><a NAME="E5:6_1_2_1_1_2_4"/>
<font color="Maroon" title="c17">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">X'</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c19">r2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> <b> such that </b><br/><a NAME="E7:6_1_2_1_1_2_4"/><i><font color="Green" title="E29">A34</font></i>: 
<font color="Maroon" title="c17">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">r2</font>
 <b>and </b><br/><a NAME="E8:6_1_2_1_1_2_4"/><i><font color="Green" title="E30">A35</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Maroon" title="c19">r2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c20">x</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Maroon" title="c17">x'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c21">y</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Maroon" title="c16">y'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  <b>by </b><i><a class="txt" href="axioms.html#E3:6_1_2_1_1_2_4"><i><font color="Green" title="E27">A32</font></i></a>, <a class="txt" href="axioms.html#E4:6_1_2_1_1_2_4"><i><font color="Green" title="E28">A33</font></i></a>, <a class="txt" href="axioms.html#E7:6_1_2_1_1_2_4"><i><font color="Green" title="E29">A34</font></i></a>, <a class="txt" href="axioms.html#E8:6_1_2_1_1_2_4"><i><font color="Green" title="E30">A35</font></i></a>, <a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E10:6_1_2_1_1_2_4"/><i><font color="Green" title="E31">A36</font></i>: 
( <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c21">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<b>by </b><i><a class="txt" href="axioms.html#E24"><i><font color="Green" title="E4">Lm4</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E11:6_1_2_1_1_2_4"/>
<font color="Maroon" title="c20">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">y</font>
 
<b>by </b><i><a class="txt" href="axioms.html#E1:6"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="axioms.html#E3:6_1_2_1_1_2_4"><i><font color="Green" title="E27">A32</font></i></a>, <a class="txt" href="axioms.html#E4:6_1_2_1_1_2_4"><i><font color="Green" title="E28">A33</font></i></a>, <a class="txt" href="axioms.html#E7:6_1_2_1_1_2_4"><i><font color="Green" title="E29">A34</font></i></a>, <a class="txt" href="axioms.html#E8:6_1_2_1_1_2_4"><i><font color="Green" title="E30">A35</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c22">x''</font>, <font color="Maroon" title="c23">y''</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> <b> such that </b><br/><a NAME="E13:6_1_2_1_1_2_4"/><i><font color="Green" title="E32">A37</font></i>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Maroon" title="c22">x''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E14:6_1_2_1_1_2_4"/><i><font color="Green" title="E33">A38</font></i>: 
<font color="Maroon" title="c21">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Maroon" title="c23">y''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E15:6_1_2_1_1_2_4"/><i><font color="Green" title="E34">A39</font></i>: 
<font color="Maroon" title="c23">y''</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c22">x''</font>
 <b>by </b><i><a class="txt" href="axioms.html#E10:6_1_2_1_1_2_4"><i><font color="Green" title="E31">A36</font></i></a>, <a class="ref" href="xxreal_0.html#D5" title="XXREAL_0:def.5">XXREAL_0:def 5</a></i>;<br/>
<a NAME="E16:6_1_2_1_1_2_4"/>
( <font color="Maroon" title="c17">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">x''</font> &amp; <font color="Maroon" title="c16">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">y''</font> )
 
<b>by </b><i><a class="txt" href="axioms.html#E13:6_1_2_1_1_2_4"><i><font color="Green" title="E32">A37</font></i></a>, <a class="txt" href="axioms.html#E14:6_1_2_1_1_2_4"><i><font color="Green" title="E33">A38</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E17:6_1_2_1_1_2_4"/>
<font color="Maroon" title="c16">y'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c17">x'</font>
 <b>by </b><i><a class="txt" href="axioms.html#E15:6_1_2_1_1_2_4"><i><font color="Green" title="E34">A39</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c16">z'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> <b> such that </b><br/><a NAME="E30:6_1_2_1_1_2"/><i><font color="Green" title="E27">A40</font></i>: 
for <font color="Olive" title="b1">y'</font>, <font color="Olive" title="b2">x'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>   st <font color="Olive" title="b1">y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y'</font> &amp; <font color="Olive" title="b2">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">X'</font> holds <br/>( <font color="Olive" title="b1">y'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c16">z'</font> &amp; <font color="Maroon" title="c16">z'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Olive" title="b2">x'</font> )
 <b>by </b><i><a class="txt" href="axioms.html#E17:6_1_2_1_1_2"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="axioms.html#E27:6_1_2_1_1_2"><i><font color="Green" title="E26">A31</font></i></a>, <a class="ref" href="arytm_2.html#T9" title="ARYTM_2:th.9">ARYTM_2:9</a></i>;<br/><a NAME="E31:6_1_2_1_1_2"/><i><font color="Green" title="E28">A41</font></i>: 
<font color="Maroon" title="c9">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&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" title="E17">A19</font></i></a>, <a class="ref" href="arytm_0.html#T3" title="ARYTM_0:th.3">ARYTM_0:3</a></i>;<br/><a NAME="E32:6_1_2_1_1_2"/>
<font color="Maroon" title="c9">y1</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c16">z'</font>
 <b>by </b><i><a class="txt" href="axioms.html#E17:6_1_2_1_1_2"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="axioms.html#E27:6_1_2_1_1_2"><i><font color="Green" title="E26">A31</font></i></a>, <a class="txt" href="axioms.html#E30:6_1_2_1_1_2"><i><font color="Green" title="E27">A40</font></i></a></i>;<br/><a NAME="E33:6_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c16">z'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&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" title="E28">A41</font></i></a>, <a class="ref" href="arytm_1.html#T5" title="ARYTM_1:th.5">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" title="TARSKI:func.4">[</a><span class="default">0,<font color="Maroon" title="c16">z'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>by </b><i><a class="ref" href="arytm_0.html#T2" title="ARYTM_0:th.2">ARYTM_0:2</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c17">z</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Maroon" title="c16">z'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><b>take </b>
<font color="Maroon" title="c17">z</font>
;<br/><a NAME="E36:6_1_2_1_1_2"/><i><font color="Green" title="E29">A42</font></i>: 
<font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="axioms.html#E24"><i><font color="Green" title="E4">Lm4</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>let </b><font color="Maroon" title="c18">x</font>, <font color="Maroon" title="c19">y</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><b>that </b><br/><a NAME="E37:6_1_2_1_1_2"/><i><font color="Green" title="E30">A43</font></i>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <b>and </b><br/><a NAME="E38:6_1_2_1_1_2"/><i><font color="Green" title="E31">A44</font></i>: 
<font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 ;<br/><b>consider </b><font color="Maroon" title="c20">e</font>, <font color="Maroon" title="c21">x'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E40:6_1_2_1_1_2"/><i><font color="Green" title="E32">A45</font></i>: 
<font color="Maroon" title="c20">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>and </b><br/><a NAME="E41:6_1_2_1_1_2"/><i><font color="Green" title="E33">A46</font></i>: 
<font color="Maroon" title="c21">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 <b>and </b><br/><a NAME="E42:6_1_2_1_1_2"/><i><font color="Green" title="E34">A47</font></i>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">e</font>,<font color="Maroon" title="c21">x'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="axioms.html#E18:6_1_2_1_1_2"><i><font color="Green" title="E21">A23</font></i></a>, <a class="txt" href="axioms.html#E37:6_1_2_1_1_2"><i><font color="Green" title="E30">A43</font></i></a>, <a class="ref" href="zfmisc_1.html#T103" title="ZFMISC_1:th.103">ZFMISC_1:103</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c22">x'</font> = <font color="Maroon" title="c21">x'</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  <b>by </b><i><a class="txt" href="axioms.html#E41:6_1_2_1_1_2"><i><font color="Green" title="E33">A46</font></i></a></i>;<br/><a NAME="E44:6_1_2_1_1_2"/><i><font color="Green" title="E35">A48</font></i>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Maroon" title="c22">x'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="axioms.html#E40:6_1_2_1_1_2"><i><font color="Green" title="E32">A45</font></i></a>, <a class="txt" href="axioms.html#E42:6_1_2_1_1_2"><i><font color="Green" title="E34">A47</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E45:6_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E36">A49</font></i>: 
<font color="Maroon" title="c22">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">X'</font>
 <b>by </b><i><a class="txt" href="axioms.html#E37:6_1_2_1_1_2"><i><font color="Green" title="E30">A43</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" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  or <font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 </span><b>by </b><i><a class="txt" href="axioms.html#E8:6_1_2"><i><font color="Green" title="E11">A8</font></i></a>, <a class="txt" href="axioms.html#E38:6_1_2_1_1_2"><i><font color="Green" title="E31">A44</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">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" title="E37">A50</font></i>: 
<font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 ;<br/><div class="add"><a NAME="E2:6_1_2_1_1_2_5_1_1"/>
( <font color="Maroon" title="c9">y1</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c16">z'</font> &amp; <font color="Maroon" title="c16">z'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c22">x'</font> )
 <b>by </b><i><a class="txt" href="axioms.html#E17:6_1_2_1_1_2"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="axioms.html#E30:6_1_2_1_1_2"><i><font color="Green" title="E27">A40</font></i></a>, <a class="txt" href="axioms.html#E45:6_1_2_1_1_2"><i><font color="Green" title="E36">A49</font></i></a></i>;<br/><b>hence </b><a NAME="E3:6_1_2_1_1_2_5_1_1"/>
<font color="Maroon" title="c18">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">z</font>
 <b>by </b><i><a class="txt" href="axioms.html#E18:6_1_2_1_1_2"><i><font color="Green" title="E21">A23</font></i></a>, <a class="txt" href="axioms.html#E36:6_1_2_1_1_2"><i><font color="Green" title="E29">A42</font></i></a>, <a class="txt" href="axioms.html#E37:6_1_2_1_1_2"><i><font color="Green" title="E30">A43</font></i></a>, <a class="txt" href="axioms.html#E44:6_1_2_1_1_2"><i><font color="Green" title="E35">A48</font></i></a>, <a class="txt" href="axioms.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/><a NAME="E4:6_1_2_1_1_2_5_1_1"/>
( not <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  &amp; not <font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 <b>by </b><i><a class="txt" href="axioms.html#E36:6_1_2_1_1_2"><i><font color="Green" title="E29">A42</font></i></a>, <a class="txt" href="axioms.html#E1:6_1_2_1_1_2_5_1_1"><i><font color="Green" title="E37">A50</font></i></a>, <a class="ref" href="arytm_0.html#T5" title="ARYTM_0:th.5">ARYTM_0:5</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E5:6_1_2_1_1_2_5_1_1"/>
<font color="Maroon" title="c17">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">y</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#D5" title="XXREAL_0:def.5">XXREAL_0:def 5</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" title="E37">A51</font></i>: 
<font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c23">e</font>, <font color="Maroon" title="c24">y'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:6_1_2_1_1_2_5_1_2"/><i><font color="Green" title="E38">A52</font></i>: 
<font color="Maroon" title="c23">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>and </b><br/><a NAME="E4:6_1_2_1_1_2_5_1_2"/><i><font color="Green" title="E39">A53</font></i>: 
<font color="Maroon" title="c24">y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 <b>and </b><br/><a NAME="E5:6_1_2_1_1_2_5_1_2"/><i><font color="Green" title="E40">A54</font></i>: 
<font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">e</font>,<font color="Maroon" title="c24">y'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T103" title="ZFMISC_1:th.103">ZFMISC_1:103</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c25">y'</font> = <font color="Maroon" title="c24">y'</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">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" title="E39">A53</font></i></a></i>;<br/><a NAME="E7:6_1_2_1_1_2_5_1_2"/><i><font color="Green" title="E41">A55</font></i>: 
<font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Maroon" title="c25">y'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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" title="E38">A52</font></i></a>, <a class="txt" href="axioms.html#E5:6_1_2_1_1_2_5_1_2"><i><font color="Green" title="E40">A54</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E8:6_1_2_1_1_2_5_1_2"/><b>then </b>
<font color="Maroon" title="c25">y'</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="txt" href="axioms.html#E38:6_1_2_1_1_2"><i><font color="Green" title="E31">A44</font></i></a></i>;<br/><a NAME="E9:6_1_2_1_1_2_5_1_2"/><b>then </b>
( <font color="Maroon" title="c25">y'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c16">z'</font> &amp; <font color="Maroon" title="c16">z'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c22">x'</font> )
 <b>by </b><i><a class="txt" href="axioms.html#E30:6_1_2_1_1_2"><i><font color="Green" title="E27">A40</font></i></a>, <a class="txt" href="axioms.html#E45:6_1_2_1_1_2"><i><font color="Green" title="E36">A49</font></i></a></i>;<br/><b>hence </b><a NAME="E10:6_1_2_1_1_2_5_1_2"/>
( <font color="Maroon" title="c18">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">z</font> &amp; <font color="Maroon" title="c17">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">y</font> )
 <b>by </b><i><a class="txt" href="axioms.html#E18:6_1_2_1_1_2"><i><font color="Green" title="E21">A23</font></i></a>, <a class="txt" href="axioms.html#E36:6_1_2_1_1_2"><i><font color="Green" title="E29">A42</font></i></a>, <a class="txt" href="axioms.html#E37:6_1_2_1_1_2"><i><font color="Green" title="E30">A43</font></i></a>, <a class="txt" href="axioms.html#E44:6_1_2_1_1_2"><i><font color="Green" title="E35">A48</font></i></a>, <a class="txt" href="axioms.html#E1:6_1_2_1_1_2_5_1_2"><i><font color="Green" title="E37">A51</font></i></a>, <a class="txt" href="axioms.html#E7:6_1_2_1_1_2_5_1_2"><i><font color="Green" title="E41">A55</font></i></a>, <a class="txt" href="axioms.html#E1"><i><font color="Green" title="E1">Lm1</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" title="c18">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">z</font> &amp; <font color="Maroon" title="c17">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">y</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" title="c2">X</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c6">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:6_1_2_1_1_3"/><i><font color="Green" title="E12">A56</font></i>: 
<font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <b>and </b><br/><a NAME="E4:6_1_2_1_1_3"/><i><font color="Green" title="E13">A57</font></i>: 
<font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 <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="c7">x1</font> = <font color="Maroon" title="c6">x1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  <b>by </b><i><a class="txt" href="axioms.html#E4:6_1_2_1_1_3"><i><font color="Green" title="E13">A57</font></i></a></i>;<br/><a NAME="E6:6_1_2_1_1_3"/>
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 ;<br/><b>then reconsider </b><font color="Maroon" title="c8">x0</font> = <font color="Maroon" title="c7">x1</font> as  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  <b>by </b><i><a class="ref" href="arytm_0.html#T1" title="ARYTM_0:th.1">ARYTM_0:1</a>, <a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">X'</font> = <font color="Maroon" title="c2">X</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><a NAME="E9:6_1_2_1_1_3"/><i><font color="Green" title="E14">A58</font></i>: 
<font color="Maroon" title="c8">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">X'</font>
 <b>by </b><i><a class="txt" href="axioms.html#E3:6_1_2_1_1_3"><i><font color="Green" title="E12">A56</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E10:6_1_2_1_1_3"/><i><font color="Green" title="E15">A59</font></i>: 
<font color="Maroon" title="c3">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">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" title="c10">u</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="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:6_1_2_1_1_3_1"/><i><font color="Green" title="E16">A60</font></i>: 
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c11">y</font> = <font color="Maroon" title="c10">u</font> as  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">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" title="E17">A61</font></i>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/><a NAME="E2:6_1_2_1_1_3_1_1"/><i><font color="Green" title="E18">A62</font></i>: 
<font color="Maroon" title="c8">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">y</font>
 <b>by </b><i><a class="txt" href="axioms.html#E1:6"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="axioms.html#E3:6_1_2_1_1_3"><i><font color="Green" title="E12">A56</font></i></a>, <a class="txt" href="axioms.html#E1:6_1_2_1_1_3_1"><i><font color="Green" title="E16">A60</font></i></a></i>;<br/><a NAME="E3:6_1_2_1_1_3_1_1"/>
( not <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  &amp; not <font color="Maroon" title="c8">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</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" title="E17">A61</font></i></a>, <a class="ref" href="arytm_0.html#T5" title="ARYTM_0:th.5">ARYTM_0:5</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">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" title="E18">A62</font></i></a>, <a class="ref" href="xxreal_0.html#D5" title="XXREAL_0:def.5">XXREAL_0:def 5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:6_1_2_1_1_3_1"/>
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 <b>by </b><i><a class="txt" href="axioms.html#E8:6_1_2"><i><font color="Green" title="E11">A8</font></i></a>, <a class="txt" href="axioms.html#E1:6_1_2_1_1_3_1"><i><font color="Green" title="E16">A60</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c10">Y'</font> = <font color="Maroon" title="c3">Y</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  ;<br/><a NAME="E12:6_1_2_1_1_3"/>
for <font color="Olive" title="b1">x'</font>, <font color="Olive" title="b2">y'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>   st <font color="Olive" title="b1">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">X'</font> &amp; <font color="Olive" title="b2">y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Y'</font> holds <br/><font color="Olive" title="b1">x'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Olive" title="b2">y'</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" title="c11">x'</font>, <font color="Maroon" title="c12">y'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> ;<br/>

<b>assume </b><a NAME="E1:6_1_2_1_1_3_2"/><i><font color="Green" title="E16">A63</font></i>: 
( <font color="Maroon" title="c11">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">X'</font> &amp; <font color="Maroon" title="c12">y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Y'</font> )
 ;<br/>

<a NAME="E2:6_1_2_1_1_3_2"/>
( <font color="Maroon" title="c11">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  &amp; <font color="Maroon" title="c12">y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  )
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">x</font> = <font color="Maroon" title="c11">x'</font>, <font color="Maroon" title="c14">y</font> = <font color="Maroon" title="c12">y'</font> as  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  <b>by </b><i><a class="ref" href="arytm_0.html#T1" title="ARYTM_0:th.1">ARYTM_0:1</a>, <a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E4:6_1_2_1_1_3_2"/>
<font color="Maroon" title="c9">X'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E5:6_1_2_1_1_3_2"/><b>then </b>
<font color="Maroon" title="c13">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">y</font>
 
<b>by </b><i><a class="txt" href="axioms.html#E1:6"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="axioms.html#E1:6_1_2_1_1_3_2"><i><font color="Green" title="E16">A63</font></i></a></i>;<br/>
<a NAME="E6:6_1_2_1_1_3_2"/><b>then </b>
 ex <font color="Olive" title="b1">x'</font>, <font color="Olive" title="b2">y'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  st <br/>( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x'</font> &amp; <font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y'</font> &amp; <font color="Olive" title="b1">x'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Olive" title="b2">y'</font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#D5" title="XXREAL_0:def.5">XXREAL_0:def 5</a></i>;<br/>
<b>hence </b><a NAME="E7:6_1_2_1_1_3_2"/>
<font color="Maroon" title="c11">x'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c12">y'</font>
 ;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c11">z'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> <b> such that </b><br/><a NAME="E14:6_1_2_1_1_3"/><i><font color="Green" title="E16">A64</font></i>: 
for <font color="Olive" title="b1">x'</font>, <font color="Olive" title="b2">y'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>   st <font color="Olive" title="b1">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">X'</font> &amp; <font color="Olive" title="b2">y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Y'</font> holds <br/>( <font color="Olive" title="b1">x'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c11">z'</font> &amp; <font color="Maroon" title="c11">z'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Olive" title="b2">y'</font> )
 <b>by </b><i><a class="txt" href="axioms.html#E6:6_1_2"><i><font color="Green" title="E9">A6</font></i></a>, <a class="txt" href="axioms.html#E9:6_1_2_1_1_3"><i><font color="Green" title="E14">A58</font></i></a>, <a class="ref" href="arytm_2.html#T9" title="ARYTM_2:th.9">ARYTM_2:9</a></i>;<br/><a NAME="E15:6_1_2_1_1_3"/>
<font color="Maroon" title="c11">z'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 ;<br/><b>then reconsider </b><font color="Maroon" title="c12">z</font> = <font color="Maroon" title="c11">z'</font> as  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  <b>by </b><i><a class="ref" href="arytm_0.html#T1" title="ARYTM_0:th.1">ARYTM_0:1</a>, <a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><b>take </b>
<font color="Maroon" title="c12">z</font>
;<br/><b>let </b><font color="Maroon" title="c13">x</font>, <font color="Maroon" title="c14">y</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><b>that </b><br/><a NAME="E17:6_1_2_1_1_3"/><i><font color="Green" title="E17">A65</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <b>and </b><br/><a NAME="E18:6_1_2_1_1_3"/><i><font color="Green" title="E18">A66</font></i>: 
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c15">y'</font> = <font color="Maroon" title="c14">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  <b>by </b><i><a class="txt" href="axioms.html#E10:6_1_2_1_1_3"><i><font color="Green" title="E15">A59</font></i></a>, <a class="txt" href="axioms.html#E18:6_1_2_1_1_3"><i><font color="Green" title="E18">A66</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" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  or <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 </span><b>by </b><i><a class="txt" href="axioms.html#E7:6_1_2"><i><font color="Green" title="E10">A7</font></i></a>, <a class="txt" href="axioms.html#E17:6_1_2_1_1_3"><i><font color="Green" title="E17">A65</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">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" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c16">x'</font> = <font color="Maroon" title="c13">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  ;<br/><a NAME="E3:6_1_2_1_1_3_3_1_1"/>
( <font color="Maroon" title="c16">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">X'</font> &amp; <font color="Maroon" title="c15">y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Y'</font> )
 <b>by </b><i><a class="txt" href="axioms.html#E17:6_1_2_1_1_3"><i><font color="Green" title="E17">A65</font></i></a>, <a class="txt" href="axioms.html#E18:6_1_2_1_1_3"><i><font color="Green" title="E18">A66</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">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" title="c16">x'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c11">z'</font> &amp; <font color="Maroon" title="c11">z'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c15">y'</font> )
 <b>by </b><i><a class="txt" href="axioms.html#E14:6_1_2_1_1_3"><i><font color="Green" title="E16">A64</font></i></a></i>;<br/><b>hence </b><a NAME="E5:6_1_2_1_1_3_3_1_1"/>
( <font color="Maroon" title="c13">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">z</font> &amp; <font color="Maroon" title="c12">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">y</font> )
 <b>by </b><i><a class="txt" href="axioms.html#E1"><i><font color="Green" title="E1">Lm1</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" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</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" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  &amp; not <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 <b>by </b><i><a class="ref" href="arytm_0.html#T5" title="ARYTM_0:th.5">ARYTM_0:5</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E3:6_1_2_1_1_3_3_1_2"/>
<font color="Maroon" title="c13">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">z</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#D5" title="XXREAL_0:def.5">XXREAL_0:def 5</a></i>;<br/><a NAME="E4:6_1_2_1_1_3_3_1_2"/>
( <font color="Maroon" title="c7">x1</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c11">z'</font> &amp; <font color="Maroon" title="c11">z'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c15">y'</font> )
 <b>by </b><i><a class="txt" href="axioms.html#E9:6_1_2_1_1_3"><i><font color="Green" title="E14">A58</font></i></a>, <a class="txt" href="axioms.html#E14:6_1_2_1_1_3"><i><font color="Green" title="E16">A64</font></i></a>, <a class="txt" href="axioms.html#E18:6_1_2_1_1_3"><i><font color="Green" title="E18">A66</font></i></a></i>;<br/><b>hence </b><a NAME="E5:6_1_2_1_1_3_3_1_2"/>
<font color="Maroon" title="c12">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">y</font>
 <b>by </b><i><a class="txt" href="axioms.html#E1"><i><font color="Green" title="E1">Lm1</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" title="c13">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">z</font> &amp; <font color="Maroon" title="c12">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">y</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>
