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<span class="kw">set </span><font color="Maroon" title="c1">PROJ</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="comput_1.html#K5" title="COMPUT_1:func.5">proj</a> <font color="Olive" title="b2">i</font></span>)</span> where <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">i</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) </span>}</span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c2">prd</font> =  <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">UP</font> =  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a> ;<br/>
<a NAME="E1:136"/><span class="lab"><font color="Green" title="E86">A1</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E2:136"/><span class="lab"><font color="Green" title="E87">A2</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="comput_1.html#K3" title="COMPUT_1:func.3">const</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default">1 <a href="comput_1.html#K4" title="COMPUT_1:func.4">succ</a> 1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="comput_1.html#K5" title="COMPUT_1:func.5">proj</a> <font color="Olive" title="b2">i</font></span>)</span> where <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">i</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) </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="E3:136"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="comput_1.html#K3" title="COMPUT_1:func.3">const</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="comput_1.html#K3" title="COMPUT_1:func.3">const</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default">1 <a href="comput_1.html#K4" title="COMPUT_1:func.4">succ</a> 1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E4:136"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="comput_1.html#K3" title="COMPUT_1:func.3">const</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:136"><span class="lab"><font color="Green" title="E87">A2</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:136"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="comput_1.html#K3" title="COMPUT_1:func.3">const</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:136"><span class="lab"><font color="Green" title="E86">A1</font></span></a>, <a class="ref" href="card_5.html#T10" title="CARD_5:th.10">CARD_5:10</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="comput_1.html#D17" target="_self" title="COMPUT_1:def.17">COMPUT_1:def 17</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="comput_1.html#K4" title="COMPUT_1:func.4">succ</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  &amp; ( for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">i</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 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><font color="Olive" title="b1">n</font> <a href="comput_1.html#K5" title="COMPUT_1:func.5">proj</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  ) &amp;  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  is  <a href="comput_1.html#V4" title="COMPUT_1:attr.4">composition_closed</a>  &amp;  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  is  <a href="comput_1.html#V5" title="COMPUT_1:attr.5">primitive-recursion_closed</a>  )</span><br/>

<a NAME="E6:136"/>
1 <a href="comput_1.html#K4" title="COMPUT_1:func.4">succ</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="comput_1.html#K3" title="COMPUT_1:func.3">const</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default">1 <a href="comput_1.html#K4" title="COMPUT_1:func.4">succ</a> 1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E7:136"/><span class="kw">then </span>
1 <a href="comput_1.html#K4" title="COMPUT_1:func.4">succ</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:136"><span class="lab"><font color="Green" title="E87">A2</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:136"/>
1 <a href="comput_1.html#K4" title="COMPUT_1:func.4">succ</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:136"><span class="lab"><font color="Green" title="E86">A1</font></span></a>, <a class="ref" href="card_5.html#T10" title="CARD_5:th.10">CARD_5:10</a></span>; <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">n</font>, <font color="Olive" title="b2">i</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 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><font color="Olive" title="b1">n</font> <a href="comput_1.html#K5" title="COMPUT_1:func.5">proj</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  ) &amp;  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  is  <a href="comput_1.html#V4" title="COMPUT_1:attr.4">composition_closed</a>  &amp;  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  is  <a href="comput_1.html#V5" title="COMPUT_1:attr.5">primitive-recursion_closed</a>  )</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  is  <a href="comput_1.html#V4" title="COMPUT_1:attr.4">composition_closed</a>  &amp;  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  is  <a href="comput_1.html#V5" title="COMPUT_1:attr.5">primitive-recursion_closed</a>  )</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c4">n</font>, <font color="Maroon" title="c5">i</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"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> implies <font color="Maroon" title="c4">n</font> <a href="comput_1.html#K5" title="COMPUT_1:func.5">proj</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:136_1"/><span class="lab"><font color="Green" title="E88">A3</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font>
 <span class="kw">and </span><br/><a NAME="E2:136_1"/><span class="lab"><font color="Green" title="E89">A4</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font>
 ; <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="c4">n</font> <a href="comput_1.html#K5" title="COMPUT_1:func.5">proj</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a> </span><br/><a NAME="E3:136_1"/>
<font color="Maroon" title="c4">n</font> <a href="comput_1.html#K5" title="COMPUT_1:func.5">proj</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="comput_1.html#K5" title="COMPUT_1:func.5">proj</a> <font color="Olive" title="b2">i</font></span>)</span> where <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">i</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:136_1"><span class="lab"><font color="Green" title="E88">A3</font></span></a>, <a class="txt" href="#E2:136_1"><span class="lab"><font color="Green" title="E89">A4</font></span></a></span>;<br/><a NAME="E4:136_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">n</font> <a href="comput_1.html#K5" title="COMPUT_1:func.5">proj</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:136"><span class="lab"><font color="Green" title="E87">A2</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E5:136_1"/>
<font color="Maroon" title="c4">n</font> <a href="comput_1.html#K5" title="COMPUT_1:func.5">proj</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:136"><span class="lab"><font color="Green" title="E86">A1</font></span></a>, <a class="ref" href="card_5.html#T10" title="CARD_5:th.10">CARD_5:10</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><span class="kw">hereby </span> <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="comput_1.html#D15" target="_self" title="COMPUT_1:def.15">COMPUT_1:def 15</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  is  <a href="comput_1.html#V5" title="COMPUT_1:attr.5">primitive-recursion_closed</a> </span>
<div class="add"><span class="kw">deffunc </span><font color="Maroon">H<sub>1</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">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Olive" title="b2">F</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span></span>)</span> where <font color="Olive" title="b1">f</font> is    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K2" title="COMPUT_1:func.2">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Olive" title="b2">F</font> is   <a href="comput_1.html#V3" title="COMPUT_1:attr.3">with_the_same_arity</a>   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="comput_1.html#K2" title="COMPUT_1:func.2">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  : ( <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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 &amp;  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">F</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b2">F</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 ) </span>}</span> ;<br/><span class="kw">let </span><font color="Maroon" title="c4">f</font> be    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K2" title="COMPUT_1:func.2">HFuncs</a> <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"> for <font color="Olive" title="b1">F</font> being  <a href="comput_1.html#V3" title="COMPUT_1:attr.3">with_the_same_arity</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="comput_1.html#K2" title="COMPUT_1:func.2">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  &amp;  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">F</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  holds <br/><font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Olive">b<sub>3</sub></font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c5">F</font> be   <a href="comput_1.html#V3" title="COMPUT_1:attr.3">with_the_same_arity</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="comput_1.html#K2" title="COMPUT_1:func.2">HFuncs</a> <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="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  &amp;  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  implies <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:136_2"/><span class="lab"><font color="Green" title="E88">A5</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a> 
 <span class="kw">and </span><br/><a NAME="E2:136_2"/><span class="lab"><font color="Green" title="E89">A6</font></span>: 
 <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font>
 <span class="kw">and </span><br/><a NAME="E3:136_2"/><span class="lab"><font color="Green" title="E90">A7</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a> </span><br/><span class="kw">consider </span><font color="Maroon" title="c6">kf</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:136_2"/><span class="lab"><font color="Green" title="E91">A8</font></span>: 
<font color="Maroon" title="c6">kf</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a> 
 <span class="kw">and </span><br/><a NAME="E6:136_2"/><span class="lab"><font color="Green" title="E92">A9</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">kf</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:136_2"><span class="lab"><font color="Green" title="E88">A5</font></span></a>, <a class="ref" href="card_5.html#T10" title="CARD_5:th.10">CARD_5:10</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">kf</font> = <font color="Maroon" title="c6">kf</font> as    <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">by</span> <span class="lab"><a class="txt" href="#E5:136_2"><span class="lab"><font color="Green" title="E91">A8</font></span></a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="136_2_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="E1:136_2_1"/>
(  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="136_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:136_2_1_1"/>
 <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a> </span><br/><div class="add"><a NAME="E2:136_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:136_2"><span class="lab"><font color="Green" title="E89">A6</font></span></a></span>;<br/><a NAME="E3:136_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A10</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">F</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="c7">kf</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a>, <a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></span>;<br/><a NAME="E4:136_2_1_1"/>
<font color="Maroon" title="c5">F</font> is   <a href="comput_1.html#V3" title="COMPUT_1:attr.3">with_the_same_arity</a>   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="comput_1.html#K2" title="COMPUT_1:func.2">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/><a NAME="E5:136_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c5">F</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">kf</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:136_2"><span class="lab"><font color="Green" title="E89">A6</font></span></a>, <a class="txt" href="#E6:136_2"><span class="lab"><font color="Green" title="E92">A9</font></span></a>, <a class="txt" href="#E3:136_2_1_1"><span class="lab"><font color="Green" title="E93">A10</font></span></a></span>;<br/><a NAME="E6:136_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c5">F</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c7">kf</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">kf</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E7:136_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c5">F</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c7">kf</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="c7">kf</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E8:136_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c5">F</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c7">kf</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="comput_1.html#D23" target="_self" title="COMPUT_1:def.23">Def23</a></span>;<br/><span class="kw">hence </span><a NAME="E9:136_2_1_1"/>
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c5">F</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:136"><span class="lab"><font color="Green" title="E86">A1</font></span></a>, <a class="ref" href="card_5.html#T10" title="CARD_5:th.10">CARD_5:10</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="136_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:136_2_1_2"/><span class="lab"><font color="Green" title="E93">A11</font></span>: 
 <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a> </span><br/><div class="add"><span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">k</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 <br/>( <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="Olive" title="b1">k</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> );<br/><a NAME="E2:136_2_1_2"/><span class="lab"><font color="Green" title="E94">A12</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">y</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">x</font>,<font color="Olive" title="b2">y</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_2_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span> implies  ex <font color="Olive" title="b1">y</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="Maroon" title="c8">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:136_2_1_2_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</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="Maroon" title="c8">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<a NAME="E2:136_2_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E3:136_2_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:136_2_1_2_1"/><span class="lab"><font color="Green" title="E95">A13</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a> 
 <span class="kw">and </span><br/><a NAME="E6:136_2_1_2_1"/><span class="lab"><font color="Green" title="E96">A14</font></span>: 
<font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:136_2"><span class="lab"><font color="Green" title="E90">A7</font></span></a>, <a class="ref" href="card_5.html#T10" title="CARD_5:th.10">CARD_5:10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">k</font> = <font color="Maroon" title="c9">k</font> as    <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">by</span> <span class="lab"><a class="txt" href="#E5:136_2_1_2_1"><span class="lab"><font color="Green" title="E95">A13</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c10">k</font>
; <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="c8">x</font>,<font color="Maroon" title="c10">k</font>]</span><br/>

<span class="kw">take </span><font color="Maroon" title="c11">k1</font> = <font color="Maroon" title="c10">k</font>; <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="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c11">k1</font> &amp; <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k1</font> )</span><br/>

<span class="kw">thus </span><a NAME="E8:136_2_1_2_1"/>
<font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c11">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:136_2_1_2_1"><span class="lab"><font color="Green" title="E96">A14</font></span></a></span>; <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="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k1</font></span><br/>

<span class="kw">thus </span><a NAME="E9:136_2_1_2_1"/>
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k1</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><span class="kw">end;</span></div><span class="kw">consider </span><font color="Maroon" title="c8">fKF</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E4:136_2_1_2"/><span class="lab"><font color="Green" title="E95">A15</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c8">fKF</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="monoid_1.html#S1" title="MONOID_1:sch.1">MONOID_1:sch 1</a>(<a class="txt" href="#E2:136_2_1_2"><span class="lab"><font color="Green" title="E94">A12</font></span></a>);<br/></span><span class="kw">set </span><font color="Maroon" title="c9">KF</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">fKF</font>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">KF</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">fKF</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:136_2"><span class="lab"><font color="Green" title="E89">A6</font></span></a>, <a class="txt" href="#E1:136_2_1_2"><span class="lab"><font color="Green" title="E93">A11</font></span></a>, <a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c11">K</font> =  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c10">KF</font>;<br/><span class="kw">set </span><font color="Maroon" title="c12">k</font> =  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c7">kf</font>,<span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c10">KF</font></span>)</span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">k</font> =  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c7">kf</font>,<span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c10">KF</font></span>)</span> as    <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">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><a NAME="E7:136_2_1_2"/><span class="lab"><font color="Green" title="E96">A16</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">fKF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E8:136_2_1_2"/><span class="lab"><font color="Green" title="E97">A17</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">F</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="c13">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_2_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">F</font> or <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c13">k</font> )</span><br/>

<a NAME="E1:136_2_1_2_2"/><span class="lab"><font color="Green" title="E98">A18</font></span>: 
 <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c10">KF</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:136_2_1_2_2"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">F</font>
 ; <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="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c13">k</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">d</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:136_2_1_2_2"/><span class="lab"><font color="Green" title="E99">A19</font></span>: 
<font color="Maroon" title="c15">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">F</font>
 <span class="kw">and </span><br/><a NAME="E5:136_2_1_2_2"/><span class="lab"><font color="Green" title="E100">A20</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E6:136_2_1_2_2"/><span class="lab"><font color="Green" title="E101">A21</font></span>: 
<font color="Maroon" title="c15">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:136_2_1_2_2"><span class="lab"><font color="Green" title="E99">A19</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">m</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> <span class="kw"> such that </span><br/><a NAME="E8:136_2_1_2_2"/><span class="lab"><font color="Green" title="E102">A22</font></span>: 
<font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c16">m</font>
 <span class="kw">and </span><br/><a NAME="E9:136_2_1_2_2"/><span class="lab"><font color="Green" title="E103">A23</font></span>: 
<font color="Maroon" title="c8">fKF</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c15">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:136_2_1_2"><span class="lab"><font color="Green" title="E95">A15</font></span></a></span>;<br/>
<a NAME="E10:136_2_1_2_2"/>
<font color="Maroon" title="c16">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">KF</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:136_2_1_2"><span class="lab"><font color="Green" title="E96">A16</font></span></a>, <a class="txt" href="#E6:136_2_1_2_2"><span class="lab"><font color="Green" title="E101">A21</font></span></a>, <a class="txt" href="#E9:136_2_1_2_2"><span class="lab"><font color="Green" title="E103">A23</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<a NAME="E11:136_2_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c10">KF</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D8" title="XXREAL_2:def.8">XXREAL_2:def 8</a></span>;<br/>
<a NAME="E12:136_2_1_2_2"/><span class="kw">then </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="c16">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="c13">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:136_2_1_2_2"><span class="lab"><font color="Green" title="E98">A18</font></span></a>, <a class="ref" href="comput_1.html#T79" target="_self" title="COMPUT_1:th.79">Th79</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:136_2_1_2_2"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c13">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:136_2_1_2_2"><span class="lab"><font color="Green" title="E100">A20</font></span></a>, <a class="txt" href="#E8:136_2_1_2_2"><span class="lab"><font color="Green" title="E102">A22</font></span></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><a NAME="E9:136_2_1_2"/><span class="lab"><font color="Green" title="E98">A24</font></span>: 
<font color="Maroon" title="c5">F</font> is   <a href="comput_1.html#V3" title="COMPUT_1:attr.3">with_the_same_arity</a>   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="comput_1.html#K2" title="COMPUT_1:func.2">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/><a NAME="E10:136_2_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="c7">kf</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="c13">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="comput_1.html#T79" target="_self" title="COMPUT_1:th.79">Th79</a>, <a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/><a NAME="E11:136_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c5">F</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:136_2"><span class="lab"><font color="Green" title="E89">A6</font></span></a>, <a class="txt" href="#E6:136_2"><span class="lab"><font color="Green" title="E92">A9</font></span></a>, <a class="txt" href="#E8:136_2_1_2"><span class="lab"><font color="Green" title="E97">A17</font></span></a>, <a class="txt" href="#E9:136_2_1_2"><span class="lab"><font color="Green" title="E98">A24</font></span></a></span>;<br/><a NAME="E12:136_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c5">F</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c13">k</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E13:136_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c5">F</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c13">k</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="c13">k</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E14:136_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c5">F</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c13">k</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="comput_1.html#D23" target="_self" title="COMPUT_1:def.23">Def23</a></span>;<br/><span class="kw">hence </span><a NAME="E15:136_2_1_2"/>
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c5">F</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:136"><span class="lab"><font color="Green" title="E86">A1</font></span></a>, <a class="ref" href="card_5.html#T10" title="CARD_5:th.10">CARD_5:10</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><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div>

<span class="kw">deffunc </span><font color="Maroon">H<sub>1</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">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">{<span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</font> is    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K2" title="COMPUT_1:func.2">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  :  ex <font color="Olive" title="b2">f1</font>, <font color="Olive" title="b3">f2</font> being   <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K2" title="COMPUT_1:func.2">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ex <font color="Olive" title="b4">i</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 <br/>( <font color="Olive" title="b2">f1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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 &amp; <font color="Olive" title="b3">f2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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 &amp; <font color="Olive" title="b1">g</font> <a href="comput_1.html#R1" title="COMPUT_1:pred.1">is_primitive-recursively_expressed_by</a> <font color="Olive" title="b2">f1</font>,<font color="Olive" title="b3">f2</font>,<font color="Olive" title="b4">i</font> ) </span>}</span> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">g</font>, <font color="Maroon" title="c5">f1</font>, <font color="Maroon" title="c6">f2</font> be    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K2" title="COMPUT_1:func.2">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="comput_1.html#D16" target="_self" title="COMPUT_1:def.16">COMPUT_1:def 16</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">i</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" title="c4">g</font> <a href="comput_1.html#R1" title="COMPUT_1:pred.1">is_primitive-recursively_expressed_by</a> <font color="Maroon" title="c5">f1</font>,<font color="Maroon" title="c6">f2</font>,<font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c5">f1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  &amp; <font color="Maroon" title="c6">f2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  holds <br/><font color="Maroon" title="c4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c7">i</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="c4">g</font> <a href="comput_1.html#R1" title="COMPUT_1:pred.1">is_primitive-recursively_expressed_by</a> <font color="Maroon" title="c5">f1</font>,<font color="Maroon" title="c6">f2</font>,<font color="Maroon" title="c7">i</font> &amp; <font color="Maroon" title="c5">f1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  &amp; <font color="Maroon" title="c6">f2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  implies <font color="Maroon" title="c4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E11:136"/><span class="lab"><font color="Green" title="E88">A25</font></span>: 
<font color="Maroon" title="c4">g</font> <a href="comput_1.html#R1" title="COMPUT_1:pred.1">is_primitive-recursively_expressed_by</a> <font color="Maroon" title="c5">f1</font>,<font color="Maroon" title="c6">f2</font>,<font color="Maroon" title="c7">i</font>
 <span class="kw">and </span><br/><a NAME="E12:136"/><span class="lab"><font color="Green" title="E89">A26</font></span>: 
<font color="Maroon" title="c5">f1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a> 
 <span class="kw">and </span><br/><a NAME="E13:136"/><span class="lab"><font color="Green" title="E90">A27</font></span>: 
<font color="Maroon" title="c6">f2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</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="c4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a> </span><br/>

<span class="kw">consider </span><font color="Maroon" title="c8">k2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E15:136"/><span class="lab"><font color="Green" title="E91">A28</font></span>: 
<font color="Maroon" title="c8">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a> 
 <span class="kw">and </span><br/><a NAME="E16:136"/><span class="lab"><font color="Green" title="E92">A29</font></span>: 
<font color="Maroon" title="c6">f2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:136"><span class="lab"><font color="Green" title="E90">A27</font></span></a>, <a class="ref" href="card_5.html#T10" title="CARD_5:th.10">CARD_5:10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">k2</font> = <font color="Maroon" title="c8">k2</font> as    <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">by</span> <span class="lab"><a class="txt" href="#E15:136"><span class="lab"><font color="Green" title="E91">A28</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">k1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E19:136"/><span class="lab"><font color="Green" title="E93">A30</font></span>: 
<font color="Maroon" title="c10">k1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a> 
 <span class="kw">and </span><br/><a NAME="E20:136"/><span class="lab"><font color="Green" title="E94">A31</font></span>: 
<font color="Maroon" title="c5">f1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:136"><span class="lab"><font color="Green" title="E89">A26</font></span></a>, <a class="ref" href="card_5.html#T10" title="CARD_5:th.10">CARD_5:10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">k1</font> = <font color="Maroon" title="c10">k1</font> as    <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">by</span> <span class="lab"><a class="txt" href="#E19:136"><span class="lab"><font color="Green" title="E93">A30</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="136_3"><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="E1:136_3"/>
( <font color="Maroon" title="c11">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k2</font> or <font color="Maroon" title="c9">k2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="136_3_1"><span class="kw">suppose </span></a><a NAME="E1:136_3_1"/>
<font color="Maroon" title="c11">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k2</font>
 ; <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="c4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a> </span><br/><div class="add"><a NAME="E2:136_3_1"/><span class="kw">then </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="c11">k1</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="c9">k2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="comput_1.html#T79" target="_self" title="COMPUT_1:th.79">Th79</a></span>;<br/><a NAME="E3:136_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c9">k2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:136"><span class="lab"><font color="Green" title="E88">A25</font></span></a>, <a class="txt" href="#E20:136"><span class="lab"><font color="Green" title="E94">A31</font></span></a>, <a class="txt" href="#E16:136"><span class="lab"><font color="Green" title="E92">A29</font></span></a></span>;<br/><a NAME="E4:136_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c9">k2</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c9">k2</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E5:136_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c9">k2</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="c9">k2</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E6:136_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c9">k2</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="comput_1.html#D23" target="_self" title="COMPUT_1:def.23">Def23</a></span>;<br/><span class="kw">hence </span><a NAME="E7:136_3_1"/>
<font color="Maroon" title="c4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:136"><span class="lab"><font color="Green" title="E86">A1</font></span></a>, <a class="ref" href="card_5.html#T10" title="CARD_5:th.10">CARD_5:10</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="136_3_2"><span class="kw">suppose </span></a><a NAME="E1:136_3_2"/>
<font color="Maroon" title="c9">k2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k1</font>
 ; <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="c4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a> </span><br/><div class="add"><a NAME="E2:136_3_2"/><span class="kw">then </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="c9">k2</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="c11">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="comput_1.html#T79" target="_self" title="COMPUT_1:th.79">Th79</a></span>;<br/><a NAME="E3:136_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c11">k1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:136"><span class="lab"><font color="Green" title="E88">A25</font></span></a>, <a class="txt" href="#E20:136"><span class="lab"><font color="Green" title="E94">A31</font></span></a>, <a class="txt" href="#E16:136"><span class="lab"><font color="Green" title="E92">A29</font></span></a></span>;<br/><a NAME="E4:136_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c11">k1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c11">k1</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E5:136_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c11">k1</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="c11">k1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E6:136_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c11">k1</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="comput_1.html#D23" target="_self" title="COMPUT_1:def.23">Def23</a></span>;<br/><span class="kw">hence </span><a NAME="E7:136_3_2"/>
<font color="Maroon" title="c4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <a href="comput_1.html#K13" title="COMPUT_1:func.13">PrimRec-Approximation</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:136"><span class="lab"><font color="Green" title="E86">A1</font></span></a>, <a class="ref" href="card_5.html#T10" title="CARD_5:th.10">CARD_5:10</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><span class="kw">end;</span></div>

</div>
