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

<b>let </b><font color="Maroon">C1</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">C2</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">C3</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_3.html#NM1">RMembership_Func</a> of <font color="Maroon">C1</font>,<font color="Maroon">C2</font>, <font color="Maroon">g</font> be   <a href="fuzzy_3.html#NM1">RMembership_Func</a> of <font color="Maroon">C1</font>,<font color="Maroon">C2</font>;<br/><b>let </b><font color="Maroon">h</font> be   <a href="fuzzy_3.html#NM1">RMembership_Func</a> of <font color="Maroon">C2</font>,<font color="Maroon">C3</font>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/>











<b>assume </b><a NAME="E1:22"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C3</font> )
 ;<br/>

<b>set </b><font color="Maroon">F</font> =  <a href="fuzzy_4.html#K2" target="_self">min</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font>;<br/>
<b>set </b><font color="Maroon">G</font> =  <a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">f</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font>;<br/>
<b>set </b><font color="Maroon">H</font> =  <a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">g</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font>;<br/>
<a NAME="E2:22"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> is <a href="seq_4.html#V3">bounded</a>
 
<b>by </b><i/>;<br/>
<a NAME="E3:22"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> is <a href="seq_4.html#V1">bounded_above</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#D3">SEQ_4:def 3</a></i>;<br/>
<a NAME="E4:22"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">f</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> is <a href="seq_4.html#V3">bounded</a>
 
<b>by </b><i/>;<br/>
<a NAME="E5:22"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">f</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> is <a href="seq_4.html#V1">bounded_above</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#D3">SEQ_4:def 3</a></i>;<br/>
<a NAME="E6:22"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">g</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> is <a href="seq_4.html#V3">bounded</a>
 
<b>by </b><i/>;<br/>
<a NAME="E7:22"/><b>then </b><i><font color="Green">E37</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">g</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> is <a href="seq_4.html#V1">bounded_above</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#D3">SEQ_4:def 3</a></i>;<br/>
<a NAME="E8:22"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">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" target="_self">min</a> <font color="Maroon">f</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">f</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">f</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:22_1"/>
<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" target="_self">min</a> <font color="Maroon">f</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span>
 ;<br/>

<a NAME="E2:22_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">f</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</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" target="_self">min</a> <font color="Maroon">f</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E3:22_1"/>
<span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">f</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">f</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:22"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">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" target="_self">min</a> <font color="Maroon">g</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">g</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">g</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:22_2"/>
<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" target="_self">min</a> <font color="Maroon">g</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span>
 ;<br/>

<a NAME="E2:22_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">g</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</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" target="_self">min</a> <font color="Maroon">g</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E3:22_2"/>
<span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">g</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">g</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:22"/><i><font color="Green">E49</font></i>: 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <span class="p3">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">f</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_3"><b>proof </b></a><div class="add">

<a NAME="E1:22_3"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">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" target="_self">min</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">f</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:22_3_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font>
 ;<br/>

<b>then consider </b><font color="Maroon">r</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:22_3_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">r</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" target="_self">min</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> )
 <b>by </b><i>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:22_3_1"/><i><font color="Green">E53</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" target="_self">min</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:22_3_1"/><i><font color="Green">E54</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">C1</font>,<font color="Maroon">C2</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font> = 
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="fuzzy_1.html#D4">FUZZY_1:def 4</a></i>
<br/>.= 
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="xxreal_0.html#T33">XXREAL_0:33</a></i>
<br/>.= 
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">f</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span></span>)</span>
<b>by </b><i>, , </i>
;<br/>
<a NAME="E8:22_3_1"/><b>then </b>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">f</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a></i>;<br/>
<a NAME="E9:22_3_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">f</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E10:22_3_1"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">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" target="_self">min</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">f</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">y</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:22_3"/>
for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> holds <br/><span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">f</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:22_3_2"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:22_3_2"/><i><font color="Green">E55</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" target="_self">min</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">f</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font> )
 <b>by </b><i/>;<br/>
<a NAME="E4:22_3_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:22_3_2"/><b>then </b>
<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" target="_self">min</a> <font color="Maroon">f</font>,<font color="Maroon">h</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></i>;<br/>
<a NAME="E6:22_3_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">f</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">f</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:22_3_2"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">f</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:22_3"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <span class="p3">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">f</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="real_2.html#T182">REAL_2:182</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:22"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <span class="p3">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">g</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_4"><b>proof </b></a><div class="add">

<a NAME="E1:22_4"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">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" target="_self">min</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">g</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:22_4_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font>
 ;<br/>

<b>then consider </b><font color="Maroon">r</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:22_4_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">r</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" target="_self">min</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> )
 <b>by </b><i>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:22_4_1"/><i><font color="Green">E65</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" target="_self">min</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:22_4_1"/><i><font color="Green">E66</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">C1</font>,<font color="Maroon">C2</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font> = 
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="fuzzy_1.html#D4">FUZZY_1:def 4</a></i>
<br/>.= 
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xxreal_0.html#T33">XXREAL_0:33</a></i>
<br/>.= 
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">g</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span></span>)</span>
<b>by </b><i>, , </i>
;<br/>
<a NAME="E8:22_4_1"/><b>then </b>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">g</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a></i>;<br/>
<a NAME="E9:22_4_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">g</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E10:22_4_1"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">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" target="_self">min</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">g</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">y</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:22_4"/>
for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> holds <br/><span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">g</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:22_4_2"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:22_4_2"/><i><font color="Green">E67</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" target="_self">min</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">g</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font> )
 <b>by </b><i/>;<br/>
<a NAME="E4:22_4_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:22_4_2"/><b>then </b>
<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" target="_self">min</a> <font color="Maroon">g</font>,<font color="Maroon">h</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></i>;<br/>
<a NAME="E6:22_4_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">g</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">g</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:22_4_2"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">g</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:22_4"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <span class="p3">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">g</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="real_2.html#T182">REAL_2:182</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:22"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <span class="p3">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">f</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">g</font>,<font color="Maroon">h</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="square_1.html#T39">SQUARE_1:39</a></i>;<br/>


</div>
