<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">zero</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">z1</font>, <font color="Olive" title="b2">z2</font> being  <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c1">n</font>, <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>   st <font color="Olive" title="b1">z1</font>,<font color="Olive" title="b2">z2</font> <a href="binarith.html#R1" title="BINARITH:pred.1">are_summable</a>  holds <br/> <a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">z1</font> <a href="binarith.html#K7" title="BINARITH:func.7">+</a> <font color="Olive" title="b2">z2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Olive" title="b1">z1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Olive" title="b2">z2</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">z1</font>, <font color="Maroon" title="c3">z2</font> be   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c1">n</font>, <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">z1</font>,<font color="Maroon" title="c3">z2</font> <a href="binarith.html#R1" title="BINARITH:pred.1">are_summable</a>  implies  <a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K7" title="BINARITH:func.7">+</a> <font color="Maroon" title="c3">z2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c2">z1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c3">z2</font></span>)</span> )</span><br/>



<span class="kw">assume </span><a NAME="E1:47"/>
<font color="Maroon" title="c2">z1</font>,<font color="Maroon" title="c3">z2</font> <a href="binarith.html#R1" title="BINARITH:pred.1">are_summable</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K7" title="BINARITH:func.7">+</a> <font color="Maroon" title="c3">z2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c2">z1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c3">z2</font></span>)</span></span><br/>

<a NAME="E2:47"/><span class="kw">then </span>
 <a href="binarith.html#K8" title="BINARITH:func.8">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>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="binarith.html#D10" target="_self" title="BINARITH:def.10">Def10</a></span>;<br/>
<a NAME="E3:47"/><span class="kw">then </span>
 <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <span class="p1">(<span class="default"><a href="binarith.html#K8" title="BINARITH:func.8">add_ovfl</a> <font color="Maroon" title="c2">z1</font>,<font color="Maroon" title="c3">z2</font></span>)</span>,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></span>;<br/>
<span class="kw">hence </span> <a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K7" title="BINARITH:func.7">+</a> <font color="Maroon" title="c3">z2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K7" title="BINARITH:func.7">+</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <span class="p2">(<span class="default"><a href="binarith.html#K8" title="BINARITH:func.8">add_ovfl</a> <font color="Maroon" title="c2">z1</font>,<font color="Maroon" title="c3">z2</font></span>)</span>,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c2">z1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c3">z2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binarith.html#T47" target="_self" title="BINARITH:th.47">Th47</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
