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

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( 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> ) implies  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> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:29"/><span class="lab"><font color="Green" title="E5">A1</font></span>: 
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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  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> )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:29_1"/>
( <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c3">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or ( <font color="Maroon" title="c2">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1"><span class="kw">suppose </span></a><a NAME="E1:29_1_1"/><span class="lab"><font color="Green" title="E6">A2</font></span>: 
( <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c3">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  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> )</span><br/><div class="add"><span class="kw">take </span>
1
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 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> )</span><br/><span class="kw">thus </span><a NAME="E2:29_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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_1_1"><span class="lab"><font color="Green" title="E6">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:29_1_2"/><span class="lab"><font color="Green" title="E6">A3</font></span>: 
<font color="Maroon" title="c2">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E2:29_1_2"/><span class="lab"><font color="Green" title="E7">A4</font></span>: 
<font color="Maroon" title="c3">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  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> )</span><br/><div class="add"><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E4:29_1_2"/><span class="lab"><font color="Green" title="E8">A5</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_1_2"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></span>;<br/><a NAME="E5:29_1_2"/><span class="lab"><font color="Green" title="E9">A6</font></span>: 
<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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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 class="kw">by</span> <span class="lab"><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></span>;<br/><a NAME="E6:29_1_2"/><span class="lab"><font color="Green" title="E10">A7</font></span>: 
<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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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 class="kw">by</span> <span class="lab"><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></span>;<br/><a NAME="E7:29_1_2"/><span class="lab"><font color="Green" title="E11">A8</font></span>: 
 ex <font color="Olive" title="b1">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>  st <font color="Olive" title="b1">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_1_2"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></span>;<br/><span class="kw">thus </span><a NAME="E8:29_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> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:29_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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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="29_1_2_1_1_1"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:29_1_2_1_1_1"/><span class="lab"><font color="Green" title="E12">A9</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E2:29_1_2_1_1_1"/><span class="lab"><font color="Green" title="E13">A10</font></span>: 
<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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  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> )</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c5">z</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 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="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c5">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">y</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> implies ( <font color="Maroon" title="c6">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c5">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">y</font> ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:29_1_2_1_1_1"/><span class="lab"><font color="Green" title="E14">A11</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">and </span><br/><a NAME="E4:29_1_2_1_1_1"/><span class="lab"><font color="Green" title="E15">A12</font></span>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c5">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">y</font> )</span><br/><a NAME="E5:29_1_2_1_1_1"/>
( not <font color="Maroon" title="c5">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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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="c6">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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_1_2_1_1_1"><span class="lab"><font color="Green" title="E12">A9</font></span></a>, <a class="txt" href="#E3:29_1_2_1_1_1"><span class="lab"><font color="Green" title="E14">A11</font></span></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></span>;<br/><span class="kw">hence </span><a NAME="E6:29_1_2_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D5" title="XXREAL_0:def.5">XXREAL_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">y</font></span><br/><a NAME="E7:29_1_2_1_1_1"/>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:29_1_2"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="txt" href="#E2:29_1_2_1_1_1"><span class="lab"><font color="Green" title="E13">A10</font></span></a>, <a class="ref" href="xboole_1.html#T73" title="XBOOLE_1:th.73">XBOOLE_1:73</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">y9</font> = <font color="Maroon" title="c7">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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_1_2_1_1_1"><span class="lab"><font color="Green" title="E15">A12</font></span></a></span>;<br/><a NAME="E9:29_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="c8">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_1.html#T6" title="ARYTM_1:th.6">ARYTM_1:6</a></span>;<br/><span class="kw">hence </span><a NAME="E10:29_1_2_1_1_1"/>
<font color="Maroon" title="c5">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="axioms.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:29_1_2_1_1_2"/><span class="lab"><font color="Green" title="E12">A13</font></span>: 
<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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  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> )</span><br/><div class="add"><a NAME="E2:29_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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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="29_1_2_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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>  or <font color="Maroon" title="c5">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>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:29_1_2_1_1_2_1"/>
<font color="Maroon" title="c5">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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">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> </span><br/>

<a NAME="E2:29_1_2_1_1_2_1"/><span class="kw">then </span>
 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="c5">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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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/>
<span class="kw">hence </span><a NAME="E3:29_1_2_1_1_2_1"/>
<font color="Maroon" title="c5">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">X9</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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="E4:29_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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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="29_1_2_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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>  or <font color="Maroon" title="c6">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>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:29_1_2_1_1_2_2"/>
<font color="Maroon" title="c6">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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">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> </span><br/>

<a NAME="E2:29_1_2_1_1_2_2"/><span class="kw">then </span>
 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="c6">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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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/>
<span class="kw">hence </span><a NAME="E3:29_1_2_1_1_2_2"/>
<font color="Maroon" title="c6">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">Y9</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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/><span class="kw">consider </span><font color="Maroon" title="c7">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:29_1_2_1_1_2"/><span class="lab"><font color="Green" title="E13">A14</font></span>: 
<font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 <span class="kw">and </span><br/><a NAME="E8:29_1_2_1_1_2"/><span class="lab"><font color="Green" title="E14">A15</font></span>: 
<font color="Maroon" title="c7">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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_1_2_1_1_2"><span class="lab"><font color="Green" title="E12">A13</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c8">u</font>, <font color="Maroon" title="c9">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E10:29_1_2_1_1_2"/><span class="lab"><font color="Green" title="E15">A16</font></span>: 
<font color="Maroon" title="c8">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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E11:29_1_2_1_1_2"/><span class="lab"><font color="Green" title="E16">A17</font></span>: 
<font color="Maroon" title="c9">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> 
 <span class="kw">and </span><br/><a NAME="E12:29_1_2_1_1_2"/><span class="lab"><font color="Green" title="E17">A18</font></span>: 
<font color="Maroon" title="c7">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="c8">u</font>,<font color="Maroon" title="c9">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:29_1_2_1_1_2"><span class="lab"><font color="Green" title="E14">A15</font></span></a>, <a class="ref" href="zfmisc_1.html#T103" title="ZFMISC_1:th.103">ZFMISC_1:103</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">y1</font> = <font color="Maroon" title="c9">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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:29_1_2_1_1_2"><span class="lab"><font color="Green" title="E16">A17</font></span></a></span>;<br/><a NAME="E14:29_1_2_1_1_2"/>
<font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:29_1_2_1_1_2"><span class="lab"><font color="Green" title="E13">A14</font></span></a></span>;<br/><a NAME="E15:29_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A19</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c10">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:29_1_2_1_1_2"><span class="lab"><font color="Green" title="E15">A16</font></span></a>, <a class="txt" href="#E12:29_1_2_1_1_2"><span class="lab"><font color="Green" title="E17">A18</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">y0</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c10">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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/><a NAME="E17:29_1_2_1_1_2"/><span class="lab"><font color="Green" title="E19">A20</font></span>: 
<font color="Maroon" title="c11">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:29_1_2_1_1_2"><span class="lab"><font color="Green" title="E13">A14</font></span></a>, <a class="txt" href="#E10:29_1_2_1_1_2"><span class="lab"><font color="Green" title="E15">A16</font></span></a>, <a class="txt" href="#E12:29_1_2_1_1_2"><span class="lab"><font color="Green" title="E17">A18</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E18:29_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A21</font></span>: 
<font color="Maroon" title="c10">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y9</font>
 ;<br/><a NAME="E19:29_1_2_1_1_2"/><span class="lab"><font color="Green" title="E21">A22</font></span>: 
<font color="Maroon" title="c11">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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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 class="kw">by</span> <span class="lab"><a class="txt" href="axioms.html#E4"><span class="lab"><font color="Green" title="E4">Lm4</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E20:29_1_2_1_1_2"/><span class="lab"><font color="Green" title="E22">A23</font></span>: 
<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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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="29_1_2_1_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <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> or <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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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><br/>

<span class="kw">assume </span><a NAME="E1:29_1_2_1_1_2_3"/><span class="lab"><font color="Green" title="E23">A24</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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><br/>

<span class="kw">then </span><span class="kw">reconsider </span><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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_1_1_2_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:29_1_2_1_1_2_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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:29_1_2_1_1_2_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A25</font></span>: 
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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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 class="kw">by</span> <span class="lab"><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></span>;<br/><a NAME="E3:29_1_2_1_1_2_3_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="c11">y0</font> &amp; not <font color="Maroon" title="c11">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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E19:29_1_2_1_1_2"><span class="lab"><font color="Green" title="E21">A22</font></span></a>, <a class="txt" href="#E17:29_1_2_1_1_2"><span class="lab"><font color="Green" title="E19">A20</font></span></a>, <a class="txt" href="#E1:29_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E23">A24</font></span></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></span>;<br/><span class="kw">hence </span><a NAME="E4:29_1_2_1_1_2_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_1_2_1_1_2_3_1"><span class="lab"><font color="Green" title="E24">A25</font></span></a>, <a class="ref" href="xxreal_0.html#D5" title="XXREAL_0:def.5">XXREAL_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:29_1_2_1_1_2_3"/>
<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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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 class="kw">by</span> <span class="lab"><a class="txt" href="#E5:29_1_2"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="txt" href="#E1:29_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E23">A24</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E22:29_1_2_1_1_2"/><span class="lab"><font color="Green" title="E23">A26</font></span>: 
<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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E23:29_1_2_1_1_2"/><span class="lab"><font color="Green" title="E24">A27</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E24:29_1_2_1_1_2"/><span class="lab"><font color="Green" title="E25">A28</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_1_2"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="ref" href="zfmisc_1.html#T103" title="ZFMISC_1:th.103">ZFMISC_1:103</a></span>;<br/><span class="kw">reconsider </span><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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:29_1_2_1_1_2"><span class="lab"><font color="Green" title="E24">A27</font></span></a></span>;<br/><a NAME="E26:29_1_2_1_1_2"/>
<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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c14">x3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:29_1_2_1_1_2"><span class="lab"><font color="Green" title="E23">A26</font></span></a>, <a class="txt" href="#E24:29_1_2_1_1_2"><span class="lab"><font color="Green" title="E25">A28</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E27:29_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A29</font></span>: 
<font color="Maroon" title="c14">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_1_2"><span class="lab"><font color="Green" title="E8">A5</font></span></a></span>;<br/><a NAME="E28:29_1_2_1_1_2"/>
for <font color="Olive" title="b1">y9</font>, <font color="Olive" title="b2">x9</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">y9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y9</font> &amp; <font color="Olive" title="b2">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X9</font> holds <br/><font color="Olive" title="b1">y9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Olive" title="b2">x9</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_1_1_2_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">y9</font>, <font color="Maroon" title="c16">x9</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">y9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y9</font> &amp; <font color="Maroon" title="c16">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X9</font> implies <font color="Maroon" title="c15">y9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c16">x9</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:29_1_2_1_1_2_4"/>
<font color="Maroon" title="c15">y9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c16">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X9</font> or <font color="Maroon" title="c15">y9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c16">x9</font> )</span><br/>

<a NAME="E2:29_1_2_1_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A30</font></span>: 
 ex <font color="Olive" title="b1">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>  st <br/>( <font color="Maroon" title="c15">y9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Olive" title="b1">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/>
<span class="kw">assume </span><a NAME="E3:29_1_2_1_1_2_4"/>
<font color="Maroon" title="c16">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">y9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c16">x9</font></span><br/>

<a NAME="E4:29_1_2_1_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A31</font></span>: 
 ex <font color="Olive" title="b1">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>  st <br/>( <font color="Maroon" title="c16">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Olive" title="b1">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/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">x</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c16">x9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c18">y</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c15">y9</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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E27">A30</font></span></a>, <a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<a NAME="E6:29_1_2_1_1_2_4"/><span class="lab"><font color="Green" title="E29">A32</font></span>: 
( <font color="Maroon" title="c17">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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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="c18">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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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 class="kw">by</span> <span class="lab"><a class="txt" href="axioms.html#E4"><span class="lab"><font color="Green" title="E4">Lm4</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E7:29_1_2_1_1_2_4"/>
<font color="Maroon" title="c17">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E2:29_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E27">A30</font></span></a>, <a class="txt" href="#E4:29_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E28">A31</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">x99</font>, <font color="Maroon" title="c20">y99</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> <span class="kw"> such that </span><br/><a NAME="E9:29_1_2_1_1_2_4"/><span class="lab"><font color="Green" title="E30">A33</font></span>: 
<font color="Maroon" title="c17">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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c19">x99</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E10:29_1_2_1_1_2_4"/><span class="lab"><font color="Green" title="E31">A34</font></span>: 
( <font color="Maroon" title="c18">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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c20">y99</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c20">y99</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c19">x99</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:29_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E29">A32</font></span></a>, <a class="ref" href="xxreal_0.html#D5" title="XXREAL_0:def.5">XXREAL_0:def 5</a></span>;<br/>
<a NAME="E11:29_1_2_1_1_2_4"/>
<font color="Maroon" title="c16">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x99</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:29_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E30">A33</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:29_1_2_1_1_2_4"/>
<font color="Maroon" title="c15">y9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c16">x9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:29_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E31">A34</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">z9</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> <span class="kw"> such that </span><br/><a NAME="E30:29_1_2_1_1_2"/><span class="lab"><font color="Green" title="E27">A35</font></span>: 
for <font color="Olive" title="b1">y9</font>, <font color="Olive" title="b2">x9</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">y9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y9</font> &amp; <font color="Olive" title="b2">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X9</font> holds <br/>( <font color="Olive" title="b1">y9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c15">z9</font> &amp; <font color="Maroon" title="c15">z9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Olive" title="b2">x9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:29_1_2_1_1_2"><span class="lab"><font color="Green" title="E26">A29</font></span></a>, <a class="ref" href="arytm_2.html#T9" title="ARYTM_2:th.9">ARYTM_2:9</a></span>;<br/><a NAME="E31:29_1_2_1_1_2"/><span class="lab"><font color="Green" title="E28">A36</font></span>: 
<font color="Maroon" title="c10">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:29_1_2_1_1_2"><span class="lab"><font color="Green" title="E18">A19</font></span></a>, <a class="ref" href="arytm_0.html#T3" title="ARYTM_0:th.3">ARYTM_0:3</a></span>;<br/><a NAME="E32:29_1_2_1_1_2"/>
<font color="Maroon" title="c10">y1</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c15">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:29_1_2_1_1_2"><span class="lab"><font color="Green" title="E20">A21</font></span></a>, <a class="txt" href="#E27:29_1_2_1_1_2"><span class="lab"><font color="Green" title="E26">A29</font></span></a>, <a class="txt" href="#E30:29_1_2_1_1_2"><span class="lab"><font color="Green" title="E27">A35</font></span></a></span>;<br/><a NAME="E33:29_1_2_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c15">z9</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:29_1_2_1_1_2"><span class="lab"><font color="Green" title="E28">A36</font></span></a>, <a class="ref" href="arytm_0.html#T2" title="ARYTM_0:th.2">ARYTM_0:2</a>, <a class="ref" href="arytm_1.html#T5" title="ARYTM_1:th.5">ARYTM_1:5</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">z</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c15">z9</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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c16">z</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 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="Maroon" title="c16">z</font> &amp; <font color="Maroon" title="c16">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">y</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c17">x</font>, <font color="Maroon" title="c18">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> implies ( <font color="Maroon" title="c17">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">z</font> &amp; <font color="Maroon" title="c16">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">y</font> ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E35:29_1_2_1_1_2"/><span class="lab"><font color="Green" title="E29">A37</font></span>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">and </span><br/><a NAME="E36:29_1_2_1_1_2"/><span class="lab"><font color="Green" title="E30">A38</font></span>: 
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">z</font> &amp; <font color="Maroon" title="c16">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">y</font> )</span><br/><span class="kw">consider </span><font color="Maroon" title="c19">e</font>, <font color="Maroon" title="c20">x9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E38:29_1_2_1_1_2"/><span class="lab"><font color="Green" title="E31">A39</font></span>: 
<font color="Maroon" title="c19">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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E39:29_1_2_1_1_2"/><span class="lab"><font color="Green" title="E32">A40</font></span>: 
<font color="Maroon" title="c20">x9</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> 
 <span class="kw">and </span><br/><a NAME="E40:29_1_2_1_1_2"/><span class="lab"><font color="Green" title="E33">A41</font></span>: 
<font color="Maroon" title="c17">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="c19">e</font>,<font color="Maroon" title="c20">x9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:29_1_2_1_1_2"><span class="lab"><font color="Green" title="E22">A23</font></span></a>, <a class="txt" href="#E35:29_1_2_1_1_2"><span class="lab"><font color="Green" title="E29">A37</font></span></a>, <a class="ref" href="zfmisc_1.html#T103" title="ZFMISC_1:th.103">ZFMISC_1:103</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c21">x9</font> = <font color="Maroon" title="c20">x9</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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:29_1_2_1_1_2"><span class="lab"><font color="Green" title="E32">A40</font></span></a></span>;<br/><a NAME="E42:29_1_2_1_1_2"/><span class="lab"><font color="Green" title="E34">A42</font></span>: 
<font color="Maroon" title="c16">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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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 class="kw">by</span> <span class="lab"><a class="txt" href="axioms.html#E4"><span class="lab"><font color="Green" title="E4">Lm4</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E43:29_1_2_1_1_2"/><span class="lab"><font color="Green" title="E35">A43</font></span>: 
<font color="Maroon" title="c17">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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c21">x9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:29_1_2_1_1_2"><span class="lab"><font color="Green" title="E31">A39</font></span></a>, <a class="txt" href="#E40:29_1_2_1_1_2"><span class="lab"><font color="Green" title="E33">A41</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E44:29_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A44</font></span>: 
<font color="Maroon" title="c21">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:29_1_2_1_1_2"><span class="lab"><font color="Green" title="E29">A37</font></span></a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_1_1_2_5"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2_1_1_2_5_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:29_1_2_1_1_2_5_1"/>
( <font color="Maroon" title="c18">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="c18">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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:29_1_2"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="txt" href="#E36:29_1_2_1_1_2"><span class="lab"><font color="Green" title="E30">A38</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2_1_1_2_5_1_1"><span class="kw">suppose </span></a><a NAME="E1:29_1_2_1_1_2_5_1_1"/><span class="lab"><font color="Green" title="E37">A45</font></span>: 
<font color="Maroon" title="c18">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">z</font> &amp; <font color="Maroon" title="c16">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">y</font> )</span><br/><div class="add"><a NAME="E2:29_1_2_1_1_2_5_1_1"/>
<font color="Maroon" title="c15">z9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c21">x9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:29_1_2_1_1_2"><span class="lab"><font color="Green" title="E20">A21</font></span></a>, <a class="txt" href="#E30:29_1_2_1_1_2"><span class="lab"><font color="Green" title="E27">A35</font></span></a>, <a class="txt" href="#E44:29_1_2_1_1_2"><span class="lab"><font color="Green" title="E36">A44</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:29_1_2_1_1_2_5_1_1"/>
<font color="Maroon" title="c17">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:29_1_2_1_1_2"><span class="lab"><font color="Green" title="E22">A23</font></span></a>, <a class="txt" href="#E42:29_1_2_1_1_2"><span class="lab"><font color="Green" title="E34">A42</font></span></a>, <a class="txt" href="#E35:29_1_2_1_1_2"><span class="lab"><font color="Green" title="E29">A37</font></span></a>, <a class="txt" href="#E43:29_1_2_1_1_2"><span class="lab"><font color="Green" title="E35">A43</font></span></a>, <a class="txt" href="axioms.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">y</font></span><br/><a NAME="E4:29_1_2_1_1_2_5_1_1"/><span class="lab"><font color="Green" title="E38">A46</font></span>: 
not <font color="Maroon" title="c18">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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_1_2_1_1_2_5_1_1"><span class="lab"><font color="Green" title="E37">A45</font></span></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></span>;<br/><a NAME="E5:29_1_2_1_1_2_5_1_1"/>
not <font color="Maroon" title="c16">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:29_1_2_1_1_2"><span class="lab"><font color="Green" title="E34">A42</font></span></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></span>;<br/><span class="kw">hence </span><a NAME="E6:29_1_2_1_1_2_5_1_1"/>
<font color="Maroon" title="c16">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_1_2_1_1_2_5_1_1"><span class="lab"><font color="Green" title="E38">A46</font></span></a>, <a class="ref" href="xxreal_0.html#D5" title="XXREAL_0:def.5">XXREAL_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2_1_1_2_5_1_2"><span class="kw">suppose </span></a><a NAME="E1:29_1_2_1_1_2_5_1_2"/><span class="lab"><font color="Green" title="E37">A47</font></span>: 
<font color="Maroon" title="c18">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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">z</font> &amp; <font color="Maroon" title="c16">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">y</font> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c22">e</font>, <font color="Maroon" title="c23">y9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:29_1_2_1_1_2_5_1_2"/><span class="lab"><font color="Green" title="E38">A48</font></span>: 
<font color="Maroon" title="c22">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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E4:29_1_2_1_1_2_5_1_2"/><span class="lab"><font color="Green" title="E39">A49</font></span>: 
<font color="Maroon" title="c23">y9</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> 
 <span class="kw">and </span><br/><a NAME="E5:29_1_2_1_1_2_5_1_2"/><span class="lab"><font color="Green" title="E40">A50</font></span>: 
<font color="Maroon" title="c18">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="c22">e</font>,<font color="Maroon" title="c23">y9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T103" title="ZFMISC_1:th.103">ZFMISC_1:103</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c24">y9</font> = <font color="Maroon" title="c23">y9</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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_1_2_1_1_2_5_1_2"><span class="lab"><font color="Green" title="E39">A49</font></span></a></span>;<br/><a NAME="E7:29_1_2_1_1_2_5_1_2"/><span class="lab"><font color="Green" title="E41">A51</font></span>: 
<font color="Maroon" title="c18">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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c24">y9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_1_2_1_1_2_5_1_2"><span class="lab"><font color="Green" title="E38">A48</font></span></a>, <a class="txt" href="#E5:29_1_2_1_1_2_5_1_2"><span class="lab"><font color="Green" title="E40">A50</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E8:29_1_2_1_1_2_5_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c24">y9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:29_1_2_1_1_2"><span class="lab"><font color="Green" title="E30">A38</font></span></a></span>;<br/><a NAME="E9:29_1_2_1_1_2_5_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c24">y9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c15">z9</font> &amp; <font color="Maroon" title="c15">z9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c21">x9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:29_1_2_1_1_2"><span class="lab"><font color="Green" title="E27">A35</font></span></a>, <a class="txt" href="#E44:29_1_2_1_1_2"><span class="lab"><font color="Green" title="E36">A44</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E10:29_1_2_1_1_2_5_1_2"/>
( <font color="Maroon" title="c17">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">z</font> &amp; <font color="Maroon" title="c16">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:29_1_2_1_1_2"><span class="lab"><font color="Green" title="E22">A23</font></span></a>, <a class="txt" href="#E42:29_1_2_1_1_2"><span class="lab"><font color="Green" title="E34">A42</font></span></a>, <a class="txt" href="#E35:29_1_2_1_1_2"><span class="lab"><font color="Green" title="E29">A37</font></span></a>, <a class="txt" href="#E43:29_1_2_1_1_2"><span class="lab"><font color="Green" title="E35">A43</font></span></a>, <a class="txt" href="#E1:29_1_2_1_1_2_5_1_2"><span class="lab"><font color="Green" title="E37">A47</font></span></a>, <a class="txt" href="#E7:29_1_2_1_1_2_5_1_2"><span class="lab"><font color="Green" title="E41">A51</font></span></a>, <a class="txt" href="axioms.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E46:29_1_2_1_1_2"/>
( <font color="Maroon" title="c17">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">z</font> &amp; <font color="Maroon" title="c16">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:29_1_2_1_1_3"/><span class="lab"><font color="Green" title="E12">A52</font></span>: 
<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  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> )</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c5">X9</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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c6">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:29_1_2_1_1_3"/><span class="lab"><font color="Green" title="E13">A53</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E5:29_1_2_1_1_3"/><span class="lab"><font color="Green" title="E14">A54</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_1_2_1_1_3"><span class="lab"><font color="Green" title="E12">A52</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><span class="kw">reconsider </span><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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:29_1_2_1_1_3"><span class="lab"><font color="Green" title="E14">A54</font></span></a></span>;<br/><a NAME="E7:29_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/><span class="kw">then </span><span class="kw">reconsider </span><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>  <span class="kw">by</span> <span class="lab"><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></span>;<br/><a NAME="E9:29_1_2_1_1_3"/><span class="lab"><font color="Green" title="E15">A55</font></span>: 
<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="29_1_2_1_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> or <font color="Maroon" title="c9">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>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:29_1_2_1_1_3_1"/><span class="lab"><font color="Green" title="E16">A56</font></span>: 
<font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">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> </span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">y</font> = <font color="Maroon" title="c9">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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_1_1_3_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:29_1_2_1_1_3_1_1"/>
<font color="Maroon" title="c10">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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:29_1_2_1_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A57</font></span>: 
not <font color="Maroon" title="c10">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> 
 <span class="kw">by</span> <span class="lab"><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></span>;<br/><a NAME="E3:29_1_2_1_1_3_1_1"/>
( <font color="Maroon" title="c8">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">y</font> &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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E4:29_1_2_1_1_3"><span class="lab"><font color="Green" title="E13">A53</font></span></a>, <a class="txt" href="#E1:29_1_2_1_1_3_1"><span class="lab"><font color="Green" title="E16">A56</font></span></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></span>;<br/><span class="kw">hence </span><a NAME="E4:29_1_2_1_1_3_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_1_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E17">A57</font></span></a>, <a class="ref" href="xxreal_0.html#D5" title="XXREAL_0:def.5">XXREAL_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:29_1_2_1_1_3_1"/>
<font color="Maroon" title="c9">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:29_1_2"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="txt" href="#E1:29_1_2_1_1_3_1"><span class="lab"><font color="Green" title="E16">A56</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">Y9</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="E11:29_1_2_1_1_3"/>
for <font color="Olive" title="b1">x9</font>, <font color="Olive" title="b2">y9</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">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X9</font> &amp; <font color="Olive" title="b2">y9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y9</font> holds <br/><font color="Olive" title="b1">x9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Olive" title="b2">y9</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_1_1_3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">x9</font>, <font color="Maroon" title="c11">y9</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X9</font> &amp; <font color="Maroon" title="c11">y9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y9</font> implies <font color="Maroon" title="c10">x9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c11">y9</font> )</span><br/>

<a NAME="E1:29_1_2_1_1_3_2"/><span class="lab"><font color="Green" title="E16">A58</font></span>: 
<font color="Maroon" title="c5">X9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<a NAME="E2:29_1_2_1_1_3_2"/>
( <font color="Maroon" title="c10">x9</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="c11">y9</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/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">x</font> = <font color="Maroon" title="c10">x9</font>, <font color="Maroon" title="c13">y</font> = <font color="Maroon" title="c11">y9</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>  <span class="kw">by</span> <span class="lab"><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></span>;<br/>
<span class="kw">assume </span><a NAME="E4:29_1_2_1_1_3_2"/>
( <font color="Maroon" title="c10">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X9</font> &amp; <font color="Maroon" title="c11">y9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y9</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">x9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c11">y9</font></span><br/>

<a NAME="E5:29_1_2_1_1_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E1:29_1_2_1_1_3_2"><span class="lab"><font color="Green" title="E16">A58</font></span></a></span>;<br/>
<a NAME="E6:29_1_2_1_1_3_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x9</font>, <font color="Olive" title="b2">y9</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="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x9</font> &amp; <font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y9</font> &amp; <font color="Olive" title="b1">x9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Olive" title="b2">y9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D5" title="XXREAL_0:def.5">XXREAL_0:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:29_1_2_1_1_3_2"/>
<font color="Maroon" title="c10">x9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c11">y9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">z9</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> <span class="kw"> such that </span><br/><a NAME="E13:29_1_2_1_1_3"/><span class="lab"><font color="Green" title="E16">A59</font></span>: 
for <font color="Olive" title="b1">x9</font>, <font color="Olive" title="b2">y9</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">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X9</font> &amp; <font color="Olive" title="b2">y9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y9</font> holds <br/>( <font color="Olive" title="b1">x9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c10">z9</font> &amp; <font color="Maroon" title="c10">z9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Olive" title="b2">y9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:29_1_2"><span class="lab"><font color="Green" title="E11">A8</font></span></a>, <a class="ref" href="arytm_2.html#T9" title="ARYTM_2:th.9">ARYTM_2:9</a></span>;<br/><a NAME="E14:29_1_2_1_1_3"/>
<font color="Maroon" title="c10">z9</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/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">z</font> = <font color="Maroon" title="c10">z9</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>  <span class="kw">by</span> <span class="lab"><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></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c11">z</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 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="Maroon" title="c11">z</font> &amp; <font color="Maroon" title="c11">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">y</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c12">x</font>, <font color="Maroon" title="c13">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> implies ( <font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">z</font> &amp; <font color="Maroon" title="c11">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">y</font> ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E16:29_1_2_1_1_3"/><span class="lab"><font color="Green" title="E17">A60</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">and </span><br/><a NAME="E17:29_1_2_1_1_3"/><span class="lab"><font color="Green" title="E18">A61</font></span>: 
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">z</font> &amp; <font color="Maroon" title="c11">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">y</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c14">y9</font> = <font color="Maroon" title="c13">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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:29_1_2_1_1_3"><span class="lab"><font color="Green" title="E15">A55</font></span></a>, <a class="txt" href="#E17:29_1_2_1_1_3"><span class="lab"><font color="Green" title="E18">A61</font></span></a></span>;<br/><a NAME="E19:29_1_2_1_1_3"/><span class="lab"><font color="Green" title="E19">A62</font></span>: 
<font color="Maroon" title="c8">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_1_2_1_1_3"><span class="lab"><font color="Green" title="E13">A53</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_1_1_3_3"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2_1_1_3_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:29_1_2_1_1_3_3_1"/>
( <font color="Maroon" title="c12">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="c12">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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:29_1_2"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="txt" href="#E16:29_1_2_1_1_3"><span class="lab"><font color="Green" title="E17">A60</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2_1_1_3_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:29_1_2_1_1_3_3_1_1"/>
<font color="Maroon" title="c12">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">z</font> &amp; <font color="Maroon" title="c11">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">y</font> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">x9</font> = <font color="Maroon" title="c12">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="E3:29_1_2_1_1_3_3_1_1"/>
<font color="Maroon" title="c15">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:29_1_2_1_1_3"><span class="lab"><font color="Green" title="E17">A60</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E4:29_1_2_1_1_3_3_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c15">x9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c10">z9</font> &amp; <font color="Maroon" title="c10">z9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c14">y9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:29_1_2_1_1_3"><span class="lab"><font color="Green" title="E16">A59</font></span></a>, <a class="txt" href="#E17:29_1_2_1_1_3"><span class="lab"><font color="Green" title="E18">A61</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:29_1_2_1_1_3_3_1_1"/>
( <font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">z</font> &amp; <font color="Maroon" title="c11">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="axioms.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2_1_1_3_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:29_1_2_1_1_3_3_1_2"/><span class="lab"><font color="Green" title="E20">A63</font></span>: 
<font color="Maroon" title="c12">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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">z</font> &amp; <font color="Maroon" title="c11">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">y</font> )</span><br/><div class="add"><a NAME="E2:29_1_2_1_1_3_3_1_2"/><span class="lab"><font color="Green" title="E21">A64</font></span>: 
not <font color="Maroon" title="c11">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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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 class="kw">by</span> <span class="lab"><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></span>;<br/><a NAME="E3:29_1_2_1_1_3_3_1_2"/>
not <font color="Maroon" title="c12">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_1_2_1_1_3_3_1_2"><span class="lab"><font color="Green" title="E20">A63</font></span></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></span>;<br/><span class="kw">hence </span><a NAME="E4:29_1_2_1_1_3_3_1_2"/>
<font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_1_2_1_1_3_3_1_2"><span class="lab"><font color="Green" title="E21">A64</font></span></a>, <a class="ref" href="xxreal_0.html#D5" title="XXREAL_0:def.5">XXREAL_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">y</font></span><br/><a NAME="E5:29_1_2_1_1_3_3_1_2"/>
<font color="Maroon" title="c10">z9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c14">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:29_1_2_1_1_3"><span class="lab"><font color="Green" title="E19">A62</font></span></a>, <a class="txt" href="#E13:29_1_2_1_1_3"><span class="lab"><font color="Green" title="E16">A59</font></span></a>, <a class="txt" href="#E17:29_1_2_1_1_3"><span class="lab"><font color="Green" title="E18">A61</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:29_1_2_1_1_3_3_1_2"/>
<font color="Maroon" title="c11">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="axioms.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E21:29_1_2_1_1_3"/>
( <font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">z</font> &amp; <font color="Maroon" title="c11">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
