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<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">e1</font> be  <a href="seqm_3.html#V6">to-naturals</a> <a href="comput_1.html#V2" target="_self">from-natural-fseqs</a> <a href="comput_1.html#V4" target="_self">homogeneous</a> <a href="funct_1.html#NM1">Function</a>;<br/><b>let </b><font color="Maroon">e2</font> be  <a href="seqm_3.html#V6">to-naturals</a> <a href="comput_1.html#V2" target="_self">from-natural-fseqs</a> <a href="comput_1.html#V4" target="_self">homogeneous</a> <a href="funct_1.html#NM1">Function</a>;<br/>





<b>set </b><font color="Maroon">f1</font> = <font color="Maroon">e1</font>;<br/>
<b>set </b><font color="Maroon">f2</font> = <font color="Maroon">e2</font>;<br/>
<b>let </b><font color="Maroon">p</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:97"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">e1</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>

<b>consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:97"/><i><font color="Green">E49</font></i>: 
 <a href="comput_1.html#K9" target="_self">primrec</a> <font color="Maroon">e1</font>,<font color="Maroon">e2</font>,<font color="Maroon">i</font>,<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
 <b>and </b><br/><a NAME="E4:97"/><i><font color="Green">E51</font></i>: 
( ( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp;  <a href="finseq_3.html#K2">Del</a> <font color="Maroon">p</font>,<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">e1</font> implies <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,0</span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">e1</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">p</font>,<font color="Maroon">i</font></span>)</span></span>)</span> ) &amp; ( (  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> or  not  <a href="finseq_3.html#K2">Del</a> <font color="Maroon">p</font>,<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">e1</font> ) implies <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 <b>and </b><br/><a NAME="E5:97"/><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 <br/> ( ( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Olive">m</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Olive">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">e2</font> implies <font color="Maroon">F</font> <a href="funct_2.html#K3">.</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"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p4">(<span class="default"><font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">e2</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Olive">m</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p4"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p0">(<span class="default"><font color="Maroon">p</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Olive">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span></span>)</span></span>)</span> ) &amp; ( (  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> or  not <font color="Maroon">p</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font></span>)</span> or  not <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Olive">m</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Olive">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">e2</font> ) implies <font color="Maroon">F</font> <a href="funct_2.html#K3">.</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">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font> ) )
 <b>by </b><i/>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> ;<br/>
<a NAME="E6:97"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i>, , <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/>
<a NAME="E7:97"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/>
<a NAME="E8:97"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E9:97"/>
 <a href="comput_1.html#K9" target="_self">primrec</a> <font color="Maroon">e1</font>,<font color="Maroon">e2</font>,<font color="Maroon">i</font>,<font color="Maroon">p</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i/>;<br/>


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