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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ,  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a> </span>)</span>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a> </span>)</span> = <span class="p1">(<span class="default"><a href="comput_1.html#K13" target="_self">PR-closure</a> <font color="Maroon">a<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="comput_1.html#K14" target="_self">composition-closure</a> <font color="Maroon">a<sub>2</sub></font></span>)</span>;<br/>
<b>let </b><font color="Maroon">it1</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a> </span>)</span>, <font color="Maroon">it2</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a> </span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:133_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">it1</font> <a href="prob_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="comput_1.html#K12" target="_self">initial-funcs</a> 
 <b>and </b><br/><a NAME="E2:133_1_2"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">it1</font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="comput_1.html#K13" target="_self">PR-closure</a> <span class="p2">(<span class="default"><font color="Maroon">it1</font> <a href="prob_1.html#K1">.</a> <font color="Olive">m</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="comput_1.html#K14" target="_self">composition-closure</a> <span class="p2">(<span class="default"><font color="Maroon">it1</font> <a href="prob_1.html#K1">.</a> <font color="Olive">m</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E3:133_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">it2</font> <a href="prob_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="comput_1.html#K12" target="_self">initial-funcs</a> 
 <b>and </b><br/><a NAME="E4:133_1_2"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">it2</font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="comput_1.html#K13" target="_self">PR-closure</a> <span class="p2">(<span class="default"><font color="Maroon">it2</font> <a href="prob_1.html#K1">.</a> <font color="Olive">m</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="comput_1.html#K14" target="_self">composition-closure</a> <span class="p2">(<span class="default"><font color="Maroon">it2</font> <a href="prob_1.html#K1">.</a> <font color="Olive">m</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E5:133_1_2"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">it1</font> <a href="prob_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="comput_1.html#K12" target="_self">initial-funcs</a>  &amp; ( for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">it1</font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">m</font>,<font color="Maroon">it1</font> <a href="prob_1.html#K1">.</a> <font color="Olive">m</font>) ) )
 
<b>by </b><i>, <a class="ref" href="comput_1.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E6:133_1_2"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">it2</font> <a href="prob_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="comput_1.html#K12" target="_self">initial-funcs</a>  &amp; ( for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">it2</font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">m</font>,<font color="Maroon">it2</font> <a href="prob_1.html#K1">.</a> <font color="Olive">m</font>) ) )
 
<b>by </b><i>, <a class="ref" href="comput_1.html#D2" target="_self">Def2</a></i>;<br/>
<b>thus </b><a NAME="E7:133_1_2"/>
<font color="Maroon">it1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">it2</font>
 <b>from </b><i><a class="ref" href="recdef_1.html#S12">RECDEF_1:sch 12</a>(<a class="ref" href="comput_1.html#D3" target="_self">Def3</a>, <a class="ref" href="comput_1.html#D4" target="_self">Def4</a>);</i><br/>


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