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<b>let </b><font color="Maroon" title="c1">C</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">h</font> being   <a href="fuzzy_1.html#M1" title="FUZZY_1:mode.1">Membership_Func</a> of <font color="Maroon" title="c1">C</font> holds <br/> ( <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <font color="Olive" title="b2">g</font>,<font color="Olive" title="b3">h</font> iff ( <font color="Olive" title="b1">f</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a>  &amp; <font color="Olive" title="b1">f</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a>  &amp; ( for <font color="Olive" title="b4">h1</font> being   <a href="fuzzy_1.html#M1" title="FUZZY_1:mode.1">Membership_Func</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b4">h1</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a>  &amp; <font color="Olive" title="b4">h1</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a>  holds <br/><font color="Olive" title="b4">h1</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a>  ) ) )</span><br/><b>let </b><font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">g</font>, <font color="Maroon" title="c4">h</font> be    <a href="fuzzy_1.html#M1" title="FUZZY_1:mode.1">Membership_Func</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font> iff ( <font color="Maroon" title="c2">f</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a>  &amp; <font color="Maroon" title="c2">f</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a>  &amp; ( for <font color="Olive" title="b1">h1</font> being   <a href="fuzzy_1.html#M1" title="FUZZY_1:mode.1">Membership_Func</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b1">h1</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a>  &amp; <font color="Olive" title="b1">h1</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a>  holds <br/><font color="Olive" title="b1">h1</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a>  ) ) )</span><br/>







<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a>  &amp; <font color="Maroon" title="c2">f</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a>  &amp; ( for <font color="Olive" title="b1">h1</font> being   <a href="fuzzy_1.html#M1" title="FUZZY_1:mode.1">Membership_Func</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b1">h1</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a>  &amp; <font color="Olive" title="b1">h1</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a>  holds <br/><font color="Olive" title="b1">h1</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a>  ) implies <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:39_1"/><i><font color="Green" title="E24">A1</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a>  &amp; <font color="Maroon" title="c2">f</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a>  &amp; ( for <font color="Olive" title="b1">h1</font> being   <a href="fuzzy_1.html#M1" title="FUZZY_1:mode.1">Membership_Func</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b1">h1</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a>  &amp; <font color="Olive" title="b1">h1</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a>  holds <br/><font color="Olive" title="b1">h1</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a>  ) )</span><br/><b>hence </b><a NAME="E2:39_1"/>
( <font color="Maroon" title="c2">f</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a>  &amp; <font color="Maroon" title="c2">f</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a>  )
 <b>by </b><i><a class="ref" href="fuzzy_1.html#T18" target="_self" title="FUZZY_1:th.18">Th18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">h1</font> being   <a href="fuzzy_1.html#M1" title="FUZZY_1:mode.1">Membership_Func</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b1">h1</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a>  &amp; <font color="Olive" title="b1">h1</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a>  holds <br/><font color="Olive" title="b1">h1</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a> </span><br/><b>let </b><font color="Maroon" title="c5">h1</font> be    <a href="fuzzy_1.html#M1" title="FUZZY_1:mode.1">Membership_Func</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">h1</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a>  &amp; <font color="Maroon" title="c5">h1</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a>  implies <font color="Maroon" title="c5">h1</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a>  )</span><br/><b>assume </b><a NAME="E3:39_1"/><i><font color="Green" title="E25">A2</font></i>: 
( <font color="Maroon" title="c5">h1</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a>  &amp; <font color="Maroon" title="c5">h1</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">h1</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a> </span><br/><b>thus </b><a NAME="E4:39_1"/>
<font color="Maroon" title="c5">h1</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="fuzzy_1.html#D3" target="_self" title="FUZZY_1:def.3">FUZZY_1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span><br/>

<a NAME="E1:39_1_1"/>
( <font color="Maroon" title="c3">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c4">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> )
 
<b>by </b><i><a class="txt" href="fuzzy_1.html#E3:39_1"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="fuzzy_1.html#D3" target="_self" title="FUZZY_1:def.3">Def3</a></i>;<br/>
<a NAME="E2:39_1_1"/><b>then </b>
 <a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
 
<b>by </b><i><a class="ref" href="square_1.html#T50" title="SQUARE_1:th.50">SQUARE_1:50</a></i>;<br/>
<b>hence </b><a NAME="E3:39_1_1"/>
<font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="fuzzy_1.html#E1:39_1"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="fuzzy_1.html#D5" target="_self" title="FUZZY_1:def.5">Def5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>assume </b><b>that </b><br/><a NAME="E2:39"/><i><font color="Green" title="E24">A3</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a> 
 <b>and </b><br/><a NAME="E3:39"/><i><font color="Green" title="E25">A4</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a> 
 <b>and </b><br/><a NAME="E4:39"/><i><font color="Green" title="E26">A5</font></i>: 
for <font color="Olive" title="b1">h1</font> being   <a href="fuzzy_1.html#M1" title="FUZZY_1:mode.1">Membership_Func</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b1">h1</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a>  &amp; <font color="Olive" title="b1">h1</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a>  holds <br/><font color="Olive" title="b1">h1</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font></span><br/>

<a NAME="E5:39"/><i><font color="Green" title="E27">A6</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a> 
 
<b>by </b><i><a class="txt" href="fuzzy_1.html#E2:39"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="fuzzy_1.html#E3:39"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="fuzzy_1.html#T20" target="_self" title="FUZZY_1:th.20">Th20</a></i>;<br/>
<a NAME="E6:39"/>
(  <a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a>  &amp;  <a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a>  )
 
<b>by </b><i><a class="ref" href="fuzzy_1.html#T18" target="_self" title="FUZZY_1:th.18">Th18</a></i>;<br/>
<a NAME="E7:39"/><b>then </b>
 <a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a> 
 
<b>by </b><i><a class="txt" href="fuzzy_1.html#E4:39"><i><font color="Green" title="E26">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:39"/>
<font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font>
 <b>by </b><i><a class="txt" href="fuzzy_1.html#E5:39"><i><font color="Green" title="E27">A6</font></i></a>, <a class="txt" href="fuzzy_1.html#E6"><i><font color="Green" title="E6">Lm1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
