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<b>let </b><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c3">f1</font>, <font color="Maroon" title="c4">f2</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="comput_1.html#NV2" title="COMPUT_1:NV.2">to-naturals</a> <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a> <a href="comput_1.html#V2" title="COMPUT_1:attr.2">from-natural-fseqs</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/><b>let </b><font color="Maroon" title="c5">p</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>









<b>assume </b><a NAME="E1:105"/><i><font color="Green" title="E69">A1</font></i>: 
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">p</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c6">G</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:105"/><i><font color="Green" title="E70">A2</font></i>: 
 <a href="comput_1.html#K9" title="COMPUT_1:func.9">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c6">G</font>
 <b>and </b><br/><a NAME="E4:105"/><i><font color="Green" title="E71">A3</font></i>: 
for <font color="Olive" title="b1">p</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comput_1.html#K8" title="COMPUT_1:func.8">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b1">p</font>
 <b>by </b><i><a class="ref" href="comput_1.html#D14" target="_self" title="COMPUT_1:def.14">Def14</a></i>;<br/>
<a NAME="E5:105"/><i><font color="Green" title="E72">A4</font></i>: 
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c6">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_3.html#D4" title="CARD_3:def.4">CARD_3:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">rngG</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">G</font> as  <a href="fraenkel.html#V1" title="FRAENKEL:attr.1">functional</a> <a href="comput_1.html#V1" title="COMPUT_1:attr.1">compatible</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="comput_1.html#E3:105"><i><font color="Green" title="E70">A2</font></i></a>, <a class="ref" href="comput_1.html#T14" target="_self" title="COMPUT_1:th.14">Th14</a></i>;<br/>
<a NAME="E7:105"/><i><font color="Green" title="E73">A5</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<div><i><font color="Green" title="E74">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:105_1"/><i><font color="Green" title="E75">A7</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K9" title="COMPUT_1:func.9">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span>
 <b>and </b><br/><a NAME="E2:105_1"/><i><font color="Green" title="E76">A8</font></i>: 
not <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c8">k</font></span>)</span></span>)</span>
 ;<br/><b>set </b><font color="Maroon" title="c9">pk</font> = <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c8">k</font>;<br/><a NAME="E3:105_1"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c7">rngG</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="comput_1.html#E3:105"><i><font color="Green" title="E70">A2</font></i></a>, <a class="txt" href="comput_1.html#E1:105_1"><i><font color="Green" title="E75">A7</font></i></a>, <a class="ref" href="card_3.html#D4" title="CARD_3:def.4">CARD_3:def 4</a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c10">rngG</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">G</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="fraenkel.html#V1" title="FRAENKEL:attr.1">functional</a> <a href="comput_1.html#V1" title="COMPUT_1:attr.1">compatible</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="ref" href="zfmisc_1.html#T2" title="ZFMISC_1:th.2">ZFMISC_1:2</a></i>;<br/><a NAME="E5:105_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c10">rngG</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">f</font></span>)</span> where <font color="Olive" title="b1">f</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c10">rngG</font> : verum </span>}</span> 
 <b>by </b><i><a class="ref" href="comput_1.html#T15" target="_self" title="COMPUT_1:th.15">Th15</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:105_1"/><i><font color="Green" title="E77">A9</font></i>: 
( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">X</font> &amp; <font color="Maroon" title="c11">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">f</font></span>)</span> where <font color="Olive" title="b1">f</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c10">rngG</font> : verum </span>}</span>  )
 <b>by </b><i><a class="txt" href="comput_1.html#E3:105"><i><font color="Green" title="E70">A2</font></i></a>, <a class="txt" href="comput_1.html#E5:105"><i><font color="Green" title="E72">A4</font></i></a>, <a class="txt" href="comput_1.html#E1:105_1"><i><font color="Green" title="E75">A7</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">f</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c10">rngG</font><b> such that </b><br/><a NAME="E9:105_1"/><i><font color="Green" title="E78">A10</font></i>: 
<font color="Maroon" title="c11">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c12">f</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E7:105_1"><i><font color="Green" title="E77">A9</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c13">pp</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:105_1"/><i><font color="Green" title="E79">A11</font></i>: 
( <font color="Maroon" title="c13">pp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">G</font> &amp; <font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">pp</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">pp</font> = <font color="Maroon" title="c13">pp</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="comput_1.html#E11:105_1"><i><font color="Green" title="E79">A11</font></i></a></i>;<br/><a NAME="E13:105_1"/>
<font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">pp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comput_1.html#K8" title="COMPUT_1:func.8">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c14">pp</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E4:105"><i><font color="Green" title="E71">A3</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c15">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E15:105_1"/><i><font color="Green" title="E80">A12</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">pp</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c15">m</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E7:105_1"><i><font color="Green" title="E77">A9</font></i></a>, <a class="txt" href="comput_1.html#E9:105_1"><i><font color="Green" title="E78">A10</font></i></a>, <a class="txt" href="comput_1.html#E11:105_1"><i><font color="Green" title="E79">A11</font></i></a>, <a class="ref" href="comput_1.html#T53" target="_self" title="COMPUT_1:th.53">Th53</a></i>;<br/><b>set </b><font color="Maroon" title="c16">ppi</font> = <font color="Maroon" title="c14">pp</font> <a href="comput_1.html#K2" title="COMPUT_1:func.2">.</a> <font color="Maroon" title="c1">i</font>;<br/><i><font color="Green" title="E81">A13</font></i>: <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">pp</font> <a href="comput_1.html#K2" title="COMPUT_1:func.2">.</a> <font color="Maroon" title="c1">i</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c8">k</font></span>)</span> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">pp</font> <a href="comput_1.html#K2" title="COMPUT_1:func.2">.</a> <font color="Maroon" title="c1">i</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#T36" title="FUNCT_7:th.36">FUNCT_7:36</a></i>
<br/>.= 
<font color="Maroon" title="c14">pp</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">pp</font> <a href="comput_1.html#K2" title="COMPUT_1:func.2">.</a> <font color="Maroon" title="c1">i</font></span>)</span>
<b>by </b><i><a class="txt" href="comput_1.html#E15:105_1"><i><font color="Green" title="E80">A12</font></i></a>, <a class="ref" href="funct_7.html#T36" title="FUNCT_7:th.36">FUNCT_7:36</a></i>
<br/>.= 
<font color="Maroon" title="c14">pp</font>
<b>by </b><i><a class="ref" href="funct_7.html#T37" title="FUNCT_7:th.37">FUNCT_7:37</a></i>
;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:105_1_2"/>
( <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">pp</font> <a href="comput_1.html#K2" title="COMPUT_1:func.2">.</a> <font color="Maroon" title="c1">i</font> or <font color="Maroon" title="c14">pp</font> <a href="comput_1.html#K2" title="COMPUT_1:func.2">.</a> <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">k</font> or <font color="Maroon" title="c14">pp</font> <a href="comput_1.html#K2" title="COMPUT_1:func.2">.</a> <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c8">k</font> )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_1"><b>suppose </b></a><a NAME="E1:105_1_2_1"/>
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">pp</font> <a href="comput_1.html#K2" title="COMPUT_1:func.2">.</a> <font color="Maroon" title="c1">i</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:105_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="comput_1.html#E2:105_1"><i><font color="Green" title="E76">A8</font></i></a>, <a class="txt" href="comput_1.html#E7:105_1"><i><font color="Green" title="E77">A9</font></i></a>, <a class="txt" href="comput_1.html#E9:105_1"><i><font color="Green" title="E78">A10</font></i></a>, <a class="txt" href="comput_1.html#E11:105_1"><i><font color="Green" title="E79">A11</font></i></a>, <a class="txt" href="comput_1.html#E16:105_1"><i><font color="Green" title="E81">A13</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_2"><b>suppose </b></a><a NAME="E1:105_1_2_2"/>
<font color="Maroon" title="c14">pp</font> <a href="comput_1.html#K2" title="COMPUT_1:func.2">.</a> <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">k</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c17">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:105_1_2_2"/><i><font color="Green" title="E82">A14</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">pp</font> <a href="comput_1.html#K2" title="COMPUT_1:func.2">.</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">m</font> = <font color="Maroon" title="c17">m</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E5:105_1_2_2"/>
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">pp</font> <a href="comput_1.html#K2" title="COMPUT_1:func.2">.</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c18">m</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E3:105_1_2_2"><i><font color="Green" title="E82">A14</font></i></a></i>;<br/><a NAME="E6:105_1_2_2"/><b>then </b>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">pp</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c8">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:105"><i><font color="Green" title="E69">A1</font></i></a>, <a class="txt" href="comput_1.html#E4:105"><i><font color="Green" title="E71">A3</font></i></a>, <a class="txt" href="comput_1.html#E16:105_1"><i><font color="Green" title="E81">A13</font></i></a>, <a class="txt" href="comput_1.html#E80"><i><font color="Green" title="E66">Lm5</font></i></a></i>;<br/><b>hence </b><a NAME="E7:105_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="comput_1.html#E2:105_1"><i><font color="Green" title="E76">A8</font></i></a>, <a class="txt" href="comput_1.html#E7:105_1"><i><font color="Green" title="E77">A9</font></i></a>, <a class="txt" href="comput_1.html#E9:105_1"><i><font color="Green" title="E78">A10</font></i></a>, <a class="txt" href="comput_1.html#E11:105_1"><i><font color="Green" title="E79">A11</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_3"><b>suppose </b></a><a NAME="E1:105_1_2_3"/>
<font color="Maroon" title="c14">pp</font> <a href="comput_1.html#K2" title="COMPUT_1:func.2">.</a> <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c8">k</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c17">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:105_1_2_3"/><i><font color="Green" title="E82">A15</font></i>: 
<font color="Maroon" title="c14">pp</font> <a href="comput_1.html#K2" title="COMPUT_1:func.2">.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">m</font> = <font color="Maroon" title="c17">m</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E5:105_1_2_3"/>
<font color="Maroon" title="c14">pp</font> <a href="comput_1.html#K2" title="COMPUT_1:func.2">.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c18">m</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E3:105_1_2_3"><i><font color="Green" title="E82">A15</font></i></a></i>;<br/><b>hence </b><a NAME="E6:105_1_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="comput_1.html#E1:105"><i><font color="Green" title="E69">A1</font></i></a>, <a class="txt" href="comput_1.html#E4:105"><i><font color="Green" title="E71">A3</font></i></a>, <a class="txt" href="comput_1.html#E2:105_1"><i><font color="Green" title="E76">A8</font></i></a>, <a class="txt" href="comput_1.html#E7:105_1"><i><font color="Green" title="E77">A9</font></i></a>, <a class="txt" href="comput_1.html#E9:105_1"><i><font color="Green" title="E78">A10</font></i></a>, <a class="txt" href="comput_1.html#E11:105_1"><i><font color="Green" title="E79">A11</font></i></a>, <a class="txt" href="comput_1.html#E16:105_1"><i><font color="Green" title="E81">A13</font></i></a>, <a class="txt" href="comput_1.html#E81"><i><font color="Green" title="E67">Lm6</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c8">p0</font> = <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0;<br/>
<a NAME="E9:105"/><i><font color="Green" title="E75">A16</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0</span>)</span>
 
<b>by </b><i><a class="ref" href="funct_7.html#T32" title="FUNCT_7:th.32">FUNCT_7:32</a></i>;<br/>
<div><i><font color="Green" title="E76">A17</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="105_2"><b>now </b></a><div class="add"><a NAME="E1:105_2"/><i><font color="Green" title="E77">A18</font></i>: 
<font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comput_1.html#K8" title="COMPUT_1:func.8">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0</span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E4:105"><i><font color="Green" title="E71">A3</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c9">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:105_2"/><i><font color="Green" title="E78">A19</font></i>: 
 <a href="comput_1.html#K8" title="COMPUT_1:func.8">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c1">i</font></span>)</span>
 <b>and </b><br/><a NAME="E4:105_2"/><i><font color="Green" title="E79">A20</font></i>: 
( <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0</span>)</span> &amp;  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0</span>)</span>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f1</font> implies <font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0</span>)</span> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0</span>)</span>,<font color="Maroon" title="c1">i</font></span>)</span></span>)</span> )
 <b>and </b><br/><a NAME="E5:105_2"/><i><font color="Green" title="E80">A21</font></i>: 
( (  not <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0</span>)</span> or  not  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0</span>)</span>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f1</font> ) implies <font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>and </b><br/><a NAME="E6:105_2"/>
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>,<font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font>,<font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c4">f2</font>]
 <b>by </b><i><a class="ref" href="comput_1.html#D13" target="_self" title="COMPUT_1:def.13">Def13</a></i>;<br/><i><font color="Green" title="E81">A22</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c1">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0</span>)</span> <a href="comput_1.html#K2" title="COMPUT_1:func.2">.</a> <font color="Maroon" title="c1">i</font>
<b>by </b><i><a class="txt" href="comput_1.html#E1:105"><i><font color="Green" title="E69">A1</font></i></a>, <a class="txt" href="comput_1.html#E9:105"><i><font color="Green" title="E75">A16</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="comput_1.html#E1:105"><i><font color="Green" title="E69">A1</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>
;<br/><b>thus </b><a NAME="E8:105_2"/>
( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0</span>)</span></span>)</span> iff  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_2_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:105_2_2"/>
( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0</span>)</span></span>)</span> implies  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f1</font> )
 <b>by </b><i><a class="txt" href="comput_1.html#E4:105"><i><font color="Green" title="E71">A3</font></i></a>, <a class="txt" href="comput_1.html#E3:105_2"><i><font color="Green" title="E78">A19</font></i></a>, <a class="txt" href="comput_1.html#E5:105_2"><i><font color="Green" title="E80">A21</font></i></a>, <a class="txt" href="comput_1.html#E7:105_2"><i><font color="Green" title="E81">A22</font></i></a>, <a class="ref" href="comput_1.html#T6" target="_self" title="COMPUT_1:th.6">Th6</a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></i>;<br/>

<b>assume </b><a NAME="E2:105_2_2"/>
 <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f1</font>
 ;<br/>

<b>then </b> <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span> = 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0</span>)</span> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="txt" href="comput_1.html#E1:105"><i><font color="Green" title="E69">A1</font></i></a>, <a class="txt" href="comput_1.html#E4:105_2"><i><font color="Green" title="E79">A20</font></i></a>, <a class="ref" href="comput_1.html#T6" target="_self" title="COMPUT_1:th.6">Th6</a>, <a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a>, <a class="ref" href="funct_7.html#T32" title="FUNCT_7:th.32">FUNCT_7:32</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="funct_7.html#T36" title="FUNCT_7:th.36">FUNCT_7:36</a></i>
;<br/>
<b>hence </b><a NAME="E4:105_2_2"/>
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:105_2"><i><font color="Green" title="E77">A18</font></i></a>, <a class="txt" href="comput_1.html#E3:105_2"><i><font color="Green" title="E78">A19</font></i></a>, <a class="txt" href="comput_1.html#E7:105_2"><i><font color="Green" title="E81">A22</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E9:105_2"/>
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0</span>)</span></span>)</span>
 ;<br/><a NAME="E10:105_2"/><b>then </b>
<font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0</span>)</span>,<font color="Maroon" title="c1">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E4:105"><i><font color="Green" title="E71">A3</font></i></a>, <a class="txt" href="comput_1.html#E3:105_2"><i><font color="Green" title="E78">A19</font></i></a>, <a class="txt" href="comput_1.html#E4:105_2"><i><font color="Green" title="E79">A20</font></i></a>, <a class="txt" href="comput_1.html#E5:105_2"><i><font color="Green" title="E80">A21</font></i></a>, <a class="txt" href="comput_1.html#E7:105_2"><i><font color="Green" title="E81">A22</font></i></a>, <a class="ref" href="funct_7.html#T36" title="FUNCT_7:th.36">FUNCT_7:36</a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></i>;<br/><a NAME="E11:105_2"/><b>then </b>
( <font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c1">i</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="ref" href="comput_1.html#T6" target="_self" title="COMPUT_1:th.6">Th6</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E12:105_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c1">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:105_2"><i><font color="Green" title="E77">A18</font></i></a>, <a class="txt" href="comput_1.html#E3:105_2"><i><font color="Green" title="E78">A19</font></i></a>, <a class="txt" href="comput_1.html#E7:105_2"><i><font color="Green" title="E81">A22</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:105"/><i><font color="Green" title="E77">A23</font></i>: 
<font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">G</font>
 
<b>by </b><i><a class="txt" href="comput_1.html#E7:105"><i><font color="Green" title="E73">A5</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">rngG</font> = <font color="Maroon" title="c7">rngG</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="fraenkel.html#V1" title="FRAENKEL:attr.1">functional</a> <a href="comput_1.html#V1" title="COMPUT_1:attr.1">compatible</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<b>thus </b><a NAME="E13:105"/>
( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K9" title="COMPUT_1:func.9">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> iff  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:105_3"/>
( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K9" title="COMPUT_1:func.9">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> implies  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f1</font> )
 <b>by </b><i><a class="txt" href="comput_1.html#E8:105"><i><font color="Green" title="E74">A6</font></i></a>, <a class="txt" href="comput_1.html#E10:105"><i><font color="Green" title="E76">A17</font></i></a></i>;<br/>

<b>assume </b><a NAME="E2:105_3"/><i><font color="Green" title="E78">A24</font></i>: 
 <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f1</font>
 ;<br/>

<a NAME="E3:105_3"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">f</font></span>)</span> where <font color="Olive" title="b1">f</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">rngG</font> : verum </span>}</span> 
 
<b>by </b><i><a class="txt" href="comput_1.html#E11:105"><i><font color="Green" title="E77">A23</font></i></a></i>;<br/>
<a NAME="E4:105_3"/><b>then </b>
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">f</font></span>)</span> where <font color="Olive" title="b1">f</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">rngG</font> : verum </span>}</span> 
 
<b>by </b><i><a class="txt" href="comput_1.html#E10:105"><i><font color="Green" title="E76">A17</font></i></a>, <a class="txt" href="comput_1.html#E2:105_3"><i><font color="Green" title="E78">A24</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E5:105_3"/>
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K9" title="COMPUT_1:func.9">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E3:105"><i><font color="Green" title="E70">A2</font></i></a>, <a class="txt" href="comput_1.html#E5:105"><i><font color="Green" title="E72">A4</font></i></a>, <a class="ref" href="comput_1.html#T15" target="_self" title="COMPUT_1:th.15">Th15</a></i>;<br/>


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

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:105_4"/><i><font color="Green" title="E78">A25</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K9" title="COMPUT_1:func.9">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span>
 ;<br/><a NAME="E2:105_4"/><b>then </b>
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E8:105"><i><font color="Green" title="E74">A6</font></i></a></i>;<br/><a NAME="E3:105_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c9">rngG</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0</span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E11:105"><i><font color="Green" title="E77">A23</font></i></a>, <a class="ref" href="comput_1.html#T16" target="_self" title="COMPUT_1:th.16">Th16</a></i>;<br/><b>hence </b><a NAME="E4:105_4"/>
<span class="p1">(<span class="default"><a href="comput_1.html#K9" title="COMPUT_1:func.9">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c1">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E3:105"><i><font color="Green" title="E70">A2</font></i></a>, <a class="txt" href="comput_1.html#E8:105"><i><font color="Green" title="E74">A6</font></i></a>, <a class="txt" href="comput_1.html#E10:105"><i><font color="Green" title="E76">A17</font></i></a>, <a class="txt" href="comput_1.html#E1:105_4"><i><font color="Green" title="E78">A25</font></i></a>, <a class="ref" href="card_3.html#D4" title="CARD_3:def.4">CARD_3:def 4</a></i>;<br/></div>
<b>end;</b></div>

<b>set </b><font color="Maroon" title="c10">pm1</font> = <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c11">pm</font> = <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font>;<br/>
<b>set </b><font color="Maroon" title="c12">pc</font> = <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>;<br/>
<a NAME="E15:105"/><i><font color="Green" title="E78">A26</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_7.html#T32" title="FUNCT_7:th.32">FUNCT_7:32</a></i>;<br/>
<a NAME="E16:105"/><i><font color="Green" title="E79">A27</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_7.html#T36" title="FUNCT_7:th.36">FUNCT_7:36</a></i>;<br/>
<div><i><font color="Green" title="E80">A28</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="105_5"><b>now </b></a><div class="add"><a NAME="E1:105_5"/><i><font color="Green" title="E81">A29</font></i>: 
( <font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comput_1.html#K8" title="COMPUT_1:func.8">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comput_1.html#K8" title="COMPUT_1:func.8">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> )
 <b>by </b><i><a class="txt" href="comput_1.html#E4:105"><i><font color="Green" title="E71">A3</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c13">F1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:105_5"/><i><font color="Green" title="E82">A30</font></i>: 
 <a href="comput_1.html#K8" title="COMPUT_1:func.8">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c1">i</font></span>)</span>
 <b>and </b><br/><a NAME="E4:105_5"/><i><font color="Green" title="E83">A31</font></i>: 
( ( <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> &amp;  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f1</font> implies <font color="Maroon" title="c13">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">i</font></span>)</span></span>)</span> ) &amp; ( (  not <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> or  not  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f1</font> ) implies <font color="Maroon" title="c13">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) )
 <b>and </b><br/><a NAME="E5:105_5"/><i><font color="Green" title="E84">A32</font></i>: 
for <font color="Olive" title="b1">M</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">M</font>,<font color="Maroon" title="c13">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">M</font>,<font color="Maroon" title="c13">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">M</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c4">f2</font>]
 <b>by </b><i><a class="ref" href="comput_1.html#D13" target="_self" title="COMPUT_1:def.13">Def13</a></i>;<br/><b>consider </b><font color="Maroon" title="c14">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E7:105_5"/><i><font color="Green" title="E85">A33</font></i>: 
 <a href="comput_1.html#K8" title="COMPUT_1:func.8">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c1">i</font></span>)</span>
 <b>and </b><br/><a NAME="E8:105_5"/><i><font color="Green" title="E86">A34</font></i>: 
( ( <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> &amp;  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f1</font> implies <font color="Maroon" title="c14">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,0</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span>,<font color="Maroon" title="c1">i</font></span>)</span></span>)</span> ) &amp; ( (  not <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> or  not  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f1</font> ) implies <font color="Maroon" title="c14">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) )
 <b>and </b><br/><a NAME="E9:105_5"/><i><font color="Green" title="E87">A35</font></i>: 
for <font color="Olive" title="b1">M</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">M</font>,<font color="Maroon" title="c14">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">M</font>,<font color="Maroon" title="c14">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">M</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c4">f2</font>]
 <b>by </b><i><a class="ref" href="comput_1.html#D13" target="_self" title="COMPUT_1:def.13">Def13</a></i>;<br/><i><font color="Green" title="E88">A36</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c1">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="comput_1.html#K2" title="COMPUT_1:func.2">.</a> <font color="Maroon" title="c1">i</font>
<b>by </b><i><a class="txt" href="comput_1.html#E1:105"><i><font color="Green" title="E69">A1</font></i></a>, <a class="txt" href="comput_1.html#E15:105"><i><font color="Green" title="E78">A26</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<b>by </b><i><a class="txt" href="comput_1.html#E1:105"><i><font color="Green" title="E69">A1</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>
;<br/><i><font color="Green" title="E89">A37</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c1">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="comput_1.html#K2" title="COMPUT_1:func.2">.</a> <font color="Maroon" title="c1">i</font>
<b>by </b><i><a class="txt" href="comput_1.html#E1:105"><i><font color="Green" title="E69">A1</font></i></a>, <a class="txt" href="comput_1.html#E15:105"><i><font color="Green" title="E78">A26</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c2">m</font>
<b>by </b><i><a class="txt" href="comput_1.html#E1:105"><i><font color="Green" title="E69">A1</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>
;<br/><a NAME="E12:105_5"/><i><font color="Green" title="E90">A38</font></i>: 
<font color="Maroon" title="c13">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">m</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E4:105_5"><i><font color="Green" title="E83">A31</font></i></a>, <a class="txt" href="comput_1.html#E5:105_5"><i><font color="Green" title="E84">A32</font></i></a>, <a class="txt" href="comput_1.html#E8:105_5"><i><font color="Green" title="E86">A34</font></i></a>, <a class="txt" href="comput_1.html#E9:105_5"><i><font color="Green" title="E87">A35</font></i></a>, <a class="txt" href="comput_1.html#E78"><i><font color="Green" title="E64">Lm3</font></i></a></i>;<br/><a NAME="E13:105_5"/><i><font color="Green" title="E91">A39</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:105"><i><font color="Green" title="E69">A1</font></i></a>, <a class="txt" href="comput_1.html#E4:105_5"><i><font color="Green" title="E83">A31</font></i></a>, <a class="txt" href="comput_1.html#E5:105_5"><i><font color="Green" title="E84">A32</font></i></a>, <a class="txt" href="comput_1.html#E79"><i><font color="Green" title="E65">Lm4</font></i></a></i>;<br/><a NAME="E14:105_5"/><i><font color="Green" title="E92">A40</font></i>: 
not <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:105"><i><font color="Green" title="E69">A1</font></i></a>, <a class="txt" href="comput_1.html#E4:105_5"><i><font color="Green" title="E83">A31</font></i></a>, <a class="txt" href="comput_1.html#E5:105_5"><i><font color="Green" title="E84">A32</font></i></a>, <a class="txt" href="comput_1.html#E79"><i><font color="Green" title="E65">Lm4</font></i></a></i>;<br/><b>thus </b><a NAME="E15:105_5"/><i><font color="Green" title="E93">A41</font></i>: 
( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> iff ( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f2</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_5_3"><b>proof </b></a><div class="add">

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:105_5_3_1"/><i><font color="Green" title="E94">A42</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E2:105_5_3_1"/><b>then </b><i><font color="Green" title="E95">A43</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E4:105"><i><font color="Green" title="E71">A3</font></i></a>, <a class="txt" href="comput_1.html#E3:105_5"><i><font color="Green" title="E82">A30</font></i></a>, <a class="txt" href="comput_1.html#E10:105_5"><i><font color="Green" title="E88">A36</font></i></a></i>;<br/><b>assume </b><a NAME="E3:105_5_3_1"/><i><font color="Green" title="E96">A44</font></i>: 
(  not <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span> or  not <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f2</font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_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="E1:105_5_3_1_1"/>
(  not <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span> or  not <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f2</font> )
 </span><b>by </b><i><a class="txt" href="comput_1.html#E3:105_5_3_1"><i><font color="Green" title="E96">A44</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_5_3_1_1_1"><b>suppose </b></a><a NAME="E1:105_5_3_1_1_1"/>
not <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:105_5_3_1_1_1"/><b>then </b>
not <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E4:105"><i><font color="Green" title="E71">A3</font></i></a>, <a class="txt" href="comput_1.html#E7:105_5"><i><font color="Green" title="E85">A33</font></i></a>, <a class="txt" href="comput_1.html#E11:105_5"><i><font color="Green" title="E89">A37</font></i></a>, <a class="txt" href="comput_1.html#E12:105_5"><i><font color="Green" title="E90">A38</font></i></a></i>;<br/><b>hence </b><a NAME="E3:105_5_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="comput_1.html#E16:105"><i><font color="Green" title="E79">A27</font></i></a>, <a class="txt" href="comput_1.html#E5:105_5"><i><font color="Green" title="E84">A32</font></i></a>, <a class="txt" href="comput_1.html#E14:105_5"><i><font color="Green" title="E92">A40</font></i></a>, <a class="txt" href="comput_1.html#E2:105_5_3_1"><i><font color="Green" title="E95">A43</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_5_3_1_1_2"><b>suppose </b></a><a NAME="E1:105_5_3_1_1_2"/>
not <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:105_5_3_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="comput_1.html#E16:105"><i><font color="Green" title="E79">A27</font></i></a>, <a class="txt" href="comput_1.html#E1:105_5"><i><font color="Green" title="E81">A29</font></i></a>, <a class="txt" href="comput_1.html#E3:105_5"><i><font color="Green" title="E82">A30</font></i></a>, <a class="txt" href="comput_1.html#E5:105_5"><i><font color="Green" title="E84">A32</font></i></a>, <a class="txt" href="comput_1.html#E10:105_5"><i><font color="Green" title="E88">A36</font></i></a>, <a class="txt" href="comput_1.html#E13:105_5"><i><font color="Green" title="E91">A39</font></i></a>, <a class="txt" href="comput_1.html#E14:105_5"><i><font color="Green" title="E92">A40</font></i></a>, <a class="txt" href="comput_1.html#E1:105_5_3_1"><i><font color="Green" title="E94">A42</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:105_5_3"/>
( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f2</font> )
 ;<br/>

<a NAME="E3:105_5_3"/><b>then </b><i><font color="Green" title="E94">A45</font></i>: 
<font color="Maroon" title="c13">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p4"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c13">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="comput_1.html#E1:105"><i><font color="Green" title="E69">A1</font></i></a>, <a class="txt" href="comput_1.html#E15:105"><i><font color="Green" title="E78">A26</font></i></a>, <a class="txt" href="comput_1.html#E16:105"><i><font color="Green" title="E79">A27</font></i></a>, <a class="txt" href="comput_1.html#E1:105_5"><i><font color="Green" title="E81">A29</font></i></a>, <a class="txt" href="comput_1.html#E3:105_5"><i><font color="Green" title="E82">A30</font></i></a>, <a class="txt" href="comput_1.html#E5:105_5"><i><font color="Green" title="E84">A32</font></i></a>, <a class="txt" href="comput_1.html#E7:105_5"><i><font color="Green" title="E85">A33</font></i></a>, <a class="txt" href="comput_1.html#E10:105_5"><i><font color="Green" title="E88">A36</font></i></a>, <a class="txt" href="comput_1.html#E11:105_5"><i><font color="Green" title="E89">A37</font></i></a>, <a class="txt" href="comput_1.html#E12:105_5"><i><font color="Green" title="E90">A38</font></i></a>, <a class="txt" href="comput_1.html#E13:105_5"><i><font color="Green" title="E91">A39</font></i></a></i>;<br/>
<a NAME="E4:105_5_3"/>
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:105_5_3"/><b>then </b>
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p4"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c13">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E6:105_5_3"/><b>then </b>
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p5"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c13">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E7:105_5_3"/>
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:105_5"><i><font color="Green" title="E81">A29</font></i></a>, <a class="txt" href="comput_1.html#E3:105_5"><i><font color="Green" title="E82">A30</font></i></a>, <a class="txt" href="comput_1.html#E10:105_5"><i><font color="Green" title="E88">A36</font></i></a>, <a class="txt" href="comput_1.html#E3:105_5_3"><i><font color="Green" title="E94">A45</font></i></a>, <a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E16:105_5"/><i><font color="Green" title="E94">A46</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E17:105_5"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f2</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E16:105"><i><font color="Green" title="E79">A27</font></i></a>, <a class="txt" href="comput_1.html#E1:105_5"><i><font color="Green" title="E81">A29</font></i></a>, <a class="txt" href="comput_1.html#E3:105_5"><i><font color="Green" title="E82">A30</font></i></a>, <a class="txt" href="comput_1.html#E5:105_5"><i><font color="Green" title="E84">A32</font></i></a>, <a class="txt" href="comput_1.html#E10:105_5"><i><font color="Green" title="E88">A36</font></i></a>, <a class="txt" href="comput_1.html#E15:105_5"><i><font color="Green" title="E93">A41</font></i></a></i>;<br/><a NAME="E18:105_5"/><b>then </b><i><font color="Green" title="E95">A47</font></i>: 
<font color="Maroon" title="c13">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p4"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c13">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:105"><i><font color="Green" title="E69">A1</font></i></a>, <a class="txt" href="comput_1.html#E15:105"><i><font color="Green" title="E78">A26</font></i></a>, <a class="txt" href="comput_1.html#E16:105"><i><font color="Green" title="E79">A27</font></i></a>, <a class="txt" href="comput_1.html#E1:105_5"><i><font color="Green" title="E81">A29</font></i></a>, <a class="txt" href="comput_1.html#E5:105_5"><i><font color="Green" title="E84">A32</font></i></a>, <a class="txt" href="comput_1.html#E7:105_5"><i><font color="Green" title="E85">A33</font></i></a>, <a class="txt" href="comput_1.html#E11:105_5"><i><font color="Green" title="E89">A37</font></i></a>, <a class="txt" href="comput_1.html#E12:105_5"><i><font color="Green" title="E90">A38</font></i></a>, <a class="txt" href="comput_1.html#E15:105_5"><i><font color="Green" title="E93">A41</font></i></a>, <a class="txt" href="comput_1.html#E16:105_5"><i><font color="Green" title="E94">A46</font></i></a></i>;<br/><a NAME="E19:105_5"/><i><font color="Green" title="E96">A48</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E20:105_5"/><b>then </b>
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p4"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c13">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p4"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c13">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="comput_1.html#E1:105_5"><i><font color="Green" title="E81">A29</font></i></a>, <a class="txt" href="comput_1.html#E3:105_5"><i><font color="Green" title="E82">A30</font></i></a>, <a class="txt" href="comput_1.html#E10:105_5"><i><font color="Green" title="E88">A36</font></i></a>, <a class="txt" href="comput_1.html#E18:105_5"><i><font color="Green" title="E95">A47</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon" title="c4">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="comput_1.html#E16:105"><i><font color="Green" title="E79">A27</font></i></a>, <a class="txt" href="comput_1.html#E1:105_5"><i><font color="Green" title="E81">A29</font></i></a>, <a class="txt" href="comput_1.html#E3:105_5"><i><font color="Green" title="E82">A30</font></i></a>, <a class="txt" href="comput_1.html#E10:105_5"><i><font color="Green" title="E88">A36</font></i></a>, <a class="txt" href="comput_1.html#E13:105_5"><i><font color="Green" title="E91">A39</font></i></a>, <a class="txt" href="comput_1.html#E19:105_5"><i><font color="Green" title="E96">A48</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/><br/></div><b>end;</b></div>
<b>thus </b><a NAME="E18:105"/>
( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K9" title="COMPUT_1:func.9">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> iff ( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K9" title="COMPUT_1:func.9">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comput_1.html#K9" title="COMPUT_1:func.9">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f2</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_6"><b>proof </b></a><div class="add">

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:105_6_1"/><i><font color="Green" title="E81">A49</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K9" title="COMPUT_1:func.9">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span>
 ;<br/><a NAME="E2:105_6_1"/>
<font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">G</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E7:105"><i><font color="Green" title="E73">A5</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E3:105_6_1"/><b>then </b>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">f</font></span>)</span> where <font color="Olive" title="b1">f</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">rngG</font> : verum </span>}</span> 
 ;<br/><a NAME="E4:105_6_1"/><b>then </b>
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">f</font></span>)</span> where <font color="Olive" title="b1">f</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">rngG</font> : verum </span>}</span> 
 <b>by </b><i><a class="txt" href="comput_1.html#E8:105"><i><font color="Green" title="E74">A6</font></i></a>, <a class="txt" href="comput_1.html#E17:105"><i><font color="Green" title="E80">A28</font></i></a>, <a class="txt" href="comput_1.html#E1:105_6_1"><i><font color="Green" title="E81">A49</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E5:105_6_1"/>
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K9" title="COMPUT_1:func.9">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E3:105"><i><font color="Green" title="E70">A2</font></i></a>, <a class="txt" href="comput_1.html#E5:105"><i><font color="Green" title="E72">A4</font></i></a>, <a class="ref" href="comput_1.html#T15" target="_self" title="COMPUT_1:th.15">Th15</a></i>;<br/><a NAME="E6:105_6_1"/><i><font color="Green" title="E82">A50</font></i>: 
<font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">G</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E7:105"><i><font color="Green" title="E73">A5</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E7:105_6_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:105"><i><font color="Green" title="E69">A1</font></i></a>, <a class="txt" href="comput_1.html#E4:105"><i><font color="Green" title="E71">A3</font></i></a>, <a class="txt" href="comput_1.html#E80"><i><font color="Green" title="E66">Lm5</font></i></a></i>;<br/><a NAME="E8:105_6_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c9">rngG</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E8:105"><i><font color="Green" title="E74">A6</font></i></a>, <a class="txt" href="comput_1.html#E17:105"><i><font color="Green" title="E80">A28</font></i></a>, <a class="txt" href="comput_1.html#E1:105_6_1"><i><font color="Green" title="E81">A49</font></i></a>, <a class="txt" href="comput_1.html#E6:105_6_1"><i><font color="Green" title="E82">A50</font></i></a>, <a class="ref" href="comput_1.html#T16" target="_self" title="COMPUT_1:th.16">Th16</a></i>;<br/><b>hence </b><a NAME="E9:105_6_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comput_1.html#K9" title="COMPUT_1:func.9">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f2</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E3:105"><i><font color="Green" title="E70">A2</font></i></a>, <a class="txt" href="comput_1.html#E8:105"><i><font color="Green" title="E74">A6</font></i></a>, <a class="txt" href="comput_1.html#E16:105"><i><font color="Green" title="E79">A27</font></i></a>, <a class="txt" href="comput_1.html#E17:105"><i><font color="Green" title="E80">A28</font></i></a>, <a class="txt" href="comput_1.html#E1:105_6_1"><i><font color="Green" title="E81">A49</font></i></a>, <a class="ref" href="card_3.html#D4" title="CARD_3:def.4">CARD_3:def 4</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:105_6"/><i><font color="Green" title="E81">A51</font></i>: 
( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K9" title="COMPUT_1:func.9">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comput_1.html#K9" title="COMPUT_1:func.9">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f2</font> )
 ;<br/>

<a NAME="E3:105_6"/><b>then </b><i><font color="Green" title="E82">A52</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="comput_1.html#E8:105"><i><font color="Green" title="E74">A6</font></i></a></i>;<br/>
<a NAME="E4:105_6"/><i><font color="Green" title="E83">A53</font></i>: 
<font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">G</font>
 
<b>by </b><i><a class="txt" href="comput_1.html#E7:105"><i><font color="Green" title="E73">A5</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:105_6"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="comput_1.html#E1:105"><i><font color="Green" title="E69">A1</font></i></a>, <a class="txt" href="comput_1.html#E4:105"><i><font color="Green" title="E71">A3</font></i></a>, <a class="txt" href="comput_1.html#E80"><i><font color="Green" title="E66">Lm5</font></i></a></i>;<br/>
<a NAME="E6:105_6"/><b>then </b><i><font color="Green" title="E84">A54</font></i>: 
<span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c9">rngG</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="comput_1.html#E3:105_6"><i><font color="Green" title="E82">A52</font></i></a>, <a class="txt" href="comput_1.html#E4:105_6"><i><font color="Green" title="E83">A53</font></i></a>, <a class="ref" href="comput_1.html#T16" target="_self" title="COMPUT_1:th.16">Th16</a></i>;<br/>
<a NAME="E7:105_6"/>
<font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">G</font>
 
<b>by </b><i><a class="txt" href="comput_1.html#E7:105"><i><font color="Green" title="E73">A5</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:105_6"/><b>then </b>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">f</font></span>)</span> where <font color="Olive" title="b1">f</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">rngG</font> : verum </span>}</span> 
 
;<br/>
<a NAME="E9:105_6"/><b>then </b>
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">f</font></span>)</span> where <font color="Olive" title="b1">f</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">rngG</font> : verum </span>}</span> 
 
<b>by </b><i><a class="txt" href="comput_1.html#E3:105"><i><font color="Green" title="E70">A2</font></i></a>, <a class="txt" href="comput_1.html#E8:105"><i><font color="Green" title="E74">A6</font></i></a>, <a class="txt" href="comput_1.html#E16:105"><i><font color="Green" title="E79">A27</font></i></a>, <a class="txt" href="comput_1.html#E17:105"><i><font color="Green" title="E80">A28</font></i></a>, <a class="txt" href="comput_1.html#E2:105_6"><i><font color="Green" title="E81">A51</font></i></a>, <a class="txt" href="comput_1.html#E6:105_6"><i><font color="Green" title="E84">A54</font></i></a>, <a class="ref" href="card_3.html#D4" title="CARD_3:def.4">CARD_3:def 4</a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E10:105_6"/>
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K9" title="COMPUT_1:func.9">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E3:105"><i><font color="Green" title="E70">A2</font></i></a>, <a class="txt" href="comput_1.html#E5:105"><i><font color="Green" title="E72">A4</font></i></a>, <a class="ref" href="comput_1.html#T15" target="_self" title="COMPUT_1:th.15">Th15</a></i>;<br/>


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

<b>assume </b><a NAME="E19:105"/><i><font color="Green" title="E81">A55</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K9" title="COMPUT_1:func.9">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span>
 ;<br/>

<a NAME="E20:105"/><i><font color="Green" title="E82">A56</font></i>: 
<font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">G</font>
 
<b>by </b><i><a class="txt" href="comput_1.html#E7:105"><i><font color="Green" title="E73">A5</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E21:105"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="comput_1.html#E1:105"><i><font color="Green" title="E69">A1</font></i></a>, <a class="txt" href="comput_1.html#E4:105"><i><font color="Green" title="E71">A3</font></i></a>, <a class="txt" href="comput_1.html#E80"><i><font color="Green" title="E66">Lm5</font></i></a></i>;<br/>
<a NAME="E22:105"/><b>then </b>
( <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c9">rngG</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c9">rngG</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="comput_1.html#E8:105"><i><font color="Green" title="E74">A6</font></i></a>, <a class="txt" href="comput_1.html#E17:105"><i><font color="Green" title="E80">A28</font></i></a>, <a class="txt" href="comput_1.html#E19:105"><i><font color="Green" title="E81">A55</font></i></a>, <a class="txt" href="comput_1.html#E20:105"><i><font color="Green" title="E82">A56</font></i></a>, <a class="ref" href="comput_1.html#T16" target="_self" title="COMPUT_1:th.16">Th16</a></i>;<br/>
<b>hence </b><a NAME="E23:105"/>
<span class="p1">(<span class="default"><a href="comput_1.html#K9" title="COMPUT_1:func.9">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comput_1.html#K9" title="COMPUT_1:func.9">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E3:105"><i><font color="Green" title="E70">A2</font></i></a>, <a class="txt" href="comput_1.html#E5:105"><i><font color="Green" title="E72">A4</font></i></a>, <a class="txt" href="comput_1.html#E8:105"><i><font color="Green" title="E74">A6</font></i></a>, <a class="txt" href="comput_1.html#E17:105"><i><font color="Green" title="E80">A28</font></i></a>, <a class="txt" href="comput_1.html#E19:105"><i><font color="Green" title="E81">A55</font></i></a>, <a class="ref" href="funct_7.html#T36" title="FUNCT_7:th.36">FUNCT_7:36</a></i>;<br/>


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