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

<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">Y</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">Z</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">R</font> be   <a href="fuzzy_3.html#NM1">RMembership_Func</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">S</font> be   <a href="fuzzy_3.html#NM1">RMembership_Func</a> of <font color="Maroon">Y</font>,<font color="Maroon">Z</font>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">z</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Z</font>;<br/>









<a NAME="E1:16"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K2">min</a> <font color="Maroon">R</font>,<font color="Maroon">S</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> &amp;  <a href="fuzzy_4.html#K2">min</a> <font color="Maroon">R</font>,<font color="Maroon">S</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font> is   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K2">min</a> <font color="Maroon">R</font>,<font color="Maroon">S</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span>, <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K2">min</a> <font color="Maroon">R</font>,<font color="Maroon">S</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="fuzzy_1.html#D1">FUZZY_1:def 1</a>, <a class="ref" href="partfun1.html#T24">PARTFUN1:24</a></i>;<br/>
<b>set </b><font color="Maroon">L</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">R</font> <a href="lfuzzy_0.html#K7">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Olive">y</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="lfuzzy_0.html#K7">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span> where <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : verum </span>}</span> ;<br/>
<a NAME="E2:16"/>
for <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">R</font> <a href="lfuzzy_0.html#K7">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Olive">y</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="lfuzzy_0.html#K7">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span> where <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : verum </span>}</span>  iff <font color="Olive">c</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K2">min</a> <font color="Maroon">R</font>,<font color="Maroon">S</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:16_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">R</font> <a href="lfuzzy_0.html#K7">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Olive">y</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="lfuzzy_0.html#K7">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span> where <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : verum </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E3:16_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>
 ;<br/><font color="Maroon">c</font> = 
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="lfuzzy_0.html#T5">LFUZZY_0:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fuzzy_4.html#K2">min</a> <font color="Maroon">R</font>,<font color="Maroon">S</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="fuzzy_4.html#D2">FUZZY_4:def 2</a></i>
;<br/><b>hence </b><a NAME="E5:16_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K2">min</a> <font color="Maroon">R</font>,<font color="Maroon">S</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="partfun1.html#T27">PARTFUN1:27</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:16_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K2">min</a> <font color="Maroon">R</font>,<font color="Maroon">S</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E4:16_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K2">min</a> <font color="Maroon">R</font>,<font color="Maroon">S</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K2">min</a> <font color="Maroon">R</font>,<font color="Maroon">S</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<font color="Maroon">c</font> = 
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="fuzzy_4.html#D2">FUZZY_4:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="lfuzzy_0.html#T5">LFUZZY_0:5</a></i>
;<br/>
<b>hence </b><a NAME="E6:16_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">R</font> <a href="lfuzzy_0.html#K7">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Olive">y</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="lfuzzy_0.html#K7">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span> where <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : verum </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:16"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K2">min</a> <font color="Maroon">R</font>,<font color="Maroon">S</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">R</font> <a href="lfuzzy_0.html#K7">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Olive">y</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="lfuzzy_0.html#K7">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span> where <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : verum </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>

<b>thus </b><a NAME="E4:16"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K2">min</a> <font color="Maroon">R</font>,<font color="Maroon">S</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><b>proof </b></a><div class="add">

<a NAME="E1:16_2"/>
 ex <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">d</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K2">min</a> <font color="Maroon">R</font>,<font color="Maroon">S</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/>
<b>take </b><font color="Maroon">d</font> = <span class="p1">(<span class="default"><a href="fuzzy_4.html#K2">min</a> <font color="Maroon">R</font>,<font color="Maroon">S</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <font color="Maroon">y</font>;<br/>

<b>thus </b><a NAME="E2:16_2_1"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K2">min</a> <font color="Maroon">R</font>,<font color="Maroon">S</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="partfun1.html#T27">PARTFUN1:27</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:16_2"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K2">min</a> <font color="Maroon">R</font>,<font color="Maroon">S</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>


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


</div>
