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

<span class="kw">set </span><font color="Maroon" title="c1">N</font> =  <a href="fuzimpl3.html#K3" title="FUZIMPL3:func.3">N_CC</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c2">I</font> =  <a href="fuzimpl2.html#NK9" title="FUZIMPL2:NK.9">I_FD</a> ;<br/>
<a NAME="E1:69"/>
 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  <a href="fuzimpl2.html#NK9" title="FUZIMPL2:NK.9">I_FD</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><a href="fuzimpl3.html#K3" title="FUZIMPL3:func.3">N_CC</a> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span>,<font color="Olive" title="b2">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="fuzimpl2.html#NK9" title="FUZIMPL2:NK.9">I_FD</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><a href="fuzimpl3.html#K3" title="FUZIMPL3:func.3">N_CC</a> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font></span>)</span>,<font color="Olive" title="b1">x</font>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="69_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"> <a href="fuzimpl2.html#NK9" title="FUZIMPL2:NK.9">I_FD</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><a href="fuzimpl3.html#K3" title="FUZIMPL3:func.3">N_CC</a> <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="c4">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="fuzimpl2.html#NK9" title="FUZIMPL2:NK.9">I_FD</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><a href="fuzimpl3.html#K3" title="FUZIMPL3:func.3">N_CC</a> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font></span>)</span>,<font color="Maroon" title="c3">x</font>)</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:69_1_1"/>
( <font color="Maroon" title="c3">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="fuzimpl3.html#K3" title="FUZIMPL3:func.3">N_CC</a> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font> or <font color="Maroon" title="c3">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="fuzimpl3.html#K3" title="FUZIMPL3:func.3">N_CC</a> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:69_1_1_1"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="fuzimpl3.html#K3" title="FUZIMPL3:func.3">N_CC</a> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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"> <a href="fuzimpl2.html#NK9" title="FUZIMPL2:NK.9">I_FD</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><a href="fuzimpl3.html#K3" title="FUZIMPL3:func.3">N_CC</a> <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="c4">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="fuzimpl2.html#NK9" title="FUZIMPL2:NK.9">I_FD</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><a href="fuzimpl3.html#K3" title="FUZIMPL3:func.3">N_CC</a> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font></span>)</span>,<font color="Maroon" title="c3">x</font>)</span><br/><div class="add"><a NAME="E2:69_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fuzimpl3.html#D6" onmouseover="rs('fuzimpl3/D6')" onmouseout="rh()">FUZIMPL3:def 6</a></span>;<br/><a NAME="E3:69_1_1_1"/><span class="kw">then </span>
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T12" onmouseover="rs('xreal_1/T12')" onmouseout="rh()">XREAL_1:12</a></span>;<br/><a NAME="E4:69_1_1_1"/><span class="kw">then </span>
<a href="fuzimpl3.html#K3" title="FUZIMPL3:func.3">N_CC</a> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fuzimpl3.html#D6" onmouseover="rs('fuzimpl3/D6')" onmouseout="rh()">FUZIMPL3:def 6</a></span>;<br/><span class="kw">then </span><a href="fuzimpl2.html#NK9" title="FUZIMPL2:NK.9">I_FD</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><a href="fuzimpl3.html#K3" title="FUZIMPL3:func.3">N_CC</a> <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="c4">y</font>) = 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="fuzimpl1.html#D23" onmouseover="rs('fuzimpl1/D23')" onmouseout="rh()">FUZIMPL1:def 23</a></span>
<br/>.= 
<a href="fuzimpl2.html#NK9" title="FUZIMPL2:NK.9">I_FD</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><a href="fuzimpl3.html#K3" title="FUZIMPL3:func.3">N_CC</a> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font></span>)</span>,<font color="Maroon" title="c3">x</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1_1_1"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="ref" href="fuzimpl1.html#D23" onmouseover="rs('fuzimpl1/D23')" onmouseout="rh()">FUZIMPL1:def 23</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:69_1_1_1"/>
<a href="fuzimpl2.html#NK9" title="FUZIMPL2:NK.9">I_FD</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><a href="fuzimpl3.html#K3" title="FUZIMPL3:func.3">N_CC</a> <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="c4">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="fuzimpl2.html#NK9" title="FUZIMPL2:NK.9">I_FD</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><a href="fuzimpl3.html#K3" title="FUZIMPL3:func.3">N_CC</a> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font></span>)</span>,<font color="Maroon" title="c3">x</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:69_1_1_2"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="fuzimpl3.html#K3" title="FUZIMPL3:func.3">N_CC</a> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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"> <a href="fuzimpl2.html#NK9" title="FUZIMPL2:NK.9">I_FD</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><a href="fuzimpl3.html#K3" title="FUZIMPL3:func.3">N_CC</a> <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="c4">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="fuzimpl2.html#NK9" title="FUZIMPL2:NK.9">I_FD</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><a href="fuzimpl3.html#K3" title="FUZIMPL3:func.3">N_CC</a> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font></span>)</span>,<font color="Maroon" title="c3">x</font>)</span><br/><div class="add"><a NAME="E2:69_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fuzimpl3.html#D6" onmouseover="rs('fuzimpl3/D6')" onmouseout="rh()">FUZIMPL3:def 6</a></span>;<br/><a NAME="E3:69_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T12" onmouseover="rs('xreal_1/T12')" onmouseout="rh()">XREAL_1:12</a></span>;<br/><a NAME="E4:69_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A2</font></span>: 
<font color="Maroon" title="c4">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="fuzimpl3.html#K3" title="FUZIMPL3:func.3">N_CC</a> <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#D6" onmouseover="rs('fuzimpl3/D6')" onmouseout="rh()">FUZIMPL3:def 6</a></span>;<br/><a NAME="E5:69_1_1_2"/><span class="lab"><font color="Green" title="E32">A3</font></span>: 
( <a href="fuzimpl3.html#K3" title="FUZIMPL3:func.3">N_CC</a> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">x</font> &amp; <a href="fuzimpl3.html#K3" title="FUZIMPL3:func.3">N_CC</a> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fuzimpl3.html#D6" onmouseover="rs('fuzimpl3/D6')" onmouseout="rh()">FUZIMPL3:def 6</a></span>;<br/><a href="fuzimpl2.html#NK9" title="FUZIMPL2:NK.9">I_FD</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><a href="fuzimpl3.html#K3" title="FUZIMPL3:func.3">N_CC</a> <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="c4">y</font>) = 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="fuzimpl3.html#K3" title="FUZIMPL3:func.3">N_CC</a> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">y</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69_1_1_2"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="ref" href="fuzimpl1.html#D23" onmouseover="rs('fuzimpl1/D23')" onmouseout="rh()">FUZIMPL1:def 23</a></span>
<br/>.= 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="fuzimpl3.html#K3" title="FUZIMPL3:func.3">N_CC</a> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">x</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:69_1_1_2"><span class="lab"><font color="Green" title="E32">A3</font></span></a></span>
<br/>.= 
<a href="fuzimpl2.html#NK9" title="FUZIMPL2:NK.9">I_FD</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><a href="fuzimpl3.html#K3" title="FUZIMPL3:func.3">N_CC</a> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font></span>)</span>,<font color="Maroon" title="c3">x</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1_1_2"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="ref" href="fuzimpl1.html#D23" onmouseover="rs('fuzimpl1/D23')" onmouseout="rh()">FUZIMPL1:def 23</a></span>
;<br/><span class="kw">hence </span><a NAME="E7:69_1_1_2"/>
<a href="fuzimpl2.html#NK9" title="FUZIMPL2:NK.9">I_FD</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><a href="fuzimpl3.html#K3" title="FUZIMPL3:func.3">N_CC</a> <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="c4">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="fuzimpl2.html#NK9" title="FUZIMPL2:NK.9">I_FD</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><a href="fuzimpl3.html#K3" title="FUZIMPL3:func.3">N_CC</a> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font></span>)</span>,<font color="Maroon" title="c3">x</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:69"/>
 <a href="fuzimpl2.html#NK9" title="FUZIMPL2:NK.9">I_FD</a>  is  <a href="fuzimpl3.html#K3" title="FUZIMPL3:func.3">N_CC</a>  <a href="fuzimpl4.html#V2" title="FUZIMPL4:attr.2">-satisfying_L-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"> verum</span><br/>


</div>
