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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <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> = <span class="p1">(<span class="default"><a href="comput_1.html#K10" title="COMPUT_1:func.10">PR-closure</a> $2</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#K11" title="COMPUT_1:func.11">composition-closure</a> $2</span>)</span>;<br/>
<b>let </b><font color="Maroon" title="c1">it1</font>, <font color="Maroon" title="c2">it2</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <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 class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">it1</font> <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>  <a href="comput_1.html#K9" title="COMPUT_1:func.9">initial-funcs</a>  &amp; ( for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c1">it1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" title="COMPUT_1:func.10">PR-closure</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">it1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</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#K11" title="COMPUT_1:func.11">composition-closure</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">it1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span> ) &amp; <font color="Maroon" title="c2">it2</font> <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>  <a href="comput_1.html#K9" title="COMPUT_1:func.9">initial-funcs</a>  &amp; ( for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c2">it2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" title="COMPUT_1:func.10">PR-closure</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">it2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</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#K11" title="COMPUT_1:func.11">composition-closure</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">it2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span> ) implies <font color="Maroon" title="c1">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">it2</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:129_1_2"/><i><font color="Green" title="E85">A3</font></i>: 
<font color="Maroon" title="c1">it1</font> <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>  <a href="comput_1.html#K9" title="COMPUT_1:func.9">initial-funcs</a> 
 <b>and </b><br/><a NAME="E2:129_1_2"/><i><font color="Green" title="E86">A4</font></i>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c1">it1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" title="COMPUT_1:func.10">PR-closure</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">it1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</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#K11" title="COMPUT_1:func.11">composition-closure</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">it1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E3:129_1_2"/><i><font color="Green" title="E87">A5</font></i>: 
<font color="Maroon" title="c2">it2</font> <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>  <a href="comput_1.html#K9" title="COMPUT_1:func.9">initial-funcs</a> 
 <b>and </b><br/><a NAME="E4:129_1_2"/><i><font color="Green" title="E88">A6</font></i>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c2">it2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" title="COMPUT_1:func.10">PR-closure</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">it2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</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#K11" title="COMPUT_1:func.11">composition-closure</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">it2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">it2</font></span><br/>

<a NAME="E5:129_1_2"/><i><font color="Green" title="E89">A7</font></i>: 
<font color="Maroon" title="c1">it1</font> <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>  <a href="comput_1.html#K9" title="COMPUT_1:func.9">initial-funcs</a> 
 
<b>by </b><i><a class="txt" href="#E1:129_1_2"><i><font color="Green" title="E85">A3</font></i></a></i>;<br/>
<a NAME="E6:129_1_2"/><i><font color="Green" title="E90">B7</font></i>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c1">it1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">m</font>,<font color="Maroon" title="c1">it1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</font>)
 
<b>by </b><i><a class="txt" href="#E2:129_1_2"><i><font color="Green" title="E86">A4</font></i></a></i>;<br/>
<a NAME="E7:129_1_2"/><i><font color="Green" title="E91">A8</font></i>: 
<font color="Maroon" title="c2">it2</font> <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>  <a href="comput_1.html#K9" title="COMPUT_1:func.9">initial-funcs</a> 
 
<b>by </b><i><a class="txt" href="#E3:129_1_2"><i><font color="Green" title="E87">A5</font></i></a></i>;<br/>
<a NAME="E8:129_1_2"/><i><font color="Green" title="E92">B8</font></i>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c2">it2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">m</font>,<font color="Maroon" title="c2">it2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</font>)
 
<b>by </b><i><a class="txt" href="#E4:129_1_2"><i><font color="Green" title="E88">A6</font></i></a></i>;<br/>
<b>thus </b><a NAME="E9:129_1_2"/>
<font color="Maroon" title="c1">it1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c2">it2</font>
 <b>from </b><i><a class="ref" href="nat_1.html#S16" title="NAT_1:sch.16">NAT_1:sch 16</a>(<a class="txt" href="#E5:129_1_2"><i><font color="Green" title="E89">A7</font></i></a>, <a class="txt" href="#E6:129_1_2"><i><font color="Green" title="E90">B7</font></i></a>, <a class="txt" href="#E7:129_1_2"><i><font color="Green" title="E91">A8</font></i></a>, <a class="txt" href="#E8:129_1_2"><i><font color="Green" title="E92">B8</font></i></a>);</i> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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