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<b>let </b><font color="Maroon" title="c1">C1</font>, <font color="Maroon" title="c2">C2</font>, <font color="Maroon" title="c3">C3</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> ;<br/><b>let </b><font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">g</font> be   <a href="fuzzy_3.html#NM1" title="FUZZY_3:NM.1">RMembership_Func</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>;<br/><b>let </b><font color="Maroon" title="c6">h</font> be   <a href="fuzzy_3.html#NM1" title="FUZZY_3:NM.1">RMembership_Func</a> of <font color="Maroon" title="c2">C2</font>,<font color="Maroon" title="c3">C3</font>;<br/><b>let </b><font color="Maroon" title="c7">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c3">C3</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>; <i><font color="Red">:: according to </font></i><a class="ref" href="fuzzy_1.html#D3" title="FUZZY_1:def.3">FUZZY_1:def 3</a><br/>











<b>consider </b><font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c9">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:25"/><i><font color="Green" title="E14">A1</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font> &amp; <font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C3</font> &amp; <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E3:25"/><i><font color="Green" title="E15">A2</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">h</font></span>)</span> <a href="fuzzy_4.html#K1" title="FUZZY_4:func.1">.</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p3">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">z</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="fuzzy_4.html#E2:25"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="fuzzy_4.html#D3" target="_self" title="FUZZY_4:def.3">Def3</a></i>;<br/>
<span class="p1">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">h</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">h</font></span>)</span></span>)</span> <a href="fuzzy_4.html#K1" title="FUZZY_4:func.1">.</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">z</font> = 
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">h</font></span>)</span> <a href="fuzzy_4.html#K1" title="FUZZY_4:func.1">.</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">z</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">h</font></span>)</span> <a href="fuzzy_4.html#K1" title="FUZZY_4:func.1">.</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">z</font></span>)</span>
<b>by </b><i><a class="txt" href="fuzzy_4.html#E2:25"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="fuzzy_1.html#D4" title="FUZZY_1:def.4">FUZZY_1:def 4</a></i>
<br/>.= 
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">z</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">h</font></span>)</span> <a href="fuzzy_4.html#K1" title="FUZZY_4:func.1">.</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">z</font></span>)</span>
<b>by </b><i><a class="txt" href="fuzzy_4.html#E2:25"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="fuzzy_4.html#D3" target="_self" title="FUZZY_4:def.3">Def3</a></i>
<br/>.= 
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">z</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">z</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fuzzy_4.html#E2:25"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="fuzzy_4.html#D3" target="_self" title="FUZZY_4:def.3">Def3</a></i>
;<br/>
<b>hence </b><a NAME="E5:25"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">h</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">h</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">h</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">c</font>
 <b>by </b><i><a class="txt" href="fuzzy_4.html#E2:25"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="fuzzy_4.html#E3:25"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="fuzzy_4.html#E22"><i><font color="Green" title="E13">Lm4</font></i></a></i>;<br/>


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