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<b>let </b><font color="Maroon">g1</font> be    <a href="rfunct_3.html#M2">Element</a> of  <a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a> , <font color="Maroon">g2</font> be    <a href="rfunct_3.html#M2">Element</a> of  <a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a> ;<br/>

<b>reconsider </b><font color="Maroon">ff1</font> = <font color="Maroon">f1</font>, <font color="Maroon">ff2</font> = <font color="Maroon">f2</font> as    <a href="rfunct_3.html#M2">Element</a> of  <a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<b>given </b><font color="Maroon">F1</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><a NAME="E2:92_1_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</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="E3:92_1_2"/><i><font color="Green">E66</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">f1</font> implies <font color="Maroon">F1</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">f1</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> )
 <b>and </b><br/><a NAME="E4:92_1_2"/><i><font color="Green">E67</font></i>: 
( (  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">f1</font> ) implies <font color="Maroon">F1</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="E5:92_1_2"/><i><font color="Green">E136</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">S<sub>1</sub></font>[<font color="Olive">m</font>,<font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font>,<font color="Maroon">F1</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>,<font color="Maroon">p</font>,<font color="Maroon">i</font>,<font color="Maroon">f2</font>]
 ;<br/>

<b>given </b><font color="Maroon">F2</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><a NAME="E6:92_1_2"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">g2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</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="E7:92_1_2"/><i><font color="Green">E138</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">f1</font> implies <font color="Maroon">F2</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">f1</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> )
 <b>and </b><br/><a NAME="E8:92_1_2"/><i><font color="Green">E139</font></i>: 
( (  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">f1</font> ) implies <font color="Maroon">F2</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="E9:92_1_2"/><i><font color="Green">E142</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">S<sub>1</sub></font>[<font color="Olive">m</font>,<font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font>,<font color="Maroon">F2</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>,<font color="Maroon">p</font>,<font color="Maroon">i</font>,<font color="Maroon">f2</font>]
 ;<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">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E10:92_1_2"/><i><font color="Green">E143</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i><a class="ref" href="comput_1.html#T6" target="_self">Th6</a>, , , </i>;<br/>
<div><i><font color="Green">E153</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:92_1_2_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">m</font>]
 ;<br/><a NAME="E2:92_1_2_1"/><b>then </b>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">m</font>,<font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font>,<font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">p</font>,<font color="Maroon">i</font>,<font color="Maroon">f2</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">m</font>,<font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font>,<font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">p</font>,<font color="Maroon">i</font>,<font color="Maroon">ff2</font>] )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:92_1_2_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E12:92_1_2"/>
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">S<sub>2</sub></font>[<font color="Olive">m</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="E13:92_1_2"/>
<font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font>
 <b>by </b><i>, <a class="ref" href="comput_1.html#T7" target="_self">Th7</a></i>;<br/>


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