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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#NM1" title="FUZZY_1:NM.1">Membership_Func</a> of <font color="Maroon" title="c1">C</font> holds <br/> ( <font color="Olive" title="b1">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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#NM1" title="FUZZY_1:NM.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/><span class="kw">let </span><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#NM1" title="FUZZY_1:NM.1">Membership_Func</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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#NM1" title="FUZZY_1:NM.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><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#NM1" title="FUZZY_1:NM.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="funct_2.html#R2" title="FUNCT_2:pred.2">=</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"><span class="kw">assume </span><a NAME="E1:41_1"/><span class="lab"><font color="Green" title="E20">A1</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#NM1" title="FUZZY_1:NM.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/><span class="kw">hence </span><a NAME="E2:41_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>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fuzzy_1.html#T18" target="_self" title="FUZZY_1:th.18">Th18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">h1</font> being  <a href="fuzzy_1.html#NM1" title="FUZZY_1:NM.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/><span class="kw">let </span><font color="Maroon" title="c5">h1</font> be   <a href="fuzzy_1.html#NM1" title="FUZZY_1:NM.1">Membership_Func</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">assume </span><a NAME="E3:41_1"/><span class="lab"><font color="Green" title="E21">A2</font></span>: 
( <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">thus </span><a NAME="E4:41_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E1:41_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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41_1"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="ref" href="fuzzy_1.html#D3" target="_self" title="FUZZY_1:def.3">Def3</a></span>;<br/>
<a NAME="E2:41_1_1"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T28" title="XXREAL_0:th.28">XXREAL_0:28</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:41_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="ref" href="fuzzy_1.html#D5" target="_self" title="FUZZY_1:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:41"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
( <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>  )
 <span class="kw">and </span><br/><a NAME="E3:41"/><span class="lab"><font color="Green" title="E21">A4</font></span>: 
for <font color="Olive" title="b1">h1</font> being  <a href="fuzzy_1.html#NM1" title="FUZZY_1:NM.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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="E4:41"/>
(  <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>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fuzzy_1.html#T18" target="_self" title="FUZZY_1:th.18">Th18</a></span>;<br/>
<a NAME="E5:41"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A5</font></span>: 
 <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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41"><span class="lab"><font color="Green" title="E21">A4</font></span></a></span>;<br/>
<a NAME="E6:41"/>
<font color="Maroon" title="c2">f</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="fuzzy_1.html#T20" target="_self" title="FUZZY_1:th.20">Th20</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:41"/>
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E5:41"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="txt" href="fuzzy_1.html#E6"><span class="lab"><font color="Green" title="E3">Lm1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
