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<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> =  <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> implies <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E1:134"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<a NAME="E2:134"/><i><font color="Green">E91</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="134_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:134_1"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 <b>and </b><br/><a NAME="E2:134_1"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<a NAME="E3:134_1"/><i><font color="Green">E94</font></i>: 
<a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="comput_1.html#K13" target="_self">PR-closure</a> <span class="p2">(<span class="default"><a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="comput_1.html#K14" target="_self">composition-closure</a> <span class="p2">(<span class="default"><a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="comput_1.html#D23" target="_self">Def23</a></i>;<br/>
<a NAME="E4:134_1"/>
( <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="comput_1.html#K13" target="_self">PR-closure</a> <span class="p1">(<span class="default"><a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp;  <a href="comput_1.html#K13" target="_self">PR-closure</a> <span class="p1">(<span class="default"><a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="comput_1.html#E3:134_1"><i><font color="Green">E94</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E5:134_1"/><b>then </b><i><font color="Green">E95</font></i>: 
<a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:134_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="comput_1.html#E2:134_1"><i><font color="Green">E93</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_1_1_1"><b>suppose </b></a><a NAME="E1:134_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:134_1_1_1"/>
<a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:134_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="comput_1.html#E5:134_1"><i><font color="Green">E95</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_1_1_2"><b>suppose </b></a><a NAME="E1:134_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:134_1_1_2"/>
<a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:134"/>
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#E1:134"><i><font color="Green">E90</font></i></a>, <a class="txt" href="comput_1.html#E2:134"><i><font color="Green">E91</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:134"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> implies <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>


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