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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">p1</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">p2</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1"/>
(  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p1</font> or  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p2</font> or ( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p2</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_1"><b>suppose </b></a><a NAME="E1:95_1_1"/>
(  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p1</font> or  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p2</font> )
 ;<br/><div class="add"><a NAME="E2:95_1_1"/><b>then </b>
(  <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">p1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <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">p2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:95_1_1"/>
 <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">p1</font> <a href="partfun1.html#R1">tolerates</a>  <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">p2</font>
 <b>by </b><i><a class="ref" href="partfun1.html#T141">PARTFUN1:141</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2"><b>suppose </b></a><a NAME="E1:95_1_2"/><i><font color="Green">E48</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">p1</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p2</font> )
 ;<br/><div class="add"><b>consider </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><br/><a NAME="E3:95_1_2"/><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">p1</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">p1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
 <b>and </b><br/><a NAME="E4:95_1_2"/><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">p1</font> &amp;  <a href="finseq_3.html#K2">Del</a> <font color="Maroon">p1</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">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">p1</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">p1</font>,<font color="Maroon">i</font></span>)</span></span>)</span> )
 <b>and </b><br/><a NAME="E5:95_1_2"/><i><font color="Green">E52</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">p1</font> or  not  <a href="finseq_3.html#K2">Del</a> <font color="Maroon">p1</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">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="E6:95_1_2"/><i><font color="Green">E53</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">p1</font>,<font color="Maroon">i</font>,<font color="Maroon">e2</font>]
 <b>by </b><i><a class="txt" href="comput_1.html#E9:95_1_2"><i><font color="Green">E55</font></i></a></i>;<br/><b>consider </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><br/><a NAME="E8:95_1_2"/><i><font color="Green">E54</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">p2</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">p2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
 <b>and </b><br/><a NAME="E9:95_1_2"/><i><font color="Green">E55</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">p2</font> &amp;  <a href="finseq_3.html#K2">Del</a> <font color="Maroon">p2</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">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">p2</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">p2</font>,<font color="Maroon">i</font></span>)</span></span>)</span> )
 <b>and </b><br/><a NAME="E10:95_1_2"/><i><font color="Green">E57</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">p2</font> or  not  <a href="finseq_3.html#K2">Del</a> <font color="Maroon">p2</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">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="E11:95_1_2"/><i><font color="Green">E58</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">p2</font>,<font color="Maroon">i</font>,<font color="Maroon">e2</font>]
 <b>by </b><i><a class="txt" href="comput_1.html#E9:95_1_2"><i><font color="Green">E55</font></i></a></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>for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</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">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> holds <br/> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> );<br/><a NAME="E12:95_1_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:95_1_2_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">x</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">F1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span>
 ;<br/>

<a NAME="E2:95_1_2_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <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">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,0</span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, , , <a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/>
<a NAME="E3:95_1_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,0
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:95_1_2_1"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 ;<br/>


</div><b>end;</b></div><div><i><font color="Green">E66</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_2"><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="E1:95_1_2_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">m</font>]
 ;<br/><b>thus </b><a NAME="E2:95_1_2_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:95_1_2_2_1"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">x</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">F1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:95_1_2_2_1"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<b>set </b><font color="Maroon">p1c</font> = <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">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">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_2_2_1_1"/>
( ( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">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">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">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">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">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> ) or  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p1</font> or  not <font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">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">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> or  not <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">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">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">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> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:95_1_2_2_1_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">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">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">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">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">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> )
 ;<br/><div class="add"><a NAME="E2:95_1_2_2_1_1_1"/><b>then </b>
<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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">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">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p4">(<span class="default"><font color="Maroon">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">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">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">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p0">(<span class="default"><font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><b>then </b> <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p5">(<span class="default"><font color="Maroon">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="p3">(<span class="default"><font color="Maroon">e2</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p5"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>
;<br/><a NAME="E4:95_1_2_2_1_1_1"/><b>then </b><i><font color="Green">E138</font></i>: 
( <font color="Maroon">x</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">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="comput_1.html#T10" target="_self">Th10</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>thus </b><a NAME="E5:95_1_2_2_1_1_1"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_2_1_1_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_2_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_2_2_1_1_1_2_1"/>
( <font color="Maroon">x</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">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 </span><b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_2_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:95_1_2_2_1_1_1_2_1_1"/>
<font color="Maroon">x</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">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:95_1_2_2_1_1_1_2_1_1"/><i><font color="Green">E139</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 <b>by </b><i/>;<br/><a NAME="E4:95_1_2_2_1_1_1_2_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="comput_1.html#T11" target="_self">Th11</a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E5:95_1_2_2_1_1_1_2_1_1"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_2_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:95_1_2_2_1_1_1_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:95_1_2_2_1_1_1_2_1_2"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:95_1_2_2_1_1_2"/>
(  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p1</font> or  not <font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">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">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> or  not <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">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">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">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> )
 ;<br/><div class="add"><a NAME="E2:95_1_2_2_1_1_2"/><b>then </b>
<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> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:95_1_2_2_1_1_2"/><i><font color="Green">E142</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 <b>by </b><i>, <a class="ref" href="comput_1.html#T10" target="_self">Th10</a></i>;<br/><a NAME="E5:95_1_2_2_1_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="comput_1.html#T11" target="_self">Th11</a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E6:95_1_2_2_1_1_2"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><a NAME="E14:95_1_2"/><i><font color="Green">E143</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>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>(<a class="ref" href="comput_1.html#T8" target="_self">Th8</a>, );<br/></i><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</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">F2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> holds <br/> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> );<br/><a NAME="E15:95_1_2"/><i><font color="Green">E153</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:95_1_2_3"/><i><font color="Green">E154</font></i>: 
<font color="Maroon">x</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">F2</font> <a href="funct_2.html#K3">.</a> 0</span>)</span>
 ;<br/>

<a NAME="E2:95_1_2_3"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <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">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,0</span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, <a class="txt" href="comput_1.html#E13:95_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="comput_1.html#T11" target="_self">Th11</a>, <a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/>
<a NAME="E3:95_1_2_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,0
 
<b>by </b><i><a class="ref" href="comput_1.html#T11" target="_self">Th11</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:95_1_2_3"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 ;<br/>


</div><b>end;</b></div><div><i><font color="Green">E155</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_4"><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="E1:95_1_2_4"/><i><font color="Green">E156</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">m</font>]
 ;<br/><b>thus </b><a NAME="E2:95_1_2_4"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:95_1_2_4_1"/><i><font color="Green">E157</font></i>: 
<font color="Maroon">x</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">F2</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:95_1_2_4_1"/><i><font color="Green">E158</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<b>set </b><font color="Maroon">p2c</font> = <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">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">F2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_2_4_1_1"/>
( ( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p2</font> &amp; <font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">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">F2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">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">F2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">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> ) or  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p2</font> or  not <font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">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">F2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> or  not <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">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">F2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">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> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_4_1_1_1"><b>suppose </b></a><a NAME="E1:95_1_2_4_1_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p2</font> &amp; <font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">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">F2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">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">F2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">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> )
 ;<br/><div class="add"><a NAME="E2:95_1_2_4_1_1_1"/><b>then </b>
<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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">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">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p4">(<span class="default"><font color="Maroon">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">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">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">F2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p0">(<span class="default"><font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><b>then </b> <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p5">(<span class="default"><font color="Maroon">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="p3">(<span class="default"><font color="Maroon">e2</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p5"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>
;<br/><a NAME="E4:95_1_2_4_1_1_1"/><b>then </b><i><font color="Green">E160</font></i>: 
( <font color="Maroon">x</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">F2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>thus </b><a NAME="E5:95_1_2_4_1_1_1"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_4_1_1_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_4_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_2_4_1_1_1_2_1"/>
( <font color="Maroon">x</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">F2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 </span><b>by </b><i><a class="ref" href="comput_1.html#T13" target="_self">Th13</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_4_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:95_1_2_4_1_1_1_2_1_1"/>
<font color="Maroon">x</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">F2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:95_1_2_4_1_1_1_2_1_1"/><i><font color="Green">E161</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 <b>by </b><i/>;<br/><a NAME="E4:95_1_2_4_1_1_1_2_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="comput_1.html#T12" target="_self">Th12</a>, <a class="ref" href="comput_1.html#D1" target="_self">Def1</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E5:95_1_2_4_1_1_1_2_1_1"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="comput_1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_4_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:95_1_2_4_1_1_1_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:95_1_2_4_1_1_1_2_1_2"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_4_1_1_2"><b>suppose </b></a><a NAME="E1:95_1_2_4_1_1_2"/>
(  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p2</font> or  not <font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">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">F2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> or  not <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">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">F2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">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> )
 ;<br/><div class="add"><a NAME="E2:95_1_2_4_1_1_2"/><b>then </b>
<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> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:95_1_2_4_1_1_2"/><i><font color="Green">E162</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E5:95_1_2_4_1_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="comput_1.html#T12" target="_self">Th12</a>, <a class="ref" href="comput_1.html#T13" target="_self">Th13</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E6:95_1_2_4_1_1_2"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="comput_1.html#T13" target="_self">Th13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><a NAME="E17:95_1_2"/><i><font color="Green">E163</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>3</sub></font>[<font color="Olive">m</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="ref" href="comput_1.html#T10" target="_self">Th10</a>, <a class="ref" href="comput_1.html#T11" target="_self">Th11</a>);<br/></i><a NAME="E18:95_1_2"/><i><font color="Green">E187</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">F1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font> or <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_5"><b>proof </b></a><div class="add">

<div><i><font color="Green">E188</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:95_1_2_5_1"/><i><font color="Green">E189</font></i>: 
<font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,0 <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,0
 ;<br/><b>defpred </b><font color="Maroon">S<sub>4</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="E2:95_1_2_5_1"/><i><font color="Green">E191</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[0]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_5_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_5_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_2_5_1_1_1"/>
( ( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p1</font> &amp;  <a href="finseq_3.html#K2">Del</a> <font color="Maroon">p1</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> ) or  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p1</font> or  not  <a href="finseq_3.html#K2">Del</a> <font color="Maroon">p1</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> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_5_1_1_1_1"><b>suppose </b></a><a NAME="E1:95_1_2_5_1_1_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p1</font> &amp;  <a href="finseq_3.html#K2">Del</a> <font color="Maroon">p1</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> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:95_1_2_5_1_1_1_1"/>
<font color="Maroon">S<sub>4</sub></font>[0]
 <b>by </b><i><a class="ref" href="comput_1.html#T7" target="_self">Th7</a>, , , <a class="ref" href="comput_1.html#T13" target="_self">Th13</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_5_1_1_1_2"><b>suppose </b></a><a NAME="E1:95_1_2_5_1_1_1_2"/>
(  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p1</font> or  not  <a href="finseq_3.html#K2">Del</a> <font color="Maroon">p1</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> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:95_1_2_5_1_1_1_2"/>
<font color="Maroon">S<sub>4</sub></font>[0]
 <b>by </b><i><a class="ref" href="comput_1.html#T7" target="_self">Th7</a>, , <a class="txt" href="comput_1.html#E13:95_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="comput_1.html#T13" target="_self">Th13</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><div><i><font color="Green">E192</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_5_1_2"><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="E1:95_1_2_5_1_2"/><i><font color="Green">E193</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">m</font>]
 ;<br/><a NAME="E2:95_1_2_5_1_2"/><i><font color="Green">E194</font></i>: 
( <font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font> &amp; <font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="comput_1.html#T13" target="_self">Th13</a>, </i>;<br/><a NAME="E3:95_1_2_5_1_2"/><i><font color="Green">E195</font></i>: 
<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">p1</font>,<font color="Maroon">i</font>,<font color="Maroon">e2</font>]
 <b>by </b><i/>;<br/><a NAME="E4:95_1_2_5_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">m</font>,<font color="Maroon">F2</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">p2</font>,<font color="Maroon">i</font>,<font color="Maroon">e2</font>]
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:95_1_2_5_1_2"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="ref" href="comput_1.html#T7" target="_self">Th7</a>, , , <a class="ref" href="comput_1.html#T14" target="_self">Th14</a></i>;<br/></div><b>end;</b></div><b>thus </b><a NAME="E4:95_1_2_5_1"/>
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>4</sub></font>[<font color="Olive">m</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="ref" href="comput_1.html#D1" target="_self">Def1</a>, );</i><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_5_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:95_1_2_5_2"/><i><font color="Green">E196</font></i>: 
<font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,0
 ;<br/><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="E2:95_1_2_5_2"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:95_1_2_5_2"/><i><font color="Green">E197</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E5:95_1_2_5_2"/><i><font color="Green">E198</font></i>: 
( <font color="Maroon">x</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">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">x</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">F2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> )
 <b>by </b><i><a class="ref" href="comput_1.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">n1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:95_1_2_5_2"/><i><font color="Green">E200</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">n1</font> &amp; <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">n2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E9:95_1_2_5_2"/><i><font color="Green">E202</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">n2</font> &amp; <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 <b>by </b><i>, </i>;<br/><b>thus </b><a NAME="E10:95_1_2_5_2"/>
contradiction
 <b>by </b><i><a class="ref" href="comput_1.html#T13" target="_self">Th13</a>, , , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:95_1_2_5"/>
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">F1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font> or <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="comput_1.html#T12" target="_self">Th12</a></i>;<br/>


</div><b>end;</b></div><a NAME="E19:95_1_2"/><i><font color="Green">E203</font></i>: 
for <font color="Olive">m</font>, <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font> <a href="tarski.html#R1">c=</a> <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> <font color="Olive">n</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_6"><b>proof </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>defpred </b><font color="Maroon">S<sub>4</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">m</font> <a href="tarski.html#R1">c=</a> <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> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E1:95_1_2_6"/><i><font color="Green">E204</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[0]
 
;<br/>
<div><i><font color="Green">E205</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:95_1_2_6_1"/><i><font color="Green">E206</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">n</font>]
 ;<br/><b>set </b><font color="Maroon">k</font> = <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font>;<br/><a NAME="E2:95_1_2_6_1"/>
<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> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_6_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:95_1_2_6_1_1"/><i><font color="Green">E207</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <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> <font color="Maroon">n</font></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">p1c</font> = <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></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">F1</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_6_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_2_6_1_1_1"/>
( ( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></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">F1</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></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> ) or  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p1</font> or  not <font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> or  not <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></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">F1</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></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> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_6_1_1_1_1"><b>suppose </b></a><a NAME="E1:95_1_2_6_1_1_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></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">F1</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></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> )
 ;<br/><div class="add"><a NAME="E2:95_1_2_6_1_1_1_1"/><b>then </b><i><font color="Green">E208</font></i>: 
<font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <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">F1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></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">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <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">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></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">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> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p0">(<span class="default"><font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:95_1_2_6_1_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K1">dom</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">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <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">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></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">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> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p0">(<span class="default"><font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_6_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:95_1_2_6_1_1_1_1_1"/>
not  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K1">dom</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">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <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">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></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">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> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p0">(<span class="default"><font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:95_1_2_6_1_1_1_1_1"/><i><font color="Green">E209</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <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="p3">(<span class="default"><font color="Maroon">e2</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p5"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#T4">XBOOLE_0:4</a></i>;<br/>
<a NAME="E4:95_1_2_6_1_1_1_1_1"/><i><font color="Green">E210</font></i>: 
( <font color="Maroon">x</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">F1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</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">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <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">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></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">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> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p0">(<span class="default"><font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="comput_1.html#T14" target="_self">Th14</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:95_1_2_6_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">x</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">F1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E6:95_1_2_6_1_1_1_1_1"/><b>then </b><i><font color="Green">E211</font></i>: 
( <font color="Maroon">x</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">F1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">n1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:95_1_2_6_1_1_1_1_1"/><i><font color="Green">E212</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">n1</font> &amp; <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E9:95_1_2_6_1_1_1_1_1"/>
( <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">n1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 
<b>by </b><i><a class="ref" href="comput_1.html#T7" target="_self">Th7</a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/>
<b>hence </b><a NAME="E10:95_1_2_6_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="comput_1.html#T15" target="_self">Th15</a>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:95_1_2_6_1_1_1_1"/><b>then </b>
<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> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_4.html#T33">FUNCT_4:33</a></i>;<br/><b>hence </b><a NAME="E5:95_1_2_6_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_6_1_1_1_2"><b>suppose </b></a><a NAME="E1:95_1_2_6_1_1_1_2"/>
(  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p1</font> or  not <font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> or  not <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></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">F1</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></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> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:95_1_2_6_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E3:95_1_2_6_1"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E3:95_1_2_6"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="ref" href="comput_1.html#T13" target="_self">Th13</a>, <a class="ref" href="comput_1.html#D1" target="_self">Def1</a>);</i><br/>


</div><b>end;</b></div><a NAME="E20:95_1_2"/><i><font color="Green">E213</font></i>: 
for <font color="Olive">m</font>, <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font> <a href="tarski.html#R1">c=</a> <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> <font color="Olive">n</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_7"><b>proof </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>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font> <a href="tarski.html#R1">c=</a> <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> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E1:95_1_2_7"/><i><font color="Green">E214</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[0]
 
;<br/>
<div><i><font color="Green">E215</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_7_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:95_1_2_7_1"/><i><font color="Green">E216</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">n</font>]
 ;<br/><b>set </b><font color="Maroon">k</font> = <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font>;<br/><a NAME="E2:95_1_2_7_1"/>
<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> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_7_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:95_1_2_7_1_1"/><i><font color="Green">E217</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <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> <font color="Maroon">n</font></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">p2c</font> = <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></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">F2</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_7_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_2_7_1_1_1"/>
( ( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p2</font> &amp; <font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></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">F2</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></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> ) or  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p2</font> or  not <font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> or  not <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></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">F2</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></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> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_7_1_1_1_1"><b>suppose </b></a><a NAME="E1:95_1_2_7_1_1_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p2</font> &amp; <font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></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">F2</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></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> )
 ;<br/><div class="add"><a NAME="E2:95_1_2_7_1_1_1_1"/><b>then </b><i><font color="Green">E218</font></i>: 
<font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <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">F2</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></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">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <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">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></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">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> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p0">(<span class="default"><font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:95_1_2_7_1_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K1">dom</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">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <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">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></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">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> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p0">(<span class="default"><font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_7_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:95_1_2_7_1_1_1_1_1"/>
not  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K1">dom</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">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <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">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></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">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> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p0">(<span class="default"><font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:95_1_2_7_1_1_1_1_1"/><i><font color="Green">E219</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <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="p3">(<span class="default"><font color="Maroon">e2</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p5"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#T4">XBOOLE_0:4</a></i>;<br/>
<a NAME="E4:95_1_2_7_1_1_1_1_1"/><i><font color="Green">E220</font></i>: 
( <font color="Maroon">x</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">F2</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</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">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <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">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></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">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> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p0">(<span class="default"><font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:95_1_2_7_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">x</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">F2</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E6:95_1_2_7_1_1_1_1_1"/><b>then </b><i><font color="Green">E221</font></i>: 
( <font color="Maroon">x</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">F2</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">n2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:95_1_2_7_1_1_1_1_1"/><i><font color="Green">E222</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">n2</font> &amp; <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> )
 <b>by </b><i>, <a class="ref" href="comput_1.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E9:95_1_2_7_1_1_1_1_1"/>
( <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">n2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 
<b>by </b><i><a class="ref" href="comput_1.html#T7" target="_self">Th7</a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/>
<b>hence </b><a NAME="E10:95_1_2_7_1_1_1_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:95_1_2_7_1_1_1_1"/><b>then </b>
<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> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="comput_1.html#T14" target="_self">Th14</a>, <a class="ref" href="funct_4.html#T33">FUNCT_4:33</a></i>;<br/><b>hence </b><a NAME="E5:95_1_2_7_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_7_1_1_1_2"><b>suppose </b></a><a NAME="E1:95_1_2_7_1_1_1_2"/>
(  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p2</font> or  not <font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> or  not <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></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">F2</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></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> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:95_1_2_7_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E3:95_1_2_7_1"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E3:95_1_2_7"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="ref" href="comput_1.html#D1" target="_self">Def1</a>, );</i><br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">p1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>, <font color="Maroon">n</font> = <font color="Maroon">p2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><b>reconsider </b><font color="Maroon">F1m</font> = <font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font>, <font color="Maroon">F1n</font> = <font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>, <font color="Maroon">F2m</font> = <font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font>, <font color="Maroon">F2n</font> = <font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</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>  ;<br/><b>thus </b><a NAME="E23:95_1_2"/>
 <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">p1</font> <a href="partfun1.html#R1">tolerates</a>  <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">p2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_8"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_2_8_1"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> or <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_8_1_1"><b>suppose </b></a><a NAME="E1:95_1_2_8_1_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:95_1_2_8_1_1"/><i><font color="Green">E227</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E5:95_1_2_8_1_1"/><i><font color="Green">E228</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="comput_1.html#D1" target="_self">Def1</a></i>;<br/><b>thus </b><a NAME="E6:95_1_2_8_1_1"/>
 <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">p1</font> <a href="partfun1.html#R1">tolerates</a>  <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">p2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_8_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_8_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_2_8_1_1_1_1"/>
( <font color="Maroon">F1n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2n</font> or <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">F1n</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">F2n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_8_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:95_1_2_8_1_1_1_1_1"/><i><font color="Green">E229</font></i>: 
<font color="Maroon">F1n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2n</font>
 ;<br/><div class="add"><a NAME="E2:95_1_2_8_1_1_1_1_1"/>
<font color="Maroon">F1m</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F1n</font>
 <b>by </b><i><a class="ref" href="comput_1.html#T12" target="_self">Th12</a>, </i>;<br/><b>hence </b><a NAME="E3:95_1_2_8_1_1_1_1_1"/>
 <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">p1</font> <a href="partfun1.html#R1">tolerates</a>  <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">p2</font>
 <b>by </b><i>, <a class="txt" href="comput_1.html#E12:95_1_2"><i><font color="Green">E64</font></i></a>, , <a class="ref" href="partfun1.html#T140">PARTFUN1:140</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_8_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:95_1_2_8_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">F1n</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">F2n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:95_1_2_8_1_1_1_1_2"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">F1n</font> <a href="xboole_0.html#R1">misses</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F2n</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E3:95_1_2_8_1_1_1_1_2"/><b>then </b><i><font color="Green">E230</font></i>: 
<font color="Maroon">F1n</font> <a href="partfun1.html#R1">tolerates</a> <font color="Maroon">F2n</font>
 <b>by </b><i><a class="ref" href="partfun1.html#T138">PARTFUN1:138</a></i>;<br/><a NAME="E4:95_1_2_8_1_1_1_1_2"/>
<font color="Maroon">F1m</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F1n</font>
 <b>by </b><i><a class="ref" href="comput_1.html#T12" target="_self">Th12</a>, </i>;<br/><b>hence </b><a NAME="E5:95_1_2_8_1_1_1_1_2"/>
 <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">p1</font> <a href="partfun1.html#R1">tolerates</a>  <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">p2</font>
 <b>by </b><i>, <a class="txt" href="comput_1.html#E12:95_1_2"><i><font color="Green">E64</font></i></a>, , <a class="ref" href="partfun1.html#T140">PARTFUN1:140</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_8_1_2"><b>suppose </b></a><a NAME="E1:95_1_2_8_1_2"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:95_1_2_8_1_2"/><i><font color="Green">E231</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E5:95_1_2_8_1_2"/><i><font color="Green">E232</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="comput_1.html#D1" target="_self">Def1</a></i>;<br/><b>thus </b><a NAME="E6:95_1_2_8_1_2"/>
 <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">p1</font> <a href="partfun1.html#R1">tolerates</a>  <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">p2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_8_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_8_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_2_8_1_2_1_1"/>
( <font color="Maroon">F1m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2m</font> or <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">F1m</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">F2m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_8_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:95_1_2_8_1_2_1_1_1"/><i><font color="Green">E233</font></i>: 
<font color="Maroon">F1m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2m</font>
 ;<br/><div class="add"><a NAME="E2:95_1_2_8_1_2_1_1_1"/>
<font color="Maroon">F2n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F2m</font>
 <b>by </b><i><a class="ref" href="comput_1.html#T13" target="_self">Th13</a>, </i>;<br/><b>hence </b><a NAME="E3:95_1_2_8_1_2_1_1_1"/>
 <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">p1</font> <a href="partfun1.html#R1">tolerates</a>  <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">p2</font>
 <b>by </b><i>, <a class="txt" href="comput_1.html#E12:95_1_2"><i><font color="Green">E64</font></i></a>, , <a class="ref" href="partfun1.html#T140">PARTFUN1:140</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_8_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:95_1_2_8_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">F1m</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">F2m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:95_1_2_8_1_2_1_1_2"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">F1m</font> <a href="xboole_0.html#R1">misses</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F2m</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E3:95_1_2_8_1_2_1_1_2"/><b>then </b><i><font color="Green">E234</font></i>: 
<font color="Maroon">F1m</font> <a href="partfun1.html#R1">tolerates</a> <font color="Maroon">F2m</font>
 <b>by </b><i><a class="ref" href="partfun1.html#T138">PARTFUN1:138</a></i>;<br/><a NAME="E4:95_1_2_8_1_2_1_1_2"/>
<font color="Maroon">F2n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F2m</font>
 <b>by </b><i><a class="ref" href="comput_1.html#T13" target="_self">Th13</a>, </i>;<br/><b>hence </b><a NAME="E5:95_1_2_8_1_2_1_1_2"/>
 <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">p1</font> <a href="partfun1.html#R1">tolerates</a>  <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">p2</font>
 <b>by </b><i>, <a class="txt" href="comput_1.html#E12:95_1_2"><i><font color="Green">E64</font></i></a>, , <a class="ref" href="partfun1.html#T140">PARTFUN1:140</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div>
