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

<b>let </b><font color="Maroon">C</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">f</font> be    <a href="fuzzy_1.html#M1">Membership_Func</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">g</font> be    <a href="fuzzy_1.html#M1">Membership_Func</a> of <font color="Maroon">C</font>;<br/>





<a NAME="E1:48"/><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" target="_self">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="yellow_1.html#K5">product</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="lfuzzy_0.html#K1" target="_self">RealPoset</a> <span class="p3"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D5">YELLOW_1:def 5</a></i>;<br/>
<a NAME="E2:48"/><i><font color="Green">E42</font></i>: 
( ( for <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font> holds  <font color="Maroon">f</font> <a href="lfuzzy_0.html#K7" target="_self">.</a> <font color="Olive">c</font> <a href="lfuzzy_0.html#NR5" target="_self">&lt;R=</a> <font color="Maroon">g</font> <a href="lfuzzy_0.html#K7" target="_self">.</a> <font color="Olive">c</font> ) implies <font color="Maroon">C</font>,<font color="Maroon">f</font> <a href="lfuzzy_0.html#K6" target="_self">@</a>  <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <font color="Maroon">C</font>,<font color="Maroon">g</font> <a href="lfuzzy_0.html#K6" target="_self">@</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_1"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font> holds  <font color="Maroon">f</font> <a href="lfuzzy_0.html#K7" target="_self">.</a> <font color="Olive">c</font> <a href="lfuzzy_0.html#NR5" target="_self">&lt;R=</a> <font color="Maroon">g</font> <a href="lfuzzy_0.html#K7" target="_self">.</a> <font color="Olive">c</font>
 ;<br/>

<b>set </b><font color="Maroon">f'</font> = <font color="Maroon">C</font>,<font color="Maroon">f</font> <a href="lfuzzy_0.html#K6" target="_self">@</a> ;<br/>
<b>set </b><font color="Maroon">g'</font> = <font color="Maroon">C</font>,<font color="Maroon">g</font> <a href="lfuzzy_0.html#K6" target="_self">@</a> ;<br/>
<b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">C</font>,<font color="Maroon">f</font> <a href="lfuzzy_0.html#K6" target="_self">@</a> , <font color="Maroon">g'</font> = <font color="Maroon">C</font>,<font color="Maroon">g</font> <a href="lfuzzy_0.html#K6" target="_self">@</a>  as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="lfuzzy_0.html#K1" target="_self">RealPoset</a> <span class="p4"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span></span>)</span></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E3:48_1"/>
for <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font> holds  <font color="Maroon">f'</font> <a href="waybel_3.html#K7">.</a> <font color="Olive">c</font> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <font color="Maroon">g'</font> <a href="waybel_3.html#K7">.</a> <font color="Olive">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font>;<br/>

<a NAME="E1:48_1_1"/><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="lfuzzy_0.html#K1" target="_self">RealPoset</a> <span class="p3"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span></span>)</span> <a href="waybel_3.html#K6">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="lfuzzy_0.html#K1" target="_self">RealPoset</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<b>set </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> <a href="lfuzzy_0.html#K7" target="_self">.</a> <font color="Maroon">c</font>;<br/>
<b>set </b><font color="Maroon">g1</font> = <font color="Maroon">g</font> <a href="lfuzzy_0.html#K7" target="_self">.</a> <font color="Maroon">c</font>;<br/>
<a NAME="E2:48_1_1"/>
( <font color="Maroon">f</font> <a href="lfuzzy_0.html#K7" target="_self">.</a> <font color="Maroon">c</font> <a href="lfuzzy_0.html#NR5" target="_self">&lt;R=</a> <font color="Maroon">g</font> <a href="lfuzzy_0.html#K7" target="_self">.</a> <font color="Maroon">c</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g'</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:48_1_1"/>
<font color="Maroon">f'</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c</font> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <font color="Maroon">g'</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:48_1"/>
<font color="Maroon">C</font>,<font color="Maroon">f</font> <a href="lfuzzy_0.html#K6" target="_self">@</a>  <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <font color="Maroon">C</font>,<font color="Maroon">g</font> <a href="lfuzzy_0.html#K6" target="_self">@</a> 
 <b>by </b><i>, <a class="ref" href="waybel_3.html#T28">WAYBEL_3:28</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:48"/>
( <font color="Maroon">C</font>,<font color="Maroon">f</font> <a href="lfuzzy_0.html#K6" target="_self">@</a>  <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <font color="Maroon">C</font>,<font color="Maroon">g</font> <a href="lfuzzy_0.html#K6" target="_self">@</a>  implies for <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font> holds  <font color="Maroon">f</font> <a href="lfuzzy_0.html#K7" target="_self">.</a> <font color="Olive">c</font> <a href="lfuzzy_0.html#NR5" target="_self">&lt;R=</a> <font color="Maroon">g</font> <a href="lfuzzy_0.html#K7" target="_self">.</a> <font color="Olive">c</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">C</font>,<font color="Maroon">f</font> <a href="lfuzzy_0.html#K6" target="_self">@</a>  <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <font color="Maroon">C</font>,<font color="Maroon">g</font> <a href="lfuzzy_0.html#K6" target="_self">@</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon">ff</font> = <font color="Maroon">C</font>,<font color="Maroon">f</font> <a href="lfuzzy_0.html#K6" target="_self">@</a> , <font color="Maroon">gg</font> = <font color="Maroon">C</font>,<font color="Maroon">g</font> <a href="lfuzzy_0.html#K6" target="_self">@</a>  as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="lfuzzy_0.html#K1" target="_self">RealPoset</a> <span class="p4"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span></span>)</span></span>)</span> <b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">c</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font>;<br/>

<a NAME="E3:48_2"/><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="lfuzzy_0.html#K1" target="_self">RealPoset</a> <span class="p3"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span></span>)</span> <a href="waybel_3.html#K6">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="lfuzzy_0.html#K1" target="_self">RealPoset</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E4:48_2"/>
( <font color="Maroon">ff</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c</font> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <font color="Maroon">gg</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c</font> &amp; <font color="Maroon">C</font>,<font color="Maroon">f</font> <a href="lfuzzy_0.html#K6" target="_self">@</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; <font color="Maroon">C</font>,<font color="Maroon">g</font> <a href="lfuzzy_0.html#K6" target="_self">@</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> )
 
<b>by </b><i>, , <a class="ref" href="waybel_3.html#T28">WAYBEL_3:28</a></i>;<br/>
<b>hence </b><a NAME="E5:48_2"/>
<font color="Maroon">f</font> <a href="lfuzzy_0.html#K7" target="_self">.</a> <font color="Maroon">c</font> <a href="lfuzzy_0.html#NR5" target="_self">&lt;R=</a> <font color="Maroon">g</font> <a href="lfuzzy_0.html#K7" target="_self">.</a> <font color="Maroon">c</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:48"/>
( ( for <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font> holds  <font color="Maroon">f</font> <a href="lfuzzy_0.html#K7" target="_self">.</a> <font color="Olive">c</font> <a href="lfuzzy_0.html#NR5" target="_self">&lt;R=</a> <font color="Maroon">g</font> <a href="lfuzzy_0.html#K7" target="_self">.</a> <font color="Olive">c</font> ) iff <font color="Maroon">C</font>,<font color="Maroon">f</font> <a href="lfuzzy_0.html#K6" target="_self">@</a>  <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <font color="Maroon">C</font>,<font color="Maroon">g</font> <a href="lfuzzy_0.html#K6" target="_self">@</a>  )
 <b>by </b><i/>;<br/>


</div>
