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

<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 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="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <font color="Maroon" title="c1">n</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <font color="Olive" title="b3">z</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">z</font> be   <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <font color="Maroon" title="c1">n</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <font color="Maroon" title="c4">z</font></span>)</span></span><br/>



<a NAME="E1:14"/><i><font color="Green" title="E13">A1</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <font color="Maroon" title="c4">z</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E2:14"/><i><font color="Green" title="E14">X</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="idea_1.html#E1:14"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <font color="Maroon" title="c4">z</font></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <font color="Maroon" title="c4">z</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">j</font> )</span><br/><b>assume </b><a NAME="E1:14_1"/><i><font color="Green" title="E15">A2</font></i>: 
<font color="Maroon" title="c5">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <font color="Maroon" title="c4">z</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <font color="Maroon" title="c4">z</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">j</font></span><br/><a NAME="E2:14_1"/><b>then </b><i><font color="Green" title="E16">A3</font></i>: 
( <font color="Maroon" title="c5">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <font color="Maroon" title="c4">z</font></span>)</span> &amp; <font color="Maroon" title="c5">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <font color="Maroon" title="c4">z</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="idea_1.html#E1:14"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="txt" href="idea_1.html#E2:14"><i><font color="Green" title="E14">X</font></i></a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">j</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">j</font>
<b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">j</font></span>)</span>
<b>by </b><i><a class="txt" href="idea_1.html#E1:14_1"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="idea_1.html#D2" target="_self" title="IDEA_1:def.2">Def2</a>, <a class="txt" href="idea_1.html#E2:14"><i><font color="Green" title="E14">X</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">j</font></span>)</span></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">j</font></span>)</span>
<b>by </b><i><a class="txt" href="idea_1.html#E1:14_1"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="idea_1.html#D2" target="_self" title="IDEA_1:def.2">Def2</a>, <a class="txt" href="idea_1.html#E2:14"><i><font color="Green" title="E14">X</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboolean.html#T73" title="XBOOLEAN:th.73">XBOOLEAN:73</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">j</font></span>)</span>
<b>by </b><i><a class="txt" href="idea_1.html#E1:14_1"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="idea_1.html#D2" target="_self" title="IDEA_1:def.2">Def2</a>, <a class="txt" href="idea_1.html#E2:14"><i><font color="Green" title="E14">X</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <font color="Maroon" title="c4">z</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">j</font>
<b>by </b><i><a class="txt" href="idea_1.html#E1:14_1"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="idea_1.html#D2" target="_self" title="IDEA_1:def.2">Def2</a>, <a class="txt" href="idea_1.html#E2:14"><i><font color="Green" title="E14">X</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <font color="Maroon" title="c4">z</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">j</font>
<b>by </b><i><a class="txt" href="idea_1.html#E2:14_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:14"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <font color="Maroon" title="c4">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="idea_1.html#E1:14"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
