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

<b>thus </b><a NAME="E1:2_1_1"/>
( <font color="Maroon" title="c3">f</font> <a href="fuzzy_1.html#R2" title="FUZZY_1:pred.2">is_less_than</a> <font color="Maroon" title="c4">g</font> iff for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> holds  <font color="Maroon" title="c3">f</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">g</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_1"><b>proof </b></a><div class="add">

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:2_1_1_1_1"/><i><font color="Green" title="E1">A1</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="fuzzy_1.html#R2" title="FUZZY_1:pred.2">is_less_than</a> <font color="Maroon" title="c4">g</font>
 ;<br/><b>thus </b><a NAME="E2:2_1_1_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> holds  <font color="Maroon" title="c3">f</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">g</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c6">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font>;<br/>



<a NAME="E1:2_1_1_1_1_1"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <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"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="fuzzy_1.html#D1" title="FUZZY_1:def.1">FUZZY_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:2_1_1_1_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">g</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>
 <b>by </b><i><a class="txt" href="lfuzzy_1.html#E1:2_1_1_1_1"><i><font color="Green" title="E1">A1</font></i></a>, <a class="ref" href="fuzzy_1.html#D2" title="FUZZY_1:def.2">FUZZY_1:def 2</a></i>;<br/>


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

<b>assume </b><a NAME="E2:2_1_1_1"/><i><font color="Green" title="E1">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> holds  <font color="Maroon" title="c3">f</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">g</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>
 ;<br/>

<a NAME="E3:2_1_1_1"/><i><font color="Green" title="E2">A3</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="fuzzy_1.html#D1" title="FUZZY_1:def.1">FUZZY_1:def 1</a></i>;<br/>
<a NAME="E4:2_1_1_1"/>
for <font color="Olive" title="b1">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> holds <br/><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:2_1_1_1_2"/><i><font color="Green" title="E3">A4</font></i>: 
<font color="Maroon" title="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:2_1_1_1_2"/><i><font color="Green" title="E4">A5</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c6">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="lfuzzy_1.html#E1:2_1_1_1_2"><i><font color="Green" title="E3">A4</font></i></a>, <a class="txt" href="lfuzzy_1.html#E3:2_1_1_1_2"><i><font color="Green" title="E4">A5</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">y</font> = <font color="Maroon" title="c7">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> <b>by </b><i><a class="txt" href="lfuzzy_1.html#E1:2_1_1_1_2"><i><font color="Green" title="E3">A4</font></i></a>, <a class="txt" href="lfuzzy_1.html#E3:2_1_1_1_2"><i><font color="Green" title="E4">A5</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E6:2_1_1_1_2"/>
<font color="Maroon" title="c3">f</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">g</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font>
 
<b>by </b><i><a class="txt" href="lfuzzy_1.html#E2:2_1_1_1"><i><font color="Green" title="E1">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:2_1_1_1_2"/>
<font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">c</font>
 <b>by </b><i><a class="txt" href="lfuzzy_1.html#E3:2_1_1_1_2"><i><font color="Green" title="E4">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:2_1_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="fuzzy_1.html#R2" title="FUZZY_1:pred.2">is_less_than</a> <font color="Maroon" title="c4">g</font>
 <b>by </b><i><a class="txt" href="lfuzzy_1.html#E3:2_1_1_1"><i><font color="Green" title="E2">A3</font></i></a>, <a class="ref" href="fuzzy_1.html#D2" title="FUZZY_1:def.2">FUZZY_1:def 2</a></i>;<br/>


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


</div>
