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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>  st  <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> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a>  &amp;  <a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fuzzy_1.html#K4" title="FUZZY_1:func.4">EMF</a> <font color="Maroon" title="c1">C</font> holds <br/><font color="Olive" title="b2">g</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"> (  <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#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fuzzy_1.html#K4" title="FUZZY_1:func.4">EMF</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c3">g</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a>  )</span><br/>







<b>assume </b><a NAME="E1:41"/><i><font color="Green" title="E24">A1</font></i>: 
(  <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#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fuzzy_1.html#K4" title="FUZZY_1:func.4">EMF</a> <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="c3">g</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a> </span><br/>

<a NAME="E2:41"/><b>then </b>
 <a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><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>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="ref" href="fuzzy_1.html#T30" target="_self" title="FUZZY_1:th.30">Th30</a></i>;<br/>
<b>then </b><i><font color="Green" title="E25">A2</font></i>: <font color="Maroon" title="c2">f</font> = 
 <a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font></span>)</span>,<span class="p1">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">h</font></span>)</span>
<b>by </b><i><a class="ref" href="fuzzy_1.html#T10" target="_self" title="FUZZY_1:th.10">Th10</a></i>
<br/>.= 
 <a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font>
<b>by </b><i><a class="txt" href="fuzzy_1.html#E1:41"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="fuzzy_1.html#T19" target="_self" title="FUZZY_1:th.19">Th19</a></i>
;<br/>
<b>let </b><font color="Maroon" title="c5">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="c5">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span><br/>

<a NAME="E4:41"/>
<font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span>,<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="c5">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="fuzzy_1.html#E3:41"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="fuzzy_1.html#D4" target="_self" title="FUZZY_1:def.4">Def4</a></i>;<br/>
<b>hence </b><a NAME="E5:41"/>
<font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#D8" title="XXREAL_0:def.8">XXREAL_0:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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