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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#NM1" title="FUZIMPL3:NM.1">Fuzzy_Negation</a>  st <font color="Maroon" title="c1">I</font> is  <a href="fuzimpl2.html#NV7" title="FUZIMPL2:NV.7">satisfying_(I2)</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>  holds <br/><font color="Maroon" title="c1">I</font> is  <a href="fuzimpl2.html#NV6" title="FUZIMPL2:NV.6">satisfying_(I1)</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">N</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"> ( <font color="Maroon" title="c1">I</font> is  <a href="fuzimpl2.html#NV7" title="FUZIMPL2:NV.7">satisfying_(I2)</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>  implies <font color="Maroon" title="c1">I</font> is  <a href="fuzimpl2.html#NV6" title="FUZIMPL2:NV.6">satisfying_(I1)</a>  )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:22"/><span class="lab"><font color="Green" title="E8">A0</font></span>: 
<font color="Maroon" title="c1">I</font> is  <a href="fuzimpl2.html#NV7" title="FUZIMPL2:NV.7">satisfying_(I2)</a> 
 <span class="kw">and </span><br/><a NAME="E2:22"/><span class="lab"><font color="Green" title="E9">AA</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> 
 ; <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#NV6" title="FUZIMPL2:NV.6">satisfying_(I1)</a> </span><br/>

<a NAME="E3:22"/>
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</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>  st <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">y</font> holds <br/><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="b3">z</font>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">I</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_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>, <font color="Maroon" title="c5">z</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="c3">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">y</font> implies <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="c5">z</font>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">I</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:22_1"/>
<font color="Maroon" title="c3">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <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"> <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="c5">z</font>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">I</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font>)</span><br/>

<a NAME="E2:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">F0</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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <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 class="kw">by</span> <span class="lab"><a class="ref" href="fuzimpl3.html#T7" onmouseover="rs('fuzimpl3/T7')" onmouseout="rh()">FUZIMPL3:7</a></span>;<br/>
<a NAME="E3:22_1"/><span class="lab"><font color="Green" title="E11">A3</font></span>: 
<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="c5">z</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="c5">z</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22"><span class="lab"><font color="Green" title="E9">AA</font></span></a></span>;<br/>
<a NAME="E4:22_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="c5">z</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="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <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="c5">z</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="c4">y</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_1"><span class="lab"><font color="Green" title="E10">F0</font></span></a>, <a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E8">A0</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:22_1"/>
<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="c5">z</font>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">I</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E2:22"><span class="lab"><font color="Green" title="E9">AA</font></span></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="E4:22"/>
<font color="Maroon" title="c1">I</font> is  <a href="fuzimpl2.html#NV6" title="FUZIMPL2:NV.6">satisfying_(I1)</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"> verum</span><br/>


</div>
