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<span class="kw">let </span><font color="Maroon" title="c1">m</font>, <font color="Maroon" title="c2">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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="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#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c1">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">n</font> )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">prd</font> =  <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a> ;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span>( <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#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c1">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1 );<br/>
<a NAME="E1:135"/><span class="lab"><font color="Green" title="E85">A1</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">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#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="135_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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">S<sub>2</sub></font>[<font color="Maroon" title="c4">n</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:135_1"/><span class="lab"><font color="Green" title="E86">A2</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">n</font>]
 <span class="kw">and </span><br/><a NAME="E2:135_1"/><span class="lab"><font color="Green" title="E87">A3</font></span>: 
<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#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c1">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/>

<a NAME="E3:135_1"/>
<a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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#K11" title="COMPUT_1:func.11">PR-closure</a> <span class="p2">(<span class="default"><a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  <a href="nat_1.html#K6" title="NAT_1:func.6">.</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#K12" title="COMPUT_1:func.12">composition-closure</a> <span class="p2">(<span class="default"><a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="comput_1.html#D23" target="_self" title="COMPUT_1:def.23">Def23</a></span>;<br/>
<a NAME="E4:135_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A4</font></span>: 
 <a href="comput_1.html#K11" title="COMPUT_1:func.11">PR-closure</a> <span class="p1">(<span class="default"><a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  <a href="nat_1.html#K6" title="NAT_1:func.6">.</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#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E5:135_1"/>
<a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="comput_1.html#K11" title="COMPUT_1:func.11">PR-closure</a> <span class="p1">(<span class="default"><a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E6:135_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A5</font></span>: 
<a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:135_1"><span class="lab"><font color="Green" title="E88">A4</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="135_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:135_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#K2" title="NAT_1:func.2">+</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#K2" title="NAT_1:func.2">+</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:135_1"><span class="lab"><font color="Green" title="E87">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="135_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:135_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#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c1">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:135_1_1_1"/>
<a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c1">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:135_1"><span class="lab"><font color="Green" title="E86">A2</font></span></a>, <a class="txt" href="#E6:135_1"><span class="lab"><font color="Green" title="E89">A5</font></span></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></span>; <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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="135_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:135_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#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c1">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:135_1_1_2"/>
<a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c1">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <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><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E2:135"/><span class="lab"><font color="Green" title="E86">A6</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<a NAME="E3:135"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">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>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E2:135"><span class="lab"><font color="Green" title="E86">A6</font></span></a>, <a class="txt" href="#E1:135"><span class="lab"><font color="Green" title="E85">A1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E4:135"/>
( <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#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c1">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">n</font> )
 ; <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>
