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<b>let </b><font color="Maroon">n</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">x</font> be   <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">n</font>,<a href="margrel1.html#K6">BOOLEAN</a> , <font color="Maroon">y</font> be   <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">n</font>,<a href="margrel1.html#K6">BOOLEAN</a> ;<br/>



<b>assume </b><a NAME="E1:6"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font> )
 ;<br/>

<a NAME="E2:6"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binarith.html#K8">+</a> <font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:6_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 ;<br/>

<i><font color="Green">E41</font></i>: <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="euclid.html#D4">EUCLID:def 4</a></i>
<br/>.= 
 <a href="margrel1.html#K7">FALSE</a> 
<b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/>
<a NAME="E3:6_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="binarith.html#K5">carry</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binarith.html#K8">+</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E4:6_1"/><b>then </b><i><font color="Green">E42</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="binarith.html#K5">carry</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binarith.html#K8">+</a> <font color="Maroon">y</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E5:6_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i>, , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<i><font color="Green">E44</font></i>: <span class="p1">(<span class="default"><a href="binarith.html#K5">carry</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><a href="binarith.html#K5">carry</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
<br/>.= 
 <a href="margrel1.html#K7">FALSE</a> 
<b>by </b><i>, , </i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binarith.html#K8">+</a> <font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binarith.html#K8">+</a> <font color="Maroon">y</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
<br/>.= 
<a href="margrel1.html#K7">FALSE</a>  <a href="binarith.html#K2">'xor'</a> <a href="margrel1.html#K7">FALSE</a> 
<b>by </b><i>, , , , <a class="ref" href="binarith.html#D8">BINARITH:def 8</a></i>
<br/>.= 
 <a href="margrel1.html#K7">FALSE</a> 

;<br/>
<b>hence </b><a NAME="E8:6_1"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binarith.html#K8">+</a> <font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:6"/>
<font color="Maroon">x</font> <a href="binarith.html#K8">+</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T139">FINSEQ_2:139</a></i>;<br/>


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