<?xml version="1.0"?>
<div class="add">

<span class="kw">set </span><font color="Maroon" title="c1">L</font> =  <a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <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>;<br/>
<span class="kw">set </span><font color="Maroon" title="c2">S</font> =  <a href="fuzimpl4.html#K5" title="FUZIMPL4:func.5">Fuzzy_Negations</a> ;<br/>
<a NAME="E1:87_2_1"/>
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><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>)</span>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="fuzimpl4.html#K5" title="FUZIMPL4:func.5">Fuzzy_Negations</a> &amp; <font color="Olive" title="b2">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="fuzimpl4.html#K5" title="FUZIMPL4:func.5">Fuzzy_Negations</a> &amp;  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <span class="p1"><a href="fuzimpl4.html#K6" title="FUZIMPL4:func.6">{</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="fuzimpl4.html#K6" title="FUZIMPL4:func.6">}</a></span>, <a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <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 <br/> <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1"><a href="fuzimpl4.html#K6" title="FUZIMPL4:func.6">{</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="fuzimpl4.html#K6" title="FUZIMPL4:func.6">}</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="fuzimpl4.html#K5" title="FUZIMPL4:func.5">Fuzzy_Negations</a>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="87_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><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>)</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="c3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="fuzimpl4.html#K5" title="FUZIMPL4:func.5">Fuzzy_Negations</a> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="fuzimpl4.html#K5" title="FUZIMPL4:func.5">Fuzzy_Negations</a> &amp;  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <span class="p1"><a href="fuzimpl4.html#K6" title="FUZIMPL4:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="fuzimpl4.html#K6" title="FUZIMPL4:func.6">}</a></span>, <a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <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> implies  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1"><a href="fuzimpl4.html#K6" title="FUZIMPL4:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="fuzimpl4.html#K6" title="FUZIMPL4:func.6">}</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="fuzimpl4.html#K5" title="FUZIMPL4:func.5">Fuzzy_Negations</a> )</span><br/>

<span class="kw">assume </span><a NAME="E1:87_2_1_1"/><span class="lab"><font color="Green" title="E42">S0</font></span>: 
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="fuzimpl4.html#K5" title="FUZIMPL4:func.5">Fuzzy_Negations</a> &amp; <font color="Maroon" title="c4">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="fuzimpl4.html#K5" title="FUZIMPL4:func.5">Fuzzy_Negations</a> &amp;  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <span class="p1"><a href="fuzimpl4.html#K6" title="FUZIMPL4:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="fuzimpl4.html#K6" title="FUZIMPL4:func.6">}</a></span>, <a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <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">  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1"><a href="fuzimpl4.html#K6" title="FUZIMPL4:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="fuzimpl4.html#K6" title="FUZIMPL4:func.6">}</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="fuzimpl4.html#K5" title="FUZIMPL4:func.5">Fuzzy_Negations</a></span><br/>

<a NAME="E2:87_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="fuzimpl3.html#NM1" title="FUZIMPL3:NM.1">Fuzzy_Negation</a> : verum </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fuzimpl4.html#D11" target="_self" onmouseover="rs('fuzimpl4/D11')" onmouseout="rh()">FuzNegDef</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">f</font> being   <a href="fuzimpl3.html#NM1" title="FUZIMPL3:NM.1">Fuzzy_Negation</a><span class="kw"> such that </span><br/><a NAME="E4:87_2_1_1"/><span class="lab"><font color="Green" title="E43">S1</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font>
 ;<br/>
<a NAME="E5:87_2_1_1"/>
<font color="Maroon" title="c4">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="fuzimpl3.html#NM1" title="FUZIMPL3:NM.1">Fuzzy_Negation</a> : verum </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_2_1_1"><span class="lab"><font color="Green" title="E42">S0</font></span></a>, <a class="ref" href="fuzimpl4.html#D11" target="_self" onmouseover="rs('fuzimpl4/D11')" onmouseout="rh()">FuzNegDef</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">g</font> being   <a href="fuzimpl3.html#NM1" title="FUZIMPL3:NM.1">Fuzzy_Negation</a><span class="kw"> such that </span><br/><a NAME="E7:87_2_1_1"/><span class="lab"><font color="Green" title="E44">s1</font></span>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font>
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">xx</font> = <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c8">yy</font> = <font color="Maroon" title="c4">y</font> as   <a href="fuzimpl3.html#NM1" title="FUZIMPL3:NM.1">Fuzzy_Negation</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:87_2_1_1"><span class="lab"><font color="Green" title="E43">S1</font></span></a>, <a class="txt" href="#E7:87_2_1_1"><span class="lab"><font color="Green" title="E44">s1</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">fxy</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E10:87_2_1_1"/><span class="lab"><font color="Green" title="E45">F1</font></span>: 
( <font color="Maroon" title="c9">fxy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1"><a href="fuzimpl4.html#K6" title="FUZIMPL4:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="fuzimpl4.html#K6" title="FUZIMPL4:func.6">}</a></span> &amp;  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c9">fxy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c9">fxy</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <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="ref" href="funct_2.html#D2" onmouseover="rs('funct_2/D2')" onmouseout="rh()">FUNCT_2:def 2</a>, <a class="txt" href="fuzimpl4.html#E56"><span class="lab"><font color="Green" title="E38">LemmaCarrier</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">fxy</font> = <font color="Maroon" title="c9">fxy</font> as   <a href="binop_1.html#NM1" title="BINOP_1:NM.1">UnOp</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="#E10:87_2_1_1"><span class="lab"><font color="Green" title="E45">F1</font></span></a>, <a class="ref" href="funct_2.html#T2" onmouseover="rs('funct_2/T2')" onmouseout="rh()">FUNCT_2:2</a></span>;<br/>
<span class="lab"><font color="Green" title="E46">T3</font></span>: <font color="Maroon" title="c3">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">y</font> = 
 <a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> (<span class="p1">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <font color="Maroon" title="c3">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <font color="Maroon" title="c4">y</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="lfuzzy_0.html#T21" onmouseover="rs('lfuzzy_0/T21')" onmouseout="rh()">LFUZZY_0:21</a></span>
<br/>.= 
 <a href="fuzimpl4.html#K4" title="FUZIMPL4:func.4">min</a> (<font color="Maroon" title="c7">xx</font>,<font color="Maroon" title="c8">yy</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="fuzimpl4.html#T33" target="_self" onmouseover="rs('fuzimpl4/T33')" onmouseout="rh()">MinEqMin</a></span>
;<br/>
<a NAME="E13:87_2_1_1"/>
<font color="Maroon" title="c10">fxy</font> is   <a href="fuzimpl3.html#NM1" title="FUZIMPL3:NM.1">Fuzzy_Negation</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:87_2_1_1"><span class="lab"><font color="Green" title="E45">F1</font></span></a>, <a class="txt" href="#E12:87_2_1_1"><span class="lab"><font color="Green" title="E46">T3</font></span></a>, <a class="ref" href="yellow_0.html#T40" onmouseover="rs('yellow_0/T40')" onmouseout="rh()">YELLOW_0:40</a></span>;<br/>
<a NAME="E14:87_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">fxy</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="fuzimpl3.html#NM1" title="FUZIMPL3:NM.1">Fuzzy_Negation</a> : verum </span> } </span> 
 
;<br/>
<span class="kw">hence </span><a NAME="E15:87_2_1_1"/>
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1"><a href="fuzimpl4.html#K6" title="FUZIMPL4:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="fuzimpl4.html#K6" title="FUZIMPL4:func.6">}</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="fuzimpl4.html#K5" title="FUZIMPL4:func.5">Fuzzy_Negations</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:87_2_1_1"><span class="lab"><font color="Green" title="E45">F1</font></span></a>, <a class="ref" href="fuzimpl4.html#D11" target="_self" onmouseover="rs('fuzimpl4/D11')" onmouseout="rh()">FuzNegDef</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/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:87_2_1"/>
 <a href="fuzimpl4.html#K5" title="FUZIMPL4:func.5">Fuzzy_Negations</a>  is  <a href="yellow_0.html#V5" title="YELLOW_0:attr.5">meet-inheriting</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D16" onmouseover="rs('yellow_0/D16')" onmouseout="rh()">YELLOW_0:def 16</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/>


</div>
