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<span class="kw">let </span><font color="Maroon" title="c1">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">e1</font>, <font color="Olive" title="b2">e2</font> being  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>   <a href="comput_1.html#NV2" title="COMPUT_1:NV.2">to-naturals</a>   <a href="margrel1.html#V2" title="MARGREL1:attr.2">homogeneous</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><br/>  for <font color="Olive" title="b3">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Olive" title="b1">e1</font>,<font color="Olive" title="b2">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b3">p</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="margrel1.html#K19" title="MARGREL1:func.19">arity</a> <font color="Olive" title="b1">e1</font></span>)</span></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></span><br/><span class="kw">let </span><font color="Maroon" title="c2">e1</font>, <font color="Maroon" title="c3">e2</font> be  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>   <a href="comput_1.html#NV2" title="COMPUT_1:NV.2">to-naturals</a>   <a href="margrel1.html#V2" title="MARGREL1:attr.2">homogeneous</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b1">p</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="margrel1.html#K19" title="MARGREL1:func.19">arity</a> <font color="Maroon" title="c2">e1</font></span>)</span></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></span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">f1</font> = <font color="Maroon" title="c2">e1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">f2</font> = <font color="Maroon" title="c3">e2</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">p</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c6">p</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="margrel1.html#K19" title="MARGREL1:func.19">arity</a> <font color="Maroon" title="c2">e1</font></span>)</span></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></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:92_1"/>
( <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="c6">p</font> or  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> <font color="Maroon" title="c6">p</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1"><span class="kw">suppose </span></a><a NAME="E1:92_1_1"/><span class="lab"><font color="Green" title="E55">A1</font></span>: 
<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="c6">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c6">p</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="margrel1.html#K19" title="MARGREL1:func.19">arity</a> <font color="Maroon" title="c2">e1</font></span>)</span></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></span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c7">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>,<span class="p1">(<span class="default"><a href="comput_1.html#K2" title="COMPUT_1:func.2">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:92_1_1"/><span class="lab"><font color="Green" title="E56">A2</font></span>: 
 <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c6">p</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">i</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:92_1_1"/><span class="lab"><font color="Green" title="E57">A3</font></span>: 
( <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="c6">p</font> &amp;  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c1">i</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">e1</font> implies <font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">e1</font> <a href="recdef_1.html#K1" title="RECDEF_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="c6">p</font>,<font color="Maroon" title="c1">i</font>)</span>)</span></span>)</span> )
 <span class="kw">and </span><br/><a NAME="E5:92_1_1"/><span class="lab"><font color="Green" title="E58">A4</font></span>: 
( (  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> <font color="Maroon" title="c6">p</font> or  not  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c1">i</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">e1</font> ) implies <font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <span class="kw">and </span><br/><a NAME="E6:92_1_1"/><span class="lab"><font color="Green" title="E59">A5</font></span>: 
 for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">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="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</font>,<font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c3">e2</font>]
 <span class="kw">by</span> <span class="lab"><a class="ref" href="comput_1.html#D10" target="_self" title="COMPUT_1:def.10">Def13</a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span> <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="margrel1.html#K19" title="MARGREL1:func.19">arity</a> <font color="Maroon" title="c2">e1</font></span>)</span></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/><div><span class="lab"><font color="Green" title="E60">A6</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">m</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:92_1_1_1"/><span class="lab"><font color="Green" title="E61">A7</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">m</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><span class="kw">set </span><font color="Maroon" title="c9">pc</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c8">m</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c8">m</font>)</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:92_1_1_1_1"/>
( ( <font color="Maroon" title="c6">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c8">m</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c8">m</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c8">m</font>)</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">e2</font> ) or  not <font color="Maroon" title="c6">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c8">m</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">m</font></span>)</span> or  not <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c8">m</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c8">m</font>)</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">e2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:92_1_1_1_1_1"/><span class="lab"><font color="Green" title="E62">A8</font></span>: 
( <font color="Maroon" title="c6">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c8">m</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c8">m</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c8">m</font>)</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">e2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"> <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c8">m</font>)</span>)</span> = 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T30" title="FUNCT_7:th.30">FUNCT_7:30</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T30" title="FUNCT_7:th.30">FUNCT_7:30</a></span>
;<br/><a NAME="E3:92_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A9</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c8">m</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/><a NAME="E4:92_1_1_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c8">m</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="margrel1.html#K19" title="MARGREL1:func.19">arity</a> <font color="Maroon" title="c2">e1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92_1_1_1"><span class="lab"><font color="Green" title="E61">A7</font></span></a>, <a class="txt" href="#E1:92_1_1_1_1_1"><span class="lab"><font color="Green" title="E62">A8</font></span></a>, <a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/><a NAME="E5:92_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="margrel1.html#K19" title="MARGREL1:func.19">arity</a> <font color="Maroon" title="c2">e1</font></span>)</span></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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:92_1_1_1_1_1"><span class="lab"><font color="Green" title="E63">A9</font></span></a>, <a class="ref" href="finseq_2.html#T92" title="FINSEQ_2:th.92">FINSEQ_2:92</a></span>;<br/><a NAME="E6:92_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A10</font></span>: 
<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="c6">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="margrel1.html#K19" title="MARGREL1:func.19">arity</a> <font color="Maroon" title="c2">e1</font></span>)</span></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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>;<br/><a NAME="E7:92_1_1_1_1_1"/>
<font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">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">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">e2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c8">m</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c8">m</font>)</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92_1_1"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="txt" href="#E6:92_1_1"><span class="lab"><font color="Green" title="E59">A5</font></span></a>, <a class="txt" href="#E1:92_1_1_1_1_1"><span class="lab"><font color="Green" title="E62">A8</font></span></a></span>;<br/><a NAME="E8:92_1_1_1_1_1"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">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="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">e2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c8">m</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p5"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c8">m</font>)</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></span>;<br/><a NAME="E9:92_1_1_1_1_1"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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="c6">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E10:92_1_1_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92_1_1_1"><span class="lab"><font color="Green" title="E61">A7</font></span></a>, <a class="txt" href="#E6:92_1_1_1_1_1"><span class="lab"><font color="Green" title="E64">A10</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:92_1_1_1_1_2"/>
(  not <font color="Maroon" title="c6">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c8">m</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">m</font></span>)</span> or  not <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c8">m</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c8">m</font>)</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">e2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:92_1_1_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:92_1_1"><span class="lab"><font color="Green" title="E59">A5</font></span></a>, <a class="txt" href="#E1:92_1_1_1"><span class="lab"><font color="Green" title="E61">A7</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E8:92_1_1"/><span class="lab"><font color="Green" title="E61">A11</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_2"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:92_1_1_2_1"/>
(  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c1">i</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">e1</font> or  not  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c1">i</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">e1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:92_1_1_2_1_1"/><span class="lab"><font color="Green" title="E62">A12</font></span>: 
 <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c1">i</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">e1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]</span><br/><div class="add"><a NAME="E2:92_1_1_2_1_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">e1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K19" title="MARGREL1:func.19">arity</a> <font color="Maroon" title="c2">e1</font></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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="comput_1.html#T21" target="_self" title="COMPUT_1:th.21">Th24</a></span>;<br/><a NAME="E3:92_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A13</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c1">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K19" title="MARGREL1:func.19">arity</a> <font color="Maroon" title="c2">e1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92_1_1_2_1_1"><span class="lab"><font color="Green" title="E62">A12</font></span></a>, <a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/><a NAME="E4:92_1_1_2_1_1"/><span class="lab"><font color="Green" title="E64">A14</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">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="c6">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T30" title="FUNCT_7:th.30">FUNCT_7:30</a></span>;<br/><a NAME="E5:92_1_1_2_1_1"/>
<font color="Maroon" title="c6">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92_1_1"><span class="lab"><font color="Green" title="E55">A1</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E7:92_1_1_2_1_1"/><span class="lab"><font color="Green" title="E65">A15</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></span>;<br/><a NAME="E8:92_1_1_2_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c1">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92_1_1"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="txt" href="#E7:92_1_1_2_1_1"><span class="lab"><font color="Green" title="E65">A15</font></span></a>, <a class="ref" href="finseq_3.html#T109" title="FINSEQ_3:th.109">FINSEQ_3:109</a></span>;<br/><a NAME="E9:92_1_1_2_1_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="margrel1.html#K19" title="MARGREL1:func.19">arity</a> <font color="Maroon" title="c2">e1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:92_1_1_2_1_1"><span class="lab"><font color="Green" title="E63">A13</font></span></a>, <a class="txt" href="#E7:92_1_1_2_1_1"><span class="lab"><font color="Green" title="E65">A15</font></span></a>, <a class="txt" href="#E4:92_1_1_2_1_1"><span class="lab"><font color="Green" title="E64">A14</font></span></a>, <a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/><a NAME="E10:92_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A16</font></span>: 
<font color="Maroon" title="c6">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>) is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="margrel1.html#K19" title="MARGREL1:func.19">arity</a> <font color="Maroon" title="c2">e1</font></span>)</span></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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T92" title="FINSEQ_2:th.92">FINSEQ_2:92</a></span>;<br/><a NAME="E11:92_1_1_2_1_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <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="c6">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92_1_1"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="txt" href="#E4:92_1_1"><span class="lab"><font color="Green" title="E57">A3</font></span></a>, <a class="txt" href="#E1:92_1_1_2_1_1"><span class="lab"><font color="Green" title="E62">A12</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E12:92_1_1_2_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:92_1_1_2_1_1"><span class="lab"><font color="Green" title="E66">A16</font></span></a>, <a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:92_1_1_2_1_2"/>
 not  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c1">i</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">e1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:92_1_1_2_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:92_1_1"><span class="lab"><font color="Green" title="E58">A4</font></span></a>, <a class="ref" href="relat_1.html#T38" title="RELAT_1:th.38">RELAT_1:38</a>, <a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E9:92_1_1"/>
 for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">m</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E8:92_1_1"><span class="lab"><font color="Green" title="E61">A11</font></span></a>, <a class="txt" href="#E7:92_1_1"><span class="lab"><font color="Green" title="E60">A6</font></span></a>);<br/></span><span class="kw">hence </span><a NAME="E10:92_1_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c6">p</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="margrel1.html#K19" title="MARGREL1:func.19">arity</a> <font color="Maroon" title="c2">e1</font></span>)</span></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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:92_1_1"><span class="lab"><font color="Green" title="E56">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_2"><span class="kw">suppose </span></a><a NAME="E1:92_1_2"/>
 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> <font color="Maroon" title="c6">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c6">p</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="margrel1.html#K19" title="MARGREL1:func.19">arity</a> <font color="Maroon" title="c2">e1</font></span>)</span></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></span><br/><div class="add"><a NAME="E2:92_1_2"/><span class="kw">then </span>
 <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c6">p</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="comput_1.html#T50" target="_self" title="COMPUT_1:th.50">Th54</a></span>;<br/><span class="kw">hence </span><a NAME="E3:92_1_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c6">p</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="margrel1.html#K19" title="MARGREL1:func.19">arity</a> <font color="Maroon" title="c2">e1</font></span>)</span></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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T38" title="RELAT_1:th.38">RELAT_1:38</a>, <a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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