<?xml version="1.0"?>
<div class="add">

<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">m</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">f1</font> be  non <a href="xboole_0.html#V1">empty</a> <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">f2</font> be  non <a href="xboole_0.html#V1">empty</a> <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">p</font> be    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">f1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> ;<br/>









<b>assume </b><a NAME="E1:107"/><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">p</font>
 ;<br/>

<b>consider </b><font color="Maroon">G</font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">f1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <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="E3:107"/><i><font color="Green">E49</font></i>: 
 <a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K3">Union</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E4:107"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">p</font> being   <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">f1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="comput_1.html#K9" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font>,<font color="Olive">p</font>
 <b>by </b><i/>;<br/>
<a NAME="E5:107"/><i><font color="Green">E52</font></i>: 
 <a href="card_3.html#K3">Union</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_3.html#D4">CARD_3:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">rngG</font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font> as  <a href="fraenkel.html#V1">functional</a> <a href="comput_1.html#V1" target="_self">compatible</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="comput_1.html#T11" target="_self">Th11</a>, </i>;<br/>
<a NAME="E7:107"/><i><font color="Green">E53</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">f1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><i><font color="Green">E54</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:107_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font></span>)</span>
 <b>and </b><br/><a NAME="E2:107_1"/><i><font color="Green">E57</font></i>: 
not <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">k</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">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
 ;<br/><b>set </b><font color="Maroon">pk</font> = <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">k</font>;<br/><a NAME="E3:107_1"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">rngG</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="comput_1.html#T11" target="_self">Th11</a>, <a class="ref" href="comput_1.html#D1" target="_self">Def1</a>, <a class="ref" href="card_3.html#D4">CARD_3:def 4</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/><b>then reconsider </b><font color="Maroon">rngG</font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="fraenkel.html#V1">functional</a> <a href="comput_1.html#V1" target="_self">compatible</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="zfmisc_1.html#T2">ZFMISC_1:2</a></i>;<br/><a NAME="E5:107_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">rngG</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">f</font></span>)</span> where <font color="Olive">f</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">rngG</font> : verum </span>}</span> 
 <b>by </b><i/>;<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="E7:107_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">f</font></span>)</span> where <font color="Olive">f</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">rngG</font> : verum </span>}</span>  )
 <b>by </b><i><a class="ref" href="comput_1.html#T11" target="_self">Th11</a>, , <a class="ref" href="comput_1.html#D1" target="_self">Def1</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">f</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">rngG</font><b> such that </b><br/><a NAME="E9:107_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">pp</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:107_1"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">pp</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">pp</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">pp</font> = <font color="Maroon">pp</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">f1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="comput_1.html#T14" target="_self">Th14</a></i>;<br/><a NAME="E13:107_1"/>
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">pp</font> <a href="hidden.html#R1">=</a>  <a href="comput_1.html#K9" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font>,<font color="Maroon">pp</font>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">m</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="E15:107_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pp</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font>
 <b>by </b><i>, , <a class="ref" href="comput_1.html#T14" target="_self">Th14</a>, <a class="ref" href="comput_1.html#T6" target="_self">Th6</a></i>;<br/><b>set </b><font color="Maroon">ppi</font> = <font color="Maroon">pp</font> <a href="comput_1.html#K2" target="_self">.</a> <font color="Maroon">i</font>;<br/><i><font color="Green">E67</font></i>: <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">pp</font> <a href="comput_1.html#K2" target="_self">.</a> <font color="Maroon">i</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">k</font></span>)</span> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">pp</font> <a href="comput_1.html#K2" target="_self">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#T36">FUNCT_7:36</a></i>
<br/>.= 
<font color="Maroon">pp</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">pp</font> <a href="comput_1.html#K2" target="_self">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_7.html#T36">FUNCT_7:36</a></i>
<br/>.= 
<font color="Maroon">pp</font>
<b>by </b><i><a class="ref" href="funct_7.html#T37">FUNCT_7:37</a></i>
;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:107_1_2"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pp</font> <a href="comput_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> or <font color="Maroon">pp</font> <a href="comput_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> or <font color="Maroon">pp</font> <a href="comput_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font> )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_1_2_1"><b>suppose </b></a><a NAME="E1:107_1_2_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pp</font> <a href="comput_1.html#K2" target="_self">.</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:107_1_2_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="comput_1.html#T14" target="_self">Th14</a>, <a class="ref" href="comput_1.html#T15" target="_self">Th15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_1_2_2"><b>suppose </b></a><a NAME="E1:107_1_2_2"/>
<font color="Maroon">pp</font> <a href="comput_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">m</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:107_1_2_2"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">pp</font> <a href="comput_1.html#K2" target="_self">.</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</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">m</font> = <font color="Maroon">m</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="E5:107_1_2_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">pp</font> <a href="comput_1.html#K2" target="_self">.</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/><a NAME="E6:107_1_2_2"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">pp</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="comput_1.html#T10" target="_self">Th10</a>, , <a class="ref" href="comput_1.html#T15" target="_self">Th15</a>, <a class="ref" href="comput_1.html#T8" target="_self">Th8</a></i>;<br/><b>hence </b><a NAME="E7:107_1_2_2"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="comput_1.html#T14" target="_self">Th14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_1_2_3"><b>suppose </b></a><a NAME="E1:107_1_2_3"/>
<font color="Maroon">pp</font> <a href="comput_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">m</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:107_1_2_3"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">pp</font> <a href="comput_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</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">m</font> = <font color="Maroon">m</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="E5:107_1_2_3"/>
<font color="Maroon">pp</font> <a href="comput_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:107_1_2_3"/>
contradiction
 <b>by </b><i><a class="ref" href="comput_1.html#T10" target="_self">Th10</a>, , , , , <a class="ref" href="comput_1.html#T14" target="_self">Th14</a>, <a class="ref" href="comput_1.html#T15" target="_self">Th15</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>set </b><font color="Maroon">p0</font> = <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0;<br/>
<a NAME="E9:107"/><i><font color="Green">E138</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0</span>)</span>
 
<b>by </b><i><a class="ref" href="funct_7.html#T32">FUNCT_7:32</a></i>;<br/>
<div><i><font color="Green">E139</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="107_2"><b>now </b></a><div class="add"><a NAME="E1:107_2"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0</span>)</span> <a href="hidden.html#R1">=</a>  <a href="comput_1.html#K9" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0</span>)</span>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:107_2"/><i><font color="Green">E143</font></i>: 
 <a href="comput_1.html#K9" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0</span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
 <b>and </b><br/><a NAME="E4:107_2"/><i><font color="Green">E153</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0</span>)</span> &amp;  <a href="finseq_3.html#K2">Del</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0</span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> implies <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0</span>)</span> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0</span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_3.html#K2">Del</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0</span>)</span>,<font color="Maroon">i</font></span>)</span></span>)</span> )
 <b>and </b><br/><a NAME="E5:107_2"/><i><font color="Green">E154</font></i>: 
( (  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0</span>)</span> or  not  <a href="finseq_3.html#K2">Del</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0</span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> ) implies <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>and </b><br/><a NAME="E6:107_2"/>
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>,<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font>,<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0,<font color="Maroon">i</font>,<font color="Maroon">f2</font>]
 <b>by </b><i/>;<br/><i><font color="Green">E155</font></i>: <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0</span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0</span>)</span> <a href="comput_1.html#K2" target="_self">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="comput_1.html#T10" target="_self">Th10</a>, <a class="ref" href="comput_1.html#D1" target="_self">Def1</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="comput_1.html#T10" target="_self">Th10</a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>
;<br/><b>thus </b><a NAME="E8:107_2"/>
( <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0</span>)</span></span>)</span> iff  <a href="finseq_3.html#K2">Del</a> <font color="Maroon">p</font>,<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_2_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:107_2_2"/>
( <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0</span>)</span></span>)</span> implies  <a href="finseq_3.html#K2">Del</a> <font color="Maroon">p</font>,<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> )
 <b>by </b><i>, , , <a class="ref" href="comput_1.html#T15" target="_self">Th15</a>, , <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/>

<b>assume </b><a NAME="E2:107_2_2"/>
 <a href="finseq_3.html#K2">Del</a> <font color="Maroon">p</font>,<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font>
 ;<br/>

<b>then </b> <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> = 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0</span>)</span> <a href="comput_1.html#K1" target="_self">+*</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="comput_1.html#T10" target="_self">Th10</a>, <a class="ref" href="comput_1.html#T14" target="_self">Th14</a>, , <a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a>, <a class="ref" href="funct_7.html#T32">FUNCT_7:32</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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="funct_7.html#T36">FUNCT_7:36</a></i>
;<br/>
<b>hence </b><a NAME="E4:107_2_2"/>
<font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0</span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="comput_1.html#T15" target="_self">Th15</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E9:107_2"/>
<font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0</span>)</span></span>)</span>
 ;<br/><a NAME="E10:107_2"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0</span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_3.html#K2">Del</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0</span>)</span>,<font color="Maroon">i</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="comput_1.html#T14" target="_self">Th14</a>, , <a class="ref" href="comput_1.html#T15" target="_self">Th15</a>, <a class="ref" href="funct_7.html#T36">FUNCT_7:36</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/><a NAME="E11:107_2"/><b>then </b>
( <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0</span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">p</font>,<font color="Maroon">i</font></span>)</span></span>)</span> &amp; <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0 <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">p</font> <a href="comput_1.html#K1" target="_self">+*</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="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E12:107_2"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">p</font>,<font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="comput_1.html#T15" target="_self">Th15</a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:107"/><i><font color="Green">E156</font></i>: 
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="comput_1.html#T12" target="_self">Th12</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">rngG</font> = <font color="Maroon">rngG</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="fraenkel.html#V1">functional</a> <a href="comput_1.html#V1" target="_self">compatible</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>thus </b><a NAME="E13:107"/>
( <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font></span>)</span> iff  <a href="finseq_3.html#K2">Del</a> <font color="Maroon">p</font>,<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:107_3"/>
( <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font></span>)</span> implies  <a href="finseq_3.html#K2">Del</a> <font color="Maroon">p</font>,<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> )
 <b>by </b><i><a class="ref" href="comput_1.html#T13" target="_self">Th13</a>, </i>;<br/>

<b>assume </b><a NAME="E2:107_3"/><i><font color="Green">E157</font></i>: 
 <a href="finseq_3.html#K2">Del</a> <font color="Maroon">p</font>,<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font>
 ;<br/>

<a NAME="E3:107_3"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0</span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">f</font></span>)</span> where <font color="Olive">f</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">rngG</font> : verum </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E4:107_3"/><b>then </b>
<font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0 <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">f</font></span>)</span> where <font color="Olive">f</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">rngG</font> : verum </span>}</span> 
 
<b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E5:107_3"/>
<font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="comput_1.html#T11" target="_self">Th11</a>, , </i>;<br/>


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

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:107_4"/><i><font color="Green">E158</font></i>: 
<font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font></span>)</span>
 ;<br/><a NAME="E2:107_4"/><b>then </b>
<font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="comput_1.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E3:107_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">rngG</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0</span>)</span>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:107_4"/>
<span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">p</font>,<font color="Maroon">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="comput_1.html#T11" target="_self">Th11</a>, <a class="ref" href="comput_1.html#T13" target="_self">Th13</a>, , , <a class="ref" href="card_3.html#D4">CARD_3:def 4</a></i>;<br/></div>
<b>end;</b></div>

<b>set </b><font color="Maroon">pm1</font> = <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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/>
<b>set </b><font color="Maroon">pm</font> = <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font>;<br/>
<b>set </b><font color="Maroon">pc</font> = <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">G</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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> <a href="comput_1.html#K1" target="_self">+*</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/>
<a NAME="E15:107"/><i><font color="Green">E160</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_7.html#T32">FUNCT_7:32</a></i>;<br/>
<a NAME="E16:107"/><i><font color="Green">E161</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="comput_1.html#K1" target="_self">+*</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">p</font> <a href="comput_1.html#K1" target="_self">+*</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="funct_7.html#T36">FUNCT_7:36</a></i>;<br/>
<div><i><font color="Green">E162</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="107_5"><b>now </b></a><div class="add"><a NAME="E1:107_5"/><i><font color="Green">E163</font></i>: 
( <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="comput_1.html#K9" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="comput_1.html#K9" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span> )
 <b>by </b><i/>;<br/><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="E3:107_5"/><i><font color="Green">E187</font></i>: 
 <a href="comput_1.html#K9" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</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">p</font> <a href="comput_1.html#K1" target="_self">+*</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> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
 <b>and </b><br/><a NAME="E4:107_5"/><i><font color="Green">E188</font></i>: 
( ( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp;  <a href="finseq_3.html#K2">Del</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> implies <font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0</span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_3.html#K2">Del</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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>,<font color="Maroon">i</font></span>)</span></span>)</span> ) &amp; ( (  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> or  not  <a href="finseq_3.html#K2">Del</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> ) implies <font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 <b>and </b><br/><a NAME="E5:107_5"/><i><font color="Green">E189</font></i>: 
for <font color="Olive">M</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">M</font>,<font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">M</font>,<font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">M</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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>,<font color="Maroon">i</font>,<font color="Maroon">f2</font>]
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:107_5"/><i><font color="Green">E191</font></i>: 
 <a href="comput_1.html#K9" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
 <b>and </b><br/><a NAME="E8:107_5"/><i><font color="Green">E192</font></i>: 
( ( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span> &amp;  <a href="finseq_3.html#K2">Del</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> implies <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0</span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_3.html#K2">Del</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span></span>)</span> ) &amp; ( (  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span> or  not  <a href="finseq_3.html#K2">Del</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> ) implies <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 <b>and </b><br/><a NAME="E9:107_5"/><i><font color="Green">E193</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">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">M</font>,<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">M</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font>,<font color="Maroon">f2</font>]
 <b>by </b><i/>;<br/><i><font color="Green">E194</font></i>: <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="comput_1.html#K2" target="_self">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="comput_1.html#T10" target="_self">Th10</a>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="comput_1.html#T10" target="_self">Th10</a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>
;<br/><i><font color="Green">E195</font></i>: <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span> <a href="comput_1.html#K2" target="_self">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="comput_1.html#T10" target="_self">Th10</a>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="comput_1.html#T10" target="_self">Th10</a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>
;<br/><a NAME="E12:107_5"/><i><font color="Green">E196</font></i>: 
<font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="comput_1.html#T16" target="_self">Th16</a>, , <a class="ref" href="comput_1.html#T18" target="_self">Th18</a>, </i>;<br/><a NAME="E13:107_5"/><i><font color="Green">E197</font></i>: 
<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> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></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_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="comput_1.html#T10" target="_self">Th10</a>, , <a class="ref" href="comput_1.html#T16" target="_self">Th16</a>, </i>;<br/><a NAME="E14:107_5"/><i><font color="Green">E198</font></i>: 
not <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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#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>
 <b>by </b><i><a class="ref" href="comput_1.html#T10" target="_self">Th10</a>, , <a class="ref" href="comput_1.html#T16" target="_self">Th16</a>, </i>;<br/><b>thus </b><a NAME="E15:107_5"/><i><font color="Green">E200</font></i>: 
( <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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>)</span> iff ( <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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">G</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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> <a href="comput_1.html#K1" target="_self">+*</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">f2</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_5_3"><b>proof </b></a><div class="add">

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:107_5_3_1"/><i><font color="Green">E202</font></i>: 
<font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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>)</span>
 ;<br/><a NAME="E2:107_5_3_1"/><b>then </b><i><font color="Green">E203</font></i>: 
<font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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#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>
 <b>by </b><i>, , </i>;<br/><b>assume </b><a NAME="E3:107_5_3_1"/><i><font color="Green">E204</font></i>: 
(  not <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span></span>)</span> or  not <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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">G</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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> <a href="comput_1.html#K1" target="_self">+*</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">f2</font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_5_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:107_5_3_1_1"/>
(  not <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span></span>)</span> or  not <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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">G</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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> <a href="comput_1.html#K1" target="_self">+*</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">f2</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_5_3_1_1_1"><b>suppose </b></a><a NAME="E1:107_5_3_1_1_1"/>
not <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:107_5_3_1_1_1"/><b>then </b>
not <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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>
 <b>by </b><i>, <a class="ref" href="comput_1.html#T17" target="_self">Th17</a>, <a class="ref" href="comput_1.html#D2" target="_self">Def2</a>, <a class="ref" href="comput_1.html#D3" target="_self">Def3</a></i>;<br/><b>hence </b><a NAME="E3:107_5_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="comput_1.html#T14" target="_self">Th14</a>, <a class="ref" href="comput_1.html#T16" target="_self">Th16</a>, <a class="ref" href="comput_1.html#T19" target="_self">Th19</a>, <a class="ref" href="comput_1.html#T20" target="_self">Th20</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_5_3_1_1_2"><b>suppose </b></a><a NAME="E1:107_5_3_1_1_2"/>
not <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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">G</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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> <a href="comput_1.html#K1" target="_self">+*</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">f2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:107_5_3_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="comput_1.html#T14" target="_self">Th14</a>, <a class="ref" href="comput_1.html#T15" target="_self">Th15</a>, , <a class="ref" href="comput_1.html#T16" target="_self">Th16</a>, , <a class="ref" href="comput_1.html#D4" target="_self">Def4</a>, <a class="ref" href="comput_1.html#T19" target="_self">Th19</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:107_5_3"/>
( <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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">G</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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> <a href="comput_1.html#K1" target="_self">+*</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">f2</font> )
 ;<br/>

<a NAME="E3:107_5_3"/><b>then </b><i><font color="Green">E205</font></i>: 
<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">p</font> <a href="comput_1.html#K1" target="_self">+*</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">f2</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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">p</font> <a href="comput_1.html#K1" target="_self">+*</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><a class="ref" href="comput_1.html#T10" target="_self">Th10</a>, , <a class="ref" href="comput_1.html#T14" target="_self">Th14</a>, <a class="ref" href="comput_1.html#T15" target="_self">Th15</a>, , <a class="ref" href="comput_1.html#T16" target="_self">Th16</a>, <a class="ref" href="comput_1.html#T17" target="_self">Th17</a>, , <a class="ref" href="comput_1.html#D2" target="_self">Def2</a>, <a class="ref" href="comput_1.html#D3" target="_self">Def3</a>, <a class="ref" href="comput_1.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E4:107_5_3"/>
<font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:107_5_3"/><b>then </b>
<font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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#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">p</font> <a href="comput_1.html#K1" target="_self">+*</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">f2</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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">p</font> <a href="comput_1.html#K1" target="_self">+*</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><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E6:107_5_3"/><b>then </b>
<font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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#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#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">p</font> <a href="comput_1.html#K1" target="_self">+*</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">f2</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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">p</font> <a href="comput_1.html#K1" target="_self">+*</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="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E7:107_5_3"/>
<font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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>)</span>
 <b>by </b><i><a class="ref" href="comput_1.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E16:107_5"/><i><font color="Green">E206</font></i>: 
<font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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>)</span>
 ;<br/><a NAME="E17:107_5"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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">p</font> <a href="comput_1.html#K1" target="_self">+*</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">f2</font>
 <b>by </b><i><a class="ref" href="comput_1.html#T14" target="_self">Th14</a>, <a class="ref" href="comput_1.html#T15" target="_self">Th15</a>, , <a class="ref" href="comput_1.html#T16" target="_self">Th16</a>, , </i>;<br/><a NAME="E18:107_5"/><b>then </b><i><font color="Green">E207</font></i>: 
<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">p</font> <a href="comput_1.html#K1" target="_self">+*</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">f2</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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">p</font> <a href="comput_1.html#K1" target="_self">+*</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><a class="ref" href="comput_1.html#T10" target="_self">Th10</a>, , <a class="ref" href="comput_1.html#T14" target="_self">Th14</a>, <a class="ref" href="comput_1.html#T15" target="_self">Th15</a>, <a class="ref" href="comput_1.html#T16" target="_self">Th16</a>, <a class="ref" href="comput_1.html#T17" target="_self">Th17</a>, <a class="ref" href="comput_1.html#D2" target="_self">Def2</a>, <a class="ref" href="comput_1.html#D3" target="_self">Def3</a>, , </i>;<br/><a NAME="E19:107_5"/><i><font color="Green">E208</font></i>: 
<font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E20:107_5"/><b>then </b>
<font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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#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">p</font> <a href="comput_1.html#K1" target="_self">+*</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">f2</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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">p</font> <a href="comput_1.html#K1" target="_self">+*</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><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<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"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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">f2</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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">p</font> <a href="comput_1.html#K1" target="_self">+*</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> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comput_1.html#T15" target="_self">Th15</a>, , , <a class="ref" href="comput_1.html#T20" target="_self">Th20</a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="comput_1.html#K1" target="_self">+*</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>
<b>by </b><i><a class="ref" href="comput_1.html#T14" target="_self">Th14</a>, <a class="ref" href="comput_1.html#T15" target="_self">Th15</a>, , , <a class="ref" href="comput_1.html#D4" target="_self">Def4</a>, , <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><br/></div><b>end;</b></div>
<b>thus </b><a NAME="E18:107"/>
( <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font></span>)</span> iff ( <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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">f2</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_6"><b>proof </b></a><div class="add">

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:107_6_1"/><i><font color="Green">E209</font></i>: 
<font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font></span>)</span>
 ;<br/><a NAME="E2:107_6_1"/>
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="comput_1.html#T12" target="_self">Th12</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E3:107_6_1"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">f</font></span>)</span> where <font color="Olive">f</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">rngG</font> : verum </span>}</span> 
 ;<br/><a NAME="E4:107_6_1"/><b>then </b>
<font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">f</font></span>)</span> where <font color="Olive">f</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">rngG</font> : verum </span>}</span> 
 <b>by </b><i><a class="ref" href="comput_1.html#T13" target="_self">Th13</a>, , <a class="ref" href="comput_1.html#T15" target="_self">Th15</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E5:107_6_1"/>
<font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="comput_1.html#T11" target="_self">Th11</a>, , </i>;<br/><a NAME="E6:107_6_1"/><i><font color="Green">E210</font></i>: 
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="comput_1.html#T12" target="_self">Th12</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E7:107_6_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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>)</span>
 <b>by </b><i><a class="ref" href="comput_1.html#T10" target="_self">Th10</a>, , <a class="ref" href="comput_1.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E8:107_6_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">rngG</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="comput_1.html#T13" target="_self">Th13</a>, , <a class="ref" href="comput_1.html#T15" target="_self">Th15</a>, , </i>;<br/><b>hence </b><a NAME="E9:107_6_1"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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">f2</font>
 <b>by </b><i><a class="ref" href="comput_1.html#T11" target="_self">Th11</a>, <a class="ref" href="comput_1.html#T13" target="_self">Th13</a>, <a class="ref" href="comput_1.html#T14" target="_self">Th14</a>, , <a class="ref" href="comput_1.html#T15" target="_self">Th15</a>, <a class="ref" href="card_3.html#D4">CARD_3:def 4</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:107_6"/><i><font color="Green">E211</font></i>: 
( <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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">f2</font> )
 ;<br/>

<a NAME="E3:107_6"/><b>then </b><i><font color="Green">E212</font></i>: 
<font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="comput_1.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E4:107_6"/><i><font color="Green">E213</font></i>: 
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="comput_1.html#T12" target="_self">Th12</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:107_6"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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>)</span>
 
<b>by </b><i><a class="ref" href="comput_1.html#T10" target="_self">Th10</a>, , <a class="ref" href="comput_1.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E6:107_6"/><b>then </b><i><font color="Green">E214</font></i>: 
<span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">rngG</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E7:107_6"/>
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="comput_1.html#T12" target="_self">Th12</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:107_6"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">f</font></span>)</span> where <font color="Olive">f</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">rngG</font> : verum </span>}</span> 
 
;<br/>
<a NAME="E9:107_6"/><b>then </b>
<font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">f</font></span>)</span> where <font color="Olive">f</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">rngG</font> : verum </span>}</span> 
 
<b>by </b><i><a class="ref" href="comput_1.html#T11" target="_self">Th11</a>, <a class="ref" href="comput_1.html#T13" target="_self">Th13</a>, <a class="ref" href="comput_1.html#T14" target="_self">Th14</a>, , <a class="ref" href="comput_1.html#T15" target="_self">Th15</a>, <a class="ref" href="comput_1.html#T16" target="_self">Th16</a>, <a class="ref" href="card_3.html#D4">CARD_3:def 4</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E10:107_6"/>
<font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="comput_1.html#T11" target="_self">Th11</a>, , </i>;<br/>


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

<b>assume </b><a NAME="E19:107"/><i><font color="Green">E215</font></i>: 
<font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font></span>)</span>
 ;<br/>

<a NAME="E20:107"/><i><font color="Green">E216</font></i>: 
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="comput_1.html#T12" target="_self">Th12</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E21:107"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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>)</span>
 
<b>by </b><i><a class="ref" href="comput_1.html#T10" target="_self">Th10</a>, , <a class="ref" href="comput_1.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E22:107"/><b>then </b>
( <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">rngG</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">rngG</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="comput_1.html#T13" target="_self">Th13</a>, , <a class="ref" href="comput_1.html#T15" target="_self">Th15</a>, , </i>;<br/>
<b>hence </b><a NAME="E23:107"/>
<span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</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>
 <b>by </b><i><a class="ref" href="comput_1.html#T11" target="_self">Th11</a>, , <a class="ref" href="comput_1.html#T13" target="_self">Th13</a>, , <a class="ref" href="comput_1.html#T15" target="_self">Th15</a>, <a class="ref" href="funct_7.html#T36">FUNCT_7:36</a></i>;<br/>


</div>
