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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">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">u1</font>, <font color="Olive" title="b2">v1</font>, <font color="Olive" title="b3">w1</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> holds   <a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<span class="p1">(<span class="default"><a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Olive" title="b1">u1</font>,<font color="Olive" title="b2">v1</font>)</span>)</span>,<font color="Olive" title="b3">w1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Olive" title="b1">u1</font>,<span class="p1">(<span class="default"><a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Olive" title="b2">v1</font>,<font color="Olive" title="b3">w1</font>)</span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">u1</font>, <font color="Maroon" title="c3">v1</font>, <font color="Maroon" title="c4">w1</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <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">  <a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<span class="p1">(<span class="default"><a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Maroon" title="c2">u1</font>,<font color="Maroon" title="c3">v1</font>)</span>)</span>,<font color="Maroon" title="c4">w1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Maroon" title="c2">u1</font>,<span class="p1">(<span class="default"><a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">w1</font>)</span>)</span>)</span><br/>



<a NAME="E1:2"/><span class="lab"><font color="Green" title="E3">A1</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<span class="p2">(<span class="default"><a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Maroon" title="c2">u1</font>,<font color="Maroon" title="c3">v1</font>)</span>)</span>,<font color="Maroon" title="c4">w1</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="txt" href="nbvectsp.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/>
<a NAME="E2:2"/><span class="lab"><font color="Green" title="E4">A2</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Maroon" title="c2">u1</font>,<span class="p2">(<span class="default"><a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">w1</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="txt" href="nbvectsp.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_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">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<span class="p2">(<span class="default"><a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Maroon" title="c2">u1</font>,<font color="Maroon" title="c3">v1</font>)</span>)</span>,<font color="Maroon" title="c4">w1</font>)</span>)</span> holds <br/><span class="p1">(<span class="default"><a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<span class="p2">(<span class="default"><a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Maroon" title="c2">u1</font>,<font color="Maroon" title="c3">v1</font>)</span>)</span>,<font color="Maroon" title="c4">w1</font>)</span>)</span> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Maroon" title="c2">u1</font>,<span class="p2">(<span class="default"><a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">w1</font>)</span>)</span>)</span>)</span> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Olive" title="b1">i</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">i</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"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<span class="p2">(<span class="default"><a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Maroon" title="c2">u1</font>,<font color="Maroon" title="c3">v1</font>)</span>)</span>,<font color="Maroon" title="c4">w1</font>)</span>)</span> implies <span class="p1">(<span class="default"><a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<span class="p2">(<span class="default"><a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Maroon" title="c2">u1</font>,<font color="Maroon" title="c3">v1</font>)</span>)</span>,<font color="Maroon" title="c4">w1</font>)</span>)</span> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Maroon" title="c2">u1</font>,<span class="p2">(<span class="default"><a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">w1</font>)</span>)</span>)</span>)</span> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c5">i</font> )</span><br/><span class="kw">assume </span><a NAME="E1:2_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<span class="p2">(<span class="default"><a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Maroon" title="c2">u1</font>,<font color="Maroon" title="c3">v1</font>)</span>)</span>,<font color="Maroon" title="c4">w1</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"><a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<span class="p2">(<span class="default"><a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Maroon" title="c2">u1</font>,<font color="Maroon" title="c3">v1</font>)</span>)</span>,<font color="Maroon" title="c4">w1</font>)</span>)</span> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Maroon" title="c2">u1</font>,<span class="p2">(<span class="default"><a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">w1</font>)</span>)</span>)</span>)</span> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c5">i</font></span><br/><a NAME="E2:2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E5">A3</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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="txt" href="#E1:2"><span class="lab"><font color="Green" title="E3">A1</font></span></a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<span class="p2">(<span class="default"><a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Maroon" title="c2">u1</font>,<font color="Maroon" title="c3">v1</font>)</span>)</span>,<font color="Maroon" title="c4">w1</font>)</span>)</span> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c5">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Maroon" title="c2">u1</font>,<font color="Maroon" title="c3">v1</font>)</span>)</span> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c5">i</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">w1</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c5">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="descip_1.html#D4" onmouseover="rs('descip_1/D4')" onmouseout="rh()">DESCIP_1:def 4</a>, <a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">u1</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c5">i</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">v1</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c5">i</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">w1</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c5">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="descip_1.html#D4" onmouseover="rs('descip_1/D4')" onmouseout="rh()">DESCIP_1:def 4</a>, <a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">u1</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c5">i</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">v1</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c5">i</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">w1</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c5">i</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">u1</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c5">i</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"><a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">w1</font>)</span>)</span> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c5">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="descip_1.html#D4" onmouseover="rs('descip_1/D4')" onmouseout="rh()">DESCIP_1:def 4</a>, <a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Maroon" title="c2">u1</font>,<span class="p2">(<span class="default"><a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">w1</font>)</span>)</span>)</span>)</span> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c5">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="descip_1.html#D4" onmouseover="rs('descip_1/D4')" onmouseout="rh()">DESCIP_1:def 4</a>, <a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</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="E4:2"/>
 <a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<span class="p1">(<span class="default"><a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Maroon" title="c2">u1</font>,<font color="Maroon" title="c3">v1</font>)</span>)</span>,<font color="Maroon" title="c4">w1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Maroon" title="c2">u1</font>,<span class="p1">(<span class="default"><a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">w1</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="nbvectsp.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="txt" href="#E2:2"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="ref" href="finseq_1.html#T14" onmouseover="rs('finseq_1/T14')" onmouseout="rh()">FINSEQ_1:14</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>
