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<span class="kw">let </span><font color="Maroon" title="c1">f</font>, <font color="Maroon" title="c2">g</font> be   <a href="fuzimpl3.html#NM1" title="FUZIMPL3:NM.1">Fuzzy_Negation</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">ff</font>, <font color="Olive" title="b2">gg</font> being   <a href="fuzzy_1.html#NM1" title="FUZZY_1:NM.1">Membership_Func</a> of <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>  st <font color="Maroon" title="c1">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b1">ff</font> &amp; <font color="Maroon" title="c2">g</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b2">gg</font> holds <br/> <a href="fuzimpl4.html#K4" title="FUZIMPL4:func.4">min</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font>) <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> (<font color="Olive" title="b1">ff</font>,<font color="Olive" title="b2">gg</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">ff</font>, <font color="Maroon" title="c4">gg</font> be   <a href="fuzzy_1.html#NM1" title="FUZZY_1:NM.1">Membership_Func</a> of <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c3">ff</font> &amp; <font color="Maroon" title="c2">g</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c4">gg</font> implies  <a href="fuzimpl4.html#K4" title="FUZIMPL4:func.4">min</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font>) <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> (<font color="Maroon" title="c3">ff</font>,<font color="Maroon" title="c4">gg</font>) )</span><br/>



<span class="kw">assume </span><a NAME="E1:84"/><span class="lab"><font color="Green" title="E42">A1</font></span>: 
( <font color="Maroon" title="c1">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c3">ff</font> &amp; <font color="Maroon" title="c2">g</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c4">gg</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="fuzimpl4.html#K4" title="FUZIMPL4:func.4">min</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font>) <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> (<font color="Maroon" title="c3">ff</font>,<font color="Maroon" title="c4">gg</font>)</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">f1</font>, <font color="Maroon" title="c6">g1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E3:84"/><span class="lab"><font color="Green" title="E43">A2</font></span>: 
( <font color="Maroon" title="c5">f1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c6">g1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c2">g</font> &amp;  <a href="fuzimpl4.html#K4" title="FUZIMPL4:func.4">min</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font>) <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="cousin2.html#K1" title="COUSIN2:func.1">min</a> (<font color="Maroon" title="c5">f1</font>,<font color="Maroon" title="c6">g1</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fuzimpl4.html#D10" target="_self" onmouseover="rs('fuzimpl4/D10')" onmouseout="rh()">MinFuz</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">ff1</font> = <font color="Maroon" title="c5">f1</font>, <font color="Maroon" title="c8">gg1</font> = <font color="Maroon" title="c6">g1</font> as   <a href="fuzzy_1.html#NM1" title="FUZZY_1:NM.1">Membership_Func</a> of <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:84"><span class="lab"><font color="Green" title="E43">A2</font></span></a></span>;<br/>
<a NAME="E5:84"/>
 for <font color="Olive" title="b1">c</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> holds  <span class="p1">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> (<font color="Maroon" title="c7">ff1</font>,<font color="Maroon" title="c8">gg1</font>)</span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lfuzzy_0.html#K3" title="LFUZZY_0:func.3">min</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">ff1</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <font color="Olive" title="b1">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">gg1</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <font color="Olive" title="b1">c</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fuzzy_1.html#D3" onmouseover="rs('fuzzy_1/D3')" onmouseout="rh()">FUZZY_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:84"/>
 <a href="fuzimpl4.html#K4" title="FUZIMPL4:func.4">min</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font>) <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> (<font color="Maroon" title="c3">ff</font>,<font color="Maroon" title="c4">gg</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="txt" href="#E3:84"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="ref" href="cousin2.html#D1" onmouseover="rs('cousin2/D1')" onmouseout="rh()">COUSIN2:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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