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

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



<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default">2 <a href="comput_1.html#K8" target="_self">proj</a> 2</span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <a href="comput_1.html#K21" target="_self">[pred]</a>  <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1"><a href="funct_6.html#K6">&lt;:</a><span class="default"><span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default">2 <a href="comput_1.html#K8" target="_self">proj</a> 2</span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span><a href="funct_6.html#K6">:&gt;</a></span>;<br/>
<a NAME="E1:178"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default">2 <a href="comput_1.html#K8" target="_self">proj</a> 2</span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="tarski.html#R1">c=</a>  <a href="comput_1.html#K11" target="_self">PrimRec</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="178_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:178_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default">2 <a href="comput_1.html#K8" target="_self">proj</a> 2</span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
 ;<br/>

<a NAME="E2:178_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default">2 <a href="comput_1.html#K8" target="_self">proj</a> 2</span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E3:178_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="comput_1.html#K8" target="_self">proj</a> 2
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:178_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="comput_1.html#K11" target="_self">PrimRec</a> 
 <b>by </b><i><a class="ref" href="comput_1.html#D17" target="_self">Def17</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default">2 <a href="comput_1.html#K8" target="_self">proj</a> 2</span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> as  <a href="comput_1.html#V5" target="_self">with_the_same_arity</a>  <a href="finseq_1.html#M2">FinSequence</a> of  <a href="comput_1.html#K11" target="_self">PrimRec</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E3:178"/><i><font color="Green">E107</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>( <span class="p1">(<span class="default"><a href="comput_1.html#K21" target="_self">[pred]</a>  <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2"><a href="funct_6.html#K6">&lt;:</a><span class="default"><span class="p3"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p4">(<span class="default">2 <a href="comput_1.html#K8" target="_self">proj</a> 2</span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span><a href="funct_6.html#K6">:&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font>,0</span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1">(<span class="default"><a href="comput_1.html#K21" target="_self">[pred]</a>  <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2"><a href="funct_6.html#K6">&lt;:</a><span class="default"><span class="p3"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p4">(<span class="default">2 <a href="comput_1.html#K8" target="_self">proj</a> 2</span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span><a href="funct_6.html#K6">:&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="178_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<a NAME="E1:178_2"/><i><font color="Green">E108</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="funct_6.html#K6">&lt;:</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="funct_6.html#K6">:&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">2 <a href="comput_1.html#K8" target="_self">proj</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="funct_6.html#T61">FUNCT_6:61</a></i>;<br/>
<a NAME="E2:178_2"/><i><font color="Green">E109</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">2 <a href="comput_1.html#K8" target="_self">proj</a> 2</span>)</span> <a href="hidden.html#R1">=</a> 2 <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="comput_1.html#T40" target="_self">Th40</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,0</span><a href="finseq_4.html#K2">*&gt;</a></span> as    <a href="finseq_2.html#M2">Element</a> of 2 <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T121">FINSEQ_2:121</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="comput_1.html#K21" target="_self">[pred]</a>  <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2"><a href="funct_6.html#K6">&lt;:</a><span class="default"><span class="p3"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p4">(<span class="default">2 <a href="comput_1.html#K8" target="_self">proj</a> 2</span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span><a href="funct_6.html#K6">:&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,0</span><a href="finseq_4.html#K2">*&gt;</a></span> = 
<a href="comput_1.html#K21" target="_self">[pred]</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_6.html#K6">&lt;:</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="funct_6.html#K6">:&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="comput_1.html#E1:178_2"><i><font color="Green">E108</font></i></a>, <a class="txt" href="comput_1.html#E2:178_2"><i><font color="Green">E109</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<a href="comput_1.html#K21" target="_self">[pred]</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="comput_1.html#K8" target="_self">proj</a> 2</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
<b>by </b><i><a class="txt" href="comput_1.html#E2:178_2"><i><font color="Green">E109</font></i></a>, <a class="ref" href="funct_6.html#T61">FUNCT_6:61</a></i>
<br/>.= 
<a href="comput_1.html#K21" target="_self">[pred]</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="comput_1.html#K2" target="_self">.</a> 2</span>)</span></span><a href="binarith.html#K13">*&gt;</a></span>
<b>by </b><i><a class="ref" href="comput_1.html#T42" target="_self">Th42</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="comput_1.html#T93" target="_self">Th93</a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/>

<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> as    <a href="finseq_2.html#M2">Element</a> of 2 <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T121">FINSEQ_2:121</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="comput_1.html#K21" target="_self">[pred]</a>  <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2"><a href="funct_6.html#K6">&lt;:</a><span class="default"><span class="p3"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p4">(<span class="default">2 <a href="comput_1.html#K8" target="_self">proj</a> 2</span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span><a href="funct_6.html#K6">:&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> = 
<a href="comput_1.html#K21" target="_self">[pred]</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_6.html#K6">&lt;:</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="funct_6.html#K6">:&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="comput_1.html#E1:178_2"><i><font color="Green">E108</font></i></a>, <a class="txt" href="comput_1.html#E2:178_2"><i><font color="Green">E109</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<a href="comput_1.html#K21" target="_self">[pred]</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="comput_1.html#K8" target="_self">proj</a> 2</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
<b>by </b><i><a class="txt" href="comput_1.html#E2:178_2"><i><font color="Green">E109</font></i></a>, <a class="ref" href="funct_6.html#T61">FUNCT_6:61</a></i>
<br/>.= 
<a href="comput_1.html#K21" target="_self">[pred]</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="comput_1.html#K2" target="_self">.</a> 2</span>)</span></span><a href="binarith.html#K13">*&gt;</a></span>
<b>by </b><i><a class="ref" href="comput_1.html#T42" target="_self">Th42</a></i>
<br/>.= 
<a href="comput_1.html#K21" target="_self">[pred]</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="binarith.html#K13">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="comput_1.html#T93" target="_self">Th93</a></i>
;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <a href="comput_1.html#K22" target="_self">[-]</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">a<sub>1</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E4:178"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="178_3"><b>proof </b></a><div class="add">

<b>thus </b><a href="comput_1.html#K22" target="_self">[-]</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="finseq_4.html#K2">*&gt;</a></span> = 
<span class="p1">(<span class="default">1 <a href="comput_1.html#K8" target="_self">proj</a> 1</span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="binarith.html#K13">*&gt;</a></span>
<b>by </b><i><a class="ref" href="comput_1.html#T83" target="_self">Th83</a></i>
<br/>.= 
<span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> <a href="comput_1.html#K2" target="_self">.</a> 1
<b>by </b><i><a class="ref" href="comput_1.html#T42" target="_self">Th42</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 0</span>)</span> <a href="binarith.html#K5">-'</a> 0
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 0

;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E109</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="178_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:178_4"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>6</sub></font>]
 ;<br/><div><i><font color="Green">E111</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="178_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="178_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:178_4_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0 or ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 </span><b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="178_4_1_1_1"><b>suppose </b></a><a NAME="E1:178_4_1_1_1"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:178_4_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="binarith.html#T54">BINARITH:54</a></i>;<br/><a NAME="E3:178_4_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:178_4_1_1_1"/><b>then </b><i><font color="Green">E113</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i><a class="ref" href="real_2.html#T105">REAL_2:105</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="comput_1.html#K21" target="_self">[pred]</a>  <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2"><a href="funct_6.html#K6">&lt;:</a><span class="default"><span class="p3"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p4">(<span class="default">2 <a href="comput_1.html#K8" target="_self">proj</a> 2</span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span><a href="funct_6.html#K6">:&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> = 
0
<b>by </b><i><a class="txt" href="comput_1.html#E3:178"><i><font color="Green">E107</font></i></a>, <a class="txt" href="comput_1.html#E1:178_4_1_1_1"><i><font color="Green">E112</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="comput_1.html#E4:178_4_1_1_1"><i><font color="Green">E113</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="178_4_1_1_2"><b>suppose </b></a><a NAME="E1:178_4_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:178_4_1_1_2"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E4:178_4_1_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="comput_1.html#E3:178_4_1_1_2"><i><font color="Green">E112</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E5:178_4_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="comput_1.html#E3:178_4_1_1_2"><i><font color="Green">E112</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E6:178_4_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><a NAME="E7:178_4_1_1_2"/><b>then </b><i><font color="Green">E113</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/><b>thus </b><span class="p1">(<span class="default"><a href="comput_1.html#K21" target="_self">[pred]</a>  <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2"><a href="funct_6.html#K6">&lt;:</a><span class="default"><span class="p3"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p4">(<span class="default">2 <a href="comput_1.html#K8" target="_self">proj</a> 2</span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span><a href="funct_6.html#K6">:&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> = 
<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="comput_1.html#E3:178"><i><font color="Green">E107</font></i></a>, <a class="txt" href="comput_1.html#E3:178_4_1_1_2"><i><font color="Green">E112</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="comput_1.html#E7:178_4_1_1_2"><i><font color="Green">E113</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a href="comput_1.html#K22" target="_self">[-]</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> = 
<span class="p1">(<span class="default"><a href="comput_1.html#K16" target="_self">(1,2)-&gt;(1,?,2)</a> <span class="p2">(<span class="default"><a href="comput_1.html#K21" target="_self">[pred]</a>  <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3"><a href="funct_6.html#K6">&lt;:</a><span class="default"><span class="p4"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p5">(<span class="default">2 <a href="comput_1.html#K8" target="_self">proj</a> 2</span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span><a href="funct_6.html#K6">:&gt;</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<span class="p2">(<span class="default"><a href="comput_1.html#K22" target="_self">[-]</a>  <a href="funct_1.html#K1">.</a> <span class="p3"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span></span><a href="finseq_1.html#K11">*&gt;</a></span>
<b>by </b><i><a class="ref" href="comput_1.html#T85" target="_self">Th85</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="comput_1.html#E1:178_4"><i><font color="Green">E110</font></i></a>, <a class="txt" href="comput_1.html#E2:178_4"><i><font color="Green">E111</font></i></a>, <a class="ref" href="comput_1.html#T87" target="_self">Th87</a></i>
;<br/><b>hence </b><a NAME="E4:178_4"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E6:178"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="comput_1.html#E4:178"><i><font color="Green">E108</font></i></a>, <a class="txt" href="comput_1.html#E5:178"><i><font color="Green">E109</font></i></a>);<br/></i>
<b>hence </b><a NAME="E7:178"/>
<a href="comput_1.html#K22" target="_self">[-]</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


</div>
