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<b>let </b><font color="Maroon" title="c1">m</font>, <font color="Maroon" title="c2">n</font> be    <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>set </b><font color="Maroon" title="c3">prd</font> =  <a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <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> ] <b>means </b>( <font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 implies <a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c1">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> $1 );<br/>
<a NAME="E1:132"/><i><font color="Green" title="E88">A1</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<a NAME="E2:132"/><i><font color="Green" title="E89">A2</font></i>: 
for <font color="Olive" title="b1">n</font> being   <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>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="132_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">n</font> be    <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>assume </b><b>that </b><br/><a NAME="E1:132_1"/><i><font color="Green" title="E90">A3</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">n</font>]
 <b>and </b><br/><a NAME="E2:132_1"/><i><font color="Green" title="E91">A4</font></i>: 
<font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/>

<a NAME="E3:132_1"/><i><font color="Green" title="E92">A5</font></i>: 
<a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="comput_1.html#K12" title="COMPUT_1:func.12">PR-closure</a> <span class="p2">(<span class="default"><a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="comput_1.html#K13" title="COMPUT_1:func.13">composition-closure</a> <span class="p2">(<span class="default"><a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="comput_1.html#D23" target="_self" title="COMPUT_1:def.23">Def23</a></i>;<br/>
<a NAME="E4:132_1"/>
( <a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="comput_1.html#K12" title="COMPUT_1:func.12">PR-closure</a> <span class="p1">(<span class="default"><a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span> &amp;  <a href="comput_1.html#K12" title="COMPUT_1:func.12">PR-closure</a> <span class="p1">(<span class="default"><a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="comput_1.html#E3:132_1"><i><font color="Green" title="E92">A5</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E5:132_1"/><b>then </b><i><font color="Green" title="E93">A6</font></i>: 
<a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="132_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:132_1_1"/>
( <font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="comput_1.html#E2:132_1"><i><font color="Green" title="E91">A4</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="132_1_1_1"><b>suppose </b></a><a NAME="E1:132_1_1_1"/>
<font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:132_1_1_1"/>
<a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c1">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:132_1"><i><font color="Green" title="E90">A3</font></i></a>, <a class="txt" href="comput_1.html#E5:132_1"><i><font color="Green" title="E93">A6</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="132_1_1_2"><b>suppose </b></a><a NAME="E1:132_1_1_2"/>
<font color="Maroon" title="c1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:132_1_1_2"/>
<a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c1">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:132"/>
for <font color="Olive" title="b1">n</font> being   <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>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="comput_1.html#E1:132"><i><font color="Green" title="E88">A1</font></i></a>, <a class="txt" href="comput_1.html#E2:132"><i><font color="Green" title="E89">A2</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:132"/>
( <font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> implies <a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c1">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c2">n</font> )
 ;<br/>


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