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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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">  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="finseq_2.html#NM3" title="FINSEQ_2:NM.3">Tuple</a> of <font color="Maroon" title="c1">n</font>, <a href="margrel1.html#K5" title="MARGREL1:func.5">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/><span class="kw">let </span><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">z</font> be   <a href="finseq_2.html#NM3" title="FINSEQ_2:NM.3">Tuple</a> of <font color="Maroon" title="c1">n</font>, <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</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"> <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"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a></span>;<br/>
<a NAME="E2:14"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A2</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E3:14"/><span class="lab"><font color="Green" title="E15">A3</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="14_1"><span class="kw">now </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">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="tarski.html#R2" title="TARSKI: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> holds <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="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="Olive" title="b1">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="Olive" title="b1">j</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">j</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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="c5">j</font> <a href="tarski.html#R2" title="TARSKI: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/><span class="kw">assume </span><a NAME="E1:14_1"/><span class="lab"><font color="Green" title="E16">A4</font></span>: 
<font color="Maroon" title="c5">j</font> <a href="tarski.html#R2" title="TARSKI: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" 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"> <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"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A5</font></span>: 
<font color="Maroon" title="c5">j</font> <a href="tarski.html#R2" title="TARSKI: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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><span class="kw">thus </span><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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E1:14_1"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="ref" href="idea_1.html#D2" target="_self" onmouseover="rs('idea_1/D2')" onmouseout="rh()">Def2</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E1:14_1"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="ref" href="idea_1.html#D2" target="_self" onmouseover="rs('idea_1/D2')" onmouseout="rh()">Def2</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboolean.html#T73" onmouseover="rs('xboolean/T73')" onmouseout="rh()">XBOOLEAN:73</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E1:14_1"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="ref" href="idea_1.html#D2" target="_self" onmouseover="rs('idea_1/D2')" onmouseout="rh()">Def2</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E1:14_1"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="ref" href="idea_1.html#D2" target="_self" onmouseover="rs('idea_1/D2')" onmouseout="rh()">Def2</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14_1"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</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="E5:14"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E3:14"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="ref" href="finseq_2.html#T9" onmouseover="rs('finseq_2/T9')" onmouseout="rh()">FINSEQ_2:9</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/>


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