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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></font> be   <a href="fuzzy_3.html#NM1">RMembership_Func</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="fuzzy_3.html#NM1">RMembership_Func</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>











<b>assume </b><a NAME="E1:20"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> =  <a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> =  <a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>;<br/>
<a NAME="E2:20"/>
 <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">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> is <a href="seq_4.html#V3">bounded</a>
 
<b>by </b><i><a class="ref" href="fuzzy_4.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E3:20"/><b>then </b><i><font color="Green">E14</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">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></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="E4:20"/>
 <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">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> is <a href="seq_4.html#V3">bounded</a>
 
<b>by </b><i><a class="ref" href="fuzzy_4.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E5:20"/><b>then </b><i><font color="Green">E15</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">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></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="E6:20"/>
 <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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> is <a href="seq_4.html#V3">bounded</a>
 
<b>by </b><i><a class="ref" href="fuzzy_4.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E7:20"/><b>then </b><i><font color="Green">E16</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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></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="E8:20"/><i><font color="Green">E17</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:20_1"/>
<font color="Maroon">c<sub>12</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:20_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></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="E3:20_1"/>
<span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="fuzzy_4.html#E5:20"><i><font color="Green">E15</font></i></a>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:20"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:20_2"/>
<font color="Maroon">c<sub>12</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:20_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></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="E3:20_2"/>
<span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="fuzzy_4.html#E7:20"><i><font color="Green">E16</font></i></a>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:20"/><i><font color="Green">E19</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> <font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_3"><b>proof </b></a><div class="add">

<a NAME="E1:20_3"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></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">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></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> <font color="Maroon">c<sub>4</sub></font>,<span class="p4">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">b<sub>1</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></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="E1:20_3_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>13</sub></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="E3:20_3_1"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>13</sub></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">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></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> <font color="Maroon">c<sub>4</sub></font>,<span class="p4">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="fuzzy_4.html#E3:20"><i><font color="Green">E14</font></i></a>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:20_3_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>14</sub></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">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="fuzzy_4.html#E3:20_3_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:20_3_1"/><i><font color="Green">E23</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>8</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="fuzzy_4.html#E1:20"><i><font color="Green">E13</font></i></a>, <a class="txt" href="fuzzy_4.html#E5:20_3_1"><i><font color="Green">E22</font></i></a>, <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> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font> = 
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>14</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="fuzzy_4.html#E1:20"><i><font color="Green">E13</font></i></a>, <a class="txt" href="fuzzy_4.html#E5:20_3_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="fuzzy_4.html#D2" target="_self">Def2</a></i>
<br/>.= 
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fuzzy_4.html#E6:20_3_1"><i><font color="Green">E23</font></i></a>, <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">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>8</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="fuzzy_4.html#E1:20"><i><font color="Green">E13</font></i></a>, <a class="txt" href="fuzzy_4.html#E5:20_3_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="fuzzy_4.html#D2" target="_self">Def2</a></i>
;<br/>
<a NAME="E8:20_3_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="fuzzy_4.html#E5:20_3_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a></i>;<br/>
<a NAME="E9:20_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> <font color="Maroon">c<sub>4</sub></font>,<span class="p4">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>12</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="fuzzy_4.html#E3:20_3_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E10:20_3_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></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">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></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> <font color="Maroon">c<sub>4</sub></font>,<span class="p4">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>12</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="fuzzy_4.html#E5:20_3_1"><i><font color="Green">E22</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:20_3"/>
for <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></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> <font color="Maroon">c<sub>4</sub></font>,<span class="p4">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">b<sub>1</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></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="E1:20_3_2"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:20_3_2"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>13</sub></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">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></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> <font color="Maroon">c<sub>4</sub></font>,<span class="p4">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>12</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="fuzzy_4.html#E1:20_3"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E4:20_3_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="fuzzy_4.html#E3:20_3_2"><i><font color="Green">E21</font></i></a></i>;<br/>
<a NAME="E5:20_3_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="fuzzy_1.html#D1">FUZZY_1:def 1</a></i>;<br/>
<a NAME="E6:20_3_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="fuzzy_4.html#E8:20"><i><font color="Green">E17</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:20_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> <font color="Maroon">c<sub>4</sub></font>,<span class="p4">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>12</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="fuzzy_4.html#E3:20_3_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:20_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> <font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></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="E11:20"/>
 <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">c<sub>4</sub></font>,<span class="p3">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4"><b>proof </b></a><div class="add">

<a NAME="E1:20_4"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></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">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></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> <font color="Maroon">c<sub>4</sub></font>,<span class="p4">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">b<sub>1</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></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="E1:20_4_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>13</sub></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="E3:20_4_1"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>13</sub></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">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></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> <font color="Maroon">c<sub>4</sub></font>,<span class="p4">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="fuzzy_4.html#E3:20"><i><font color="Green">E14</font></i></a>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:20_4_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>14</sub></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">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="fuzzy_4.html#E3:20_4_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:20_4_1"/><i><font color="Green">E23</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>8</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="fuzzy_4.html#E1:20"><i><font color="Green">E13</font></i></a>, <a class="txt" href="fuzzy_4.html#E5:20_4_1"><i><font color="Green">E22</font></i></a>, <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> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font> = 
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>14</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="fuzzy_4.html#E1:20"><i><font color="Green">E13</font></i></a>, <a class="txt" href="fuzzy_4.html#E5:20_4_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="fuzzy_4.html#D2" target="_self">Def2</a></i>
<br/>.= 
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fuzzy_4.html#E6:20_4_1"><i><font color="Green">E23</font></i></a>, <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">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>8</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="fuzzy_4.html#E1:20"><i><font color="Green">E13</font></i></a>, <a class="txt" href="fuzzy_4.html#E5:20_4_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="fuzzy_4.html#D2" target="_self">Def2</a></i>
;<br/>
<a NAME="E8:20_4_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="fuzzy_4.html#E5:20_4_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a></i>;<br/>
<a NAME="E9:20_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> <font color="Maroon">c<sub>4</sub></font>,<span class="p4">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>12</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="fuzzy_4.html#E3:20_4_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E10:20_4_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></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">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></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> <font color="Maroon">c<sub>4</sub></font>,<span class="p4">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>12</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="fuzzy_4.html#E5:20_4_1"><i><font color="Green">E22</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:20_4"/>
for <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></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> <font color="Maroon">c<sub>4</sub></font>,<span class="p4">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">b<sub>1</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></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="E1:20_4_2"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:20_4_2"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>13</sub></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">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></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> <font color="Maroon">c<sub>4</sub></font>,<span class="p4">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>12</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="fuzzy_4.html#E1:20_4"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E4:20_4_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="fuzzy_4.html#E3:20_4_2"><i><font color="Green">E21</font></i></a></i>;<br/>
<a NAME="E5:20_4_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="fuzzy_1.html#D1">FUZZY_1:def 1</a></i>;<br/>
<a NAME="E6:20_4_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" target="_self">min</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="fuzzy_4.html#E9:20"><i><font color="Green">E18</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:20_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> <font color="Maroon">c<sub>4</sub></font>,<span class="p4">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>12</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="fuzzy_4.html#E3:20_4_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:20_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> <font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></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="E12:20"/>
 <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">c<sub>4</sub></font>,<span class="p3">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="fuzzy_4.html#E10:20"><i><font color="Green">E19</font></i></a>, <a class="ref" href="square_1.html#T39">SQUARE_1:39</a></i>;<br/>


</div>
