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<b>let </b><font color="Maroon" title="c1">n</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">z1</font>, <font color="Maroon" title="c3">z2</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> ;<br/><b>let </b><font color="Maroon" title="c4">d1</font>, <font color="Maroon" title="c5">d2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> ;<br/>





<a NAME="E1:22"/><i><font color="Green" title="E14">A1</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></i>;<br/>
<a NAME="E2:22"/><b>then </b><i><font color="Green" title="E15">A2</font></i>: 
<span class="p1">(<span class="default"><a href="binarith.html#K5" title="BINARITH:func.5">carry</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p3"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c4">d1</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p3"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c5">d2</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binarith.html#K5" title="BINARITH:func.5">carry</a> <font color="Maroon" title="c2">z1</font>,<font color="Maroon" title="c3">z2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="binarith.html#T43" target="_self" title="BINARITH:th.43">Th43</a></i>;<br/>
<a NAME="E3:22"/><i><font color="Green" title="E16">A3</font></i>: 
<font color="Maroon" title="c2">z1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c4">d1</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="binarith.html#E1:22"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="binarith.html#T2" target="_self" title="BINARITH:th.2">Th2</a></i>;<br/>
<a NAME="E4:22"/><i><font color="Green" title="E17">A4</font></i>: 
<font color="Maroon" title="c3">z2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c5">d2</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="binarith.html#E1:22"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="binarith.html#T2" target="_self" title="BINARITH:th.2">Th2</a></i>;<br/>
<a NAME="E5:22"/><i><font color="Green" title="E18">A5</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<b>thus </b><a NAME="E6:22"/>
 <a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c2">z1</font>,<font color="Maroon" title="c3">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binarith.html#K5" title="BINARITH:func.5">carry</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p3"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c4">d1</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p3"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c5">d2</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="binarith.html#E5:22"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="binarith.html#D5" target="_self" title="BINARITH:def.5">Def5</a>, <a class="txt" href="binarith.html#E2:22"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="binarith.html#E3:22"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="binarith.html#E4:22"><i><font color="Green" title="E17">A4</font></i></a></i>;<br/>


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