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<span class="kw">let </span><font color="Maroon" title="c1">I</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</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">  for <font color="Olive" title="b1">N</font> being   <a href="fuzimpl3.html#V9" title="FUZIMPL3:attr.9">negation-strong</a>  <a href="fuzimpl3.html#NM1" title="FUZIMPL3:NM.1">Fuzzy_Negation</a>  st <font color="Olive" title="b1">N</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="fuzimpl3.html#K4" title="FUZIMPL3:func.4">FNegation</a> <font color="Maroon" title="c1">I</font> &amp; <font color="Maroon" title="c1">I</font> is  <a href="fuzimpl2.html#V2" title="FUZIMPL2:attr.2">satisfying_(EP)</a>  holds <br/>( <font color="Maroon" title="c1">I</font> is  <a href="fuzimpl2.html#NV8" title="FUZIMPL2:NV.8">satisfying_(I3)</a>  &amp; <font color="Maroon" title="c1">I</font> is  <a href="fuzimpl2.html#NV9" title="FUZIMPL2:NV.9">satisfying_(I4)</a>  &amp; <font color="Maroon" title="c1">I</font> is  <a href="fuzimpl2.html#NV10" title="FUZIMPL2:NV.10">satisfying_(I5)</a>  &amp; <font color="Maroon" title="c1">I</font> is  <a href="fuzimpl2.html#V1" title="FUZIMPL2:attr.1">satisfying_(NP)</a>  &amp; <font color="Maroon" title="c1">I</font> is <font color="Olive" title="b1">N</font> <a href="fuzimpl4.html#V1" title="FUZIMPL4:attr.1">-satisfying_CP</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">N</font> be   <a href="fuzimpl3.html#V9" title="FUZIMPL3:attr.9">negation-strong</a>  <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"> ( <font color="Maroon" title="c2">N</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="fuzimpl3.html#K4" title="FUZIMPL3:func.4">FNegation</a> <font color="Maroon" title="c1">I</font> &amp; <font color="Maroon" title="c1">I</font> is  <a href="fuzimpl2.html#V2" title="FUZIMPL2:attr.2">satisfying_(EP)</a>  implies ( <font color="Maroon" title="c1">I</font> is  <a href="fuzimpl2.html#NV8" title="FUZIMPL2:NV.8">satisfying_(I3)</a>  &amp; <font color="Maroon" title="c1">I</font> is  <a href="fuzimpl2.html#NV9" title="FUZIMPL2:NV.9">satisfying_(I4)</a>  &amp; <font color="Maroon" title="c1">I</font> is  <a href="fuzimpl2.html#NV10" title="FUZIMPL2:NV.10">satisfying_(I5)</a>  &amp; <font color="Maroon" title="c1">I</font> is  <a href="fuzimpl2.html#V1" title="FUZIMPL2:attr.1">satisfying_(NP)</a>  &amp; <font color="Maroon" title="c1">I</font> is <font color="Maroon" title="c2">N</font> <a href="fuzimpl4.html#V1" title="FUZIMPL4:attr.1">-satisfying_CP</a>  ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:28"/><span class="lab"><font color="Green" title="E14">A0</font></span>: 
<font color="Maroon" title="c2">N</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="fuzimpl3.html#K4" title="FUZIMPL3:func.4">FNegation</a> <font color="Maroon" title="c1">I</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"> (  not <font color="Maroon" title="c1">I</font> is  <a href="fuzimpl2.html#V2" title="FUZIMPL2:attr.2">satisfying_(EP)</a>  or ( <font color="Maroon" title="c1">I</font> is  <a href="fuzimpl2.html#NV8" title="FUZIMPL2:NV.8">satisfying_(I3)</a>  &amp; <font color="Maroon" title="c1">I</font> is  <a href="fuzimpl2.html#NV9" title="FUZIMPL2:NV.9">satisfying_(I4)</a>  &amp; <font color="Maroon" title="c1">I</font> is  <a href="fuzimpl2.html#NV10" title="FUZIMPL2:NV.10">satisfying_(I5)</a>  &amp; <font color="Maroon" title="c1">I</font> is  <a href="fuzimpl2.html#V1" title="FUZIMPL2:attr.1">satisfying_(NP)</a>  &amp; <font color="Maroon" title="c1">I</font> is <font color="Maroon" title="c2">N</font> <a href="fuzimpl4.html#V1" title="FUZIMPL4:attr.1">-satisfying_CP</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:28"/><span class="lab"><font color="Green" title="E15">A1</font></span>: 
<font color="Maroon" title="c1">I</font> is  <a href="fuzimpl2.html#V2" title="FUZIMPL2:attr.2">satisfying_(EP)</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"> ( <font color="Maroon" title="c1">I</font> is  <a href="fuzimpl2.html#NV8" title="FUZIMPL2:NV.8">satisfying_(I3)</a>  &amp; <font color="Maroon" title="c1">I</font> is  <a href="fuzimpl2.html#NV9" title="FUZIMPL2:NV.9">satisfying_(I4)</a>  &amp; <font color="Maroon" title="c1">I</font> is  <a href="fuzimpl2.html#NV10" title="FUZIMPL2:NV.10">satisfying_(I5)</a>  &amp; <font color="Maroon" title="c1">I</font> is  <a href="fuzimpl2.html#V1" title="FUZIMPL2:attr.1">satisfying_(NP)</a>  &amp; <font color="Maroon" title="c1">I</font> is <font color="Maroon" title="c2">N</font> <a href="fuzimpl4.html#V1" title="FUZIMPL4:attr.1">-satisfying_CP</a>  )</span><br/>

<a NAME="E3:28"/><span class="lab"><font color="Green" title="E16">A2</font></span>: 
<font color="Maroon" title="c2">N</font> is  <a href="fuzimpl3.html#V4" title="FUZIMPL3:attr.4">involutive</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fuzimpl3.html#D13" onmouseover="rs('fuzimpl3/D13')" onmouseout="rh()">FUZIMPL3:def 13</a>, <a class="ref" href="fuzimpl3.html#D11" onmouseover="rs('fuzimpl3/D11')" onmouseout="rh()">FUZIMPL3:def 11</a></span>;<br/>
<a NAME="E4:28"/>
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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  <font color="Maroon" title="c1">I</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="28_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="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>; <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">I</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>)</span><br/>

<a NAME="E1:28_1"/><span class="lab"><font color="Green" title="E17">ZZ</font></span>: 
( <font color="Maroon" title="c2">N</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</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="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/>
<font color="Maroon" title="c1">I</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span>) = 
<font color="Maroon" title="c1">I</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c3">x</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E14">A0</font></span></a>, <a class="ref" href="fuzimpl3.html#D16" onmouseover="rs('fuzimpl3/D16')" onmouseout="rh()">FUZIMPL3:def 16</a></span>
<br/>.= 
<font color="Maroon" title="c1">I</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="fuzimpl2.html#D2" onmouseover="rs('fuzimpl2/D2')" onmouseout="rh()">FUZIMPL2:def 2</a>, <a class="txt" href="#E1:28_1"><span class="lab"><font color="Green" title="E17">ZZ</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c1">I</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="fuzimpl3.html#D16" onmouseover="rs('fuzimpl3/D16')" onmouseout="rh()">FUZIMPL3:def 16</a>, <a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E14">A0</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c1">I</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="ref" href="fuzimpl3.html#D8" onmouseover="rs('fuzimpl3/D8')" onmouseout="rh()">FUZIMPL3:def 8</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E3:28_1"/>
<font color="Maroon" title="c1">I</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">T1</font></span>: 
<font color="Maroon" title="c1">I</font> is <font color="Maroon" title="c2">N</font> <a href="fuzimpl4.html#V1" title="FUZIMPL4:attr.1">-satisfying_CP</a> 
 
;<br/>
<a NAME="E6:28"/><span class="kw">then </span>
<font color="Maroon" title="c1">I</font> is  <a href="fuzimpl2.html#V1" title="FUZIMPL2:attr.1">satisfying_(NP)</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E14">A0</font></span></a>, <a class="ref" href="fuzimpl4.html#T18" target="_self" onmouseover="rs('fuzimpl4/T18')" onmouseout="rh()">Lemma156i</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:28"/>
( <font color="Maroon" title="c1">I</font> is  <a href="fuzimpl2.html#NV8" title="FUZIMPL2:NV.8">satisfying_(I3)</a>  &amp; <font color="Maroon" title="c1">I</font> is  <a href="fuzimpl2.html#NV9" title="FUZIMPL2:NV.9">satisfying_(I4)</a>  &amp; <font color="Maroon" title="c1">I</font> is  <a href="fuzimpl2.html#NV10" title="FUZIMPL2:NV.10">satisfying_(I5)</a>  &amp; <font color="Maroon" title="c1">I</font> is  <a href="fuzimpl2.html#V1" title="FUZIMPL2:attr.1">satisfying_(NP)</a>  &amp; <font color="Maroon" title="c1">I</font> is <font color="Maroon" title="c2">N</font> <a href="fuzimpl4.html#V1" title="FUZIMPL4:attr.1">-satisfying_CP</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:28"><span class="lab"><font color="Green" title="E17">T1</font></span></a>, <a class="ref" href="fuzimpl4.html#T16" target="_self" onmouseover="rs('fuzimpl4/T16')" onmouseout="rh()">Lemma154vb</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>
