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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:11"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1
 ;<br/>

<b>set </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<a NAME="E2:11"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="binari_4.html#E1:11"><i><font color="Green">E9</font></i></a>, <a class="ref" href="binari_4.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E3:11"/>
 <a href="binarith.html#K11">add_ovfl</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i><a class="txt" href="binari_4.html#E1:11"><i><font color="Green">E9</font></i></a>, <a class="ref" href="binari_4.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E4:11"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>3</sub></font> <a href="binarith.html#R1">are_summable</a> 
 
<b>by </b><i><a class="ref" href="binarith.html#D10">BINARITH:def 10</a></i>;<br/>
<b>then </b> <a href="binarith.html#K9">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K10">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="binarith.html#K9">Absval</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="binarith.html#K9">Absval</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T48">BINARITH:48</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="binarith.html#K9">Absval</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="binari_4.html#E2:11"><i><font color="Green">E10</font></i></a>, <a class="ref" href="binari_3.html#T36">BINARI_3:36</a></i>
;<br/>
<b>hence </b><a NAME="E6:11"/>
 <a href="binarith.html#K9">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K10">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="binari_4.html#E2:11"><i><font color="Green">E10</font></i></a>, <a class="ref" href="binari_3.html#T36">BINARI_3:36</a></i>;<br/>


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