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

<span class="kw">reconsider </span><font color="Maroon" title="c5">ff1</font> = <font color="Maroon" title="c1">f1</font>, <font color="Maroon" title="c6">ff2</font> = <font color="Maroon" title="c2">f2</font> as    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <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 class="kw">by</span> <span class="lab"><a class="ref" href="comput_1.html#T28" target="_self" title="COMPUT_1:th.28">Th31</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">n</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">z</font> = <font color="Maroon" title="c5">ff1</font> <a href="recdef_1.html#K1" title="RECDEF_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="c4">p</font>,<font color="Maroon" title="c3">i</font>)</span>)</span> as    <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>  ;<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> ,   <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K2" title="COMPUT_1:func.2">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,   <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <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 class="kw">means </span><font color="Maroon">S<sub>1</sub></font>[$1,$2,$3,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c6">ff2</font>];<br/>
<a NAME="E3:88_1_1"/>
<font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">Ap</font> = <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="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">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> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> <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="E5:88_1_1"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></span>;<br/>
<a NAME="E6:88_1_1"/><span class="kw">then </span>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> (<span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T45" title="PARTFUN1:th.45">PARTFUN1:45</a></span>;<br/>
<a NAME="E7:88_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A1</font></span>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="comput_1.html#K2" title="COMPUT_1:func.2">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>
 
;<br/>
<a NAME="E8:88_1_1"/>
<font color="Maroon" title="c9">Ap</font> <a href="rfunct_3.html#K5" title="RFUNCT_3:func.5">--&gt;</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">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> <font color="Maroon" title="c8">z</font>
 
;<br/>
<a NAME="E9:88_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">Ap</font> <a href="rfunct_3.html#K5" title="RFUNCT_3:func.5">--&gt;</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="comput_1.html#K2" title="COMPUT_1:func.2">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">t</font> = <font color="Maroon" title="c9">Ap</font> <a href="rfunct_3.html#K5" title="RFUNCT_3:func.5">--&gt;</a> <font color="Maroon" title="c8">z</font>, <font color="Maroon" title="c11">e</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E7:88_1_1"><span class="lab"><font color="Green" title="E50">A1</font></span></a></span>;<br/>
<a NAME="E11:88_1_1"/>
( ( <font color="Maroon" title="c3">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="c4">p</font> &amp;  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">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="c1">f1</font> ) or  not <font color="Maroon" title="c3">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="c4">p</font> or  not  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">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="c1">f1</font> )
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">A</font> being    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <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 class="kw"> such that </span><br/><a NAME="E13:88_1_1"/><span class="lab"><font color="Green" title="E51">A2</font></span>: 
( <font color="Maroon" title="c3">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="c4">p</font> &amp;  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">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="c1">f1</font> implies <font color="Maroon" title="c12">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">t</font> )
 <span class="kw">and </span><br/><a NAME="E14:88_1_1"/><span class="lab"><font color="Green" title="E52">A3</font></span>: 
( (  not <font color="Maroon" title="c3">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="c4">p</font> or  not  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">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="c1">f1</font> ) implies <font color="Maroon" title="c12">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">e</font> )
 ;<br/>
<a NAME="E15:88_1_1"/><span class="lab"><font color="Green" title="E53">A4</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> <br/>  for <font color="Olive" title="b2">x</font> being    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K2" title="COMPUT_1:func.2">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ex <font color="Olive" title="b3">y</font> being    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K2" title="COMPUT_1:func.2">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">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">  for <font color="Olive" title="b1">x</font> being    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K2" title="COMPUT_1:func.2">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ex <font color="Olive" title="b2">y</font> being    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K2" title="COMPUT_1:func.2">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c13">m</font>,<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c14">x</font> be    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K2" title="COMPUT_1:func.2">HFuncs</a> <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">  ex <font color="Olive" title="b1">y</font> being    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K2" title="COMPUT_1:func.2">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c13">m</font>,<font color="Maroon" title="c14">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>



<span class="kw">set </span><font color="Maroon" title="c15">f</font> = <font color="Maroon" title="c14">x</font> <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="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">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="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</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="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">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"><font color="Maroon" title="c14">x</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">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>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">z</font> = <font color="Maroon" title="c6">ff2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">m</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">x</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">m</font>)</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> as    <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>  ;<br/>
<a NAME="E2:88_1_1_1"/>
( ( <font color="Maroon" title="c3">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="c4">p</font> &amp; <font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">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> <font color="Maroon" title="c14">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">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"><font color="Maroon" title="c14">x</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">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="c2">f2</font> ) or  not <font color="Maroon" title="c3">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="c4">p</font> or  not <font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">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> <font color="Maroon" title="c14">x</font> or  not <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">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"><font color="Maroon" title="c14">x</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">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="c2">f2</font> )
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">y</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E4:88_1_1_1"/><span class="lab"><font color="Green" title="E54">A5</font></span>: 
( ( <font color="Maroon" title="c3">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="c4">p</font> &amp; <font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">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> <font color="Maroon" title="c14">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">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"><font color="Maroon" title="c14">x</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">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="c2">f2</font> &amp; <font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x</font> <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="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">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="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</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="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">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"><font color="Maroon" title="c14">x</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">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> ) or ( (  not <font color="Maroon" title="c3">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="c4">p</font> or  not <font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">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> <font color="Maroon" title="c14">x</font> or  not <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">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"><font color="Maroon" title="c14">x</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">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="c2">f2</font> ) &amp; <font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x</font> ) )
 ;<br/>
<a NAME="E5:88_1_1_1"/>
<font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">B</font> = <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="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">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> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> <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="E7:88_1_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="c18">B</font> <a href="rfunct_3.html#K5" title="RFUNCT_3:func.5">--&gt;</a> <font color="Maroon" title="c16">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">B</font>
 
<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/>
<a NAME="E8:88_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A6</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">x</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</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="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c13">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="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</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="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">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"><font color="Maroon" title="c14">x</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">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> <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> <font color="Maroon" title="c14">x</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c18">B</font>
 
<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="E9:88_1_1_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">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="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">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="E10:88_1_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="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font>
 
<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="E11:88_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</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="finseq_2.html#T92" title="FINSEQ_2:th.92">FINSEQ_2:92</a></span>;<br/>
<a NAME="E12:88_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A7</font></span>: 
<font color="Maroon" title="c18">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</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="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>;<br/>
<a NAME="E13:88_1_1_1"/><span class="lab"><font color="Green" title="E57">A8</font></span>: 
<font color="Maroon" title="c14">x</font> <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="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">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="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</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="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">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"><font color="Maroon" title="c14">x</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x</font> <a href="funct_4.html#K7" title="FUNCT_4:func.7">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">B</font> <a href="rfunct_3.html#K5" title="RFUNCT_3:func.5">--&gt;</a> <font color="Maroon" title="c16">z</font></span>)</span>
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:88_1_1_1_1"/><span class="lab"><font color="Green" title="E58">A9</font></span>: 
<font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">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> <font color="Maroon" title="c14">x</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" title="c14">x</font> <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="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">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="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</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="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">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"><font color="Maroon" title="c14">x</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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><br/><a NAME="E2:88_1_1_1_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c14">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c19">a</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="c14">x</font> or <font color="Maroon" title="c19">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</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><br/>

<a NAME="E1:88_1_1_1_1_1"/><span class="lab"><font color="Green" title="E59">A10</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">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="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">m</font>)</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/>
<span class="kw">assume </span><a NAME="E2:88_1_1_1_1_1"/><span class="lab"><font color="Green" title="E60">A11</font></span>: 
<font color="Maroon" title="c19">a</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="c14">x</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" title="c19">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</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><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">q</font> = <font color="Maroon" title="c19">a</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  ;<br/>
<a NAME="E4:88_1_1_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">q</font> <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="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">m</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88_1_1_1_1"><span class="lab"><font color="Green" title="E58">A9</font></span></a>, <a class="txt" href="#E2:88_1_1_1_1_1"><span class="lab"><font color="Green" title="E60">A11</font></span></a>, <a class="ref" href="margrel1.html#D1" title="MARGREL1:def.1">MARGREL1:def 1</a></span>;<br/>
<a NAME="E5:88_1_1_1_1_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88_1_1_1_1_1"><span class="lab"><font color="Green" title="E59">A10</font></span></a>, <a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:88_1_1_1_1_1"/>
<font color="Maroon" title="c19">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</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>
 ; <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><a NAME="E3:88_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A12</font></span>: 
<font color="Maroon" title="c14">x</font> <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="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">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="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</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="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">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"><font color="Maroon" title="c14">x</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">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> is  <a href="margrel1.html#V2" title="MARGREL1:attr.2">homogeneous</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:88_1_1_1"><span class="lab"><font color="Green" title="E56">A7</font></span></a>, <a class="txt" href="#E8:88_1_1_1"><span class="lab"><font color="Green" title="E55">A6</font></span></a>, <a class="ref" href="comput_1.html#T16" target="_self" title="COMPUT_1:th.16">Th19</a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/><a NAME="E4:88_1_1_1_1"/>
<font color="Maroon" title="c14">x</font> <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="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">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="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</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="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">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"><font color="Maroon" title="c14">x</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> (<span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:88_1_1_1"><span class="lab"><font color="Green" title="E57">A8</font></span></a>, <a class="ref" href="partfun1.html#T45" title="PARTFUN1:th.45">PARTFUN1:45</a></span>;<br/><span class="kw">hence </span><a NAME="E5:88_1_1_1_1"/>
<font color="Maroon" title="c14">x</font> <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="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">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="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</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="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">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"><font color="Maroon" title="c14">x</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:88_1_1_1_1"><span class="lab"><font color="Green" title="E59">A12</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>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">y</font> = <font color="Maroon" title="c17">y</font> as    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:88_1_1_1"><span class="lab"><font color="Green" title="E54">A5</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c19">y</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="Maroon" title="c13">m</font>,<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c19">y</font>]</span><br/>

<span class="kw">thus </span><a NAME="E16:88_1_1_1"/>
( <font color="Maroon" title="c3">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="c4">p</font> &amp; <font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">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> <font color="Maroon" title="c14">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">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"><font color="Maroon" title="c14">x</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">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="c6">ff2</font> implies <font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x</font> <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="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">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="c6">ff2</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="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">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"><font color="Maroon" title="c14">x</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">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="#E4:88_1_1_1"><span class="lab"><font color="Green" title="E54">A5</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"> ( (  not <font color="Maroon" title="c3">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="c4">p</font> or  not <font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">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> <font color="Maroon" title="c14">x</font> or  not <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">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"><font color="Maroon" title="c14">x</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">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="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">ff2</font> ) implies <font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x</font> )</span><br/>

<span class="kw">thus </span><a NAME="E17:88_1_1_1"/>
( (  not <font color="Maroon" title="c3">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="c4">p</font> or  not <font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">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> <font color="Maroon" title="c14">x</font> or  not <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">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"><font color="Maroon" title="c14">x</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c13">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="c6">ff2</font> ) implies <font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:88_1_1_1"><span class="lab"><font color="Green" title="E54">A5</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>
<span class="kw">consider </span><font color="Maroon" title="c13">FF</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="E17:88_1_1"/><span class="lab"><font color="Green" title="E54">A13</font></span>: 
<font color="Maroon" title="c13">FF</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> <font color="Maroon" title="c12">A</font>
 <span class="kw">and </span><br/><a NAME="E18:88_1_1"/><span class="lab"><font color="Green" title="E55">A14</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>2</sub></font>[<font color="Olive" title="b1">m</font>,<font color="Maroon" title="c13">FF</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</font>,<font color="Maroon" title="c13">FF</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>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="recdef_1.html#S2" title="RECDEF_1:sch.2">RECDEF_1:sch 2</a>(<a class="txt" href="#E15:88_1_1"><span class="lab"><font color="Green" title="E53">A4</font></span></a>);<br/></span>
<span class="kw">take </span>
<font color="Maroon" title="c13">FF</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i</font></span>)</span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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> st <br/>( <font color="Maroon" title="c13">FF</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i</font></span>)</span> &amp; ( <font color="Maroon" title="c3">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="c4">p</font> &amp;  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">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="c1">f1</font> implies <font color="Olive" title="b1">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="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">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="c1">f1</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="c4">p</font>,<font color="Maroon" title="c3">i</font>)</span>)</span></span>)</span> ) &amp; ( (  not <font color="Maroon" title="c3">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="c4">p</font> or  not  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">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="c1">f1</font> ) implies <font color="Olive" title="b1">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>  ) &amp; (  for <font color="Olive" title="b2">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 <br/> ( ( <font color="Maroon" title="c3">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="c4">p</font> &amp; <font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Olive" title="b2">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="Olive" title="b1">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Olive" title="b2">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="Olive" title="b1">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">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="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Olive" title="b2">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="c2">f2</font> implies <font color="Olive" title="b1">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">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="Olive" title="b1">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">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="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<span class="p3">(<span class="default"><font color="Olive" title="b2">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="c2">f2</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="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Olive" title="b2">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="Olive" title="b1">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">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="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Olive" title="b2">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> ) &amp; ( (  not <font color="Maroon" title="c3">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="c4">p</font> or  not <font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Olive" title="b2">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="Olive" title="b1">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">m</font></span>)</span> or  not <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Olive" title="b2">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="Olive" title="b1">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">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="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Olive" title="b2">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="c2">f2</font> ) implies <font color="Olive" title="b1">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">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> <font color="Olive" title="b1">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">m</font> ) ) ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c13">FF</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" title="c13">FF</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">FF</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i</font></span>)</span> &amp; ( <font color="Maroon" title="c3">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="c4">p</font> &amp;  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">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="c1">f1</font> implies <font color="Maroon" title="c13">FF</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="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">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="c1">f1</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="c4">p</font>,<font color="Maroon" title="c3">i</font>)</span>)</span></span>)</span> ) &amp; ( (  not <font color="Maroon" title="c3">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="c4">p</font> or  not  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">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="c1">f1</font> ) implies <font color="Maroon" title="c13">FF</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>  ) &amp; (  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 <br/> ( ( <font color="Maroon" title="c3">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="c4">p</font> &amp; <font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Olive" title="b1">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="c13">FF</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Olive" title="b1">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="c13">FF</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">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="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Olive" title="b1">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="c2">f2</font> implies <font color="Maroon" title="c13">FF</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">FF</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">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="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<span class="p3">(<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>)</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="c2">f2</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="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Olive" title="b1">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="c13">FF</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">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="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Olive" title="b1">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> ) &amp; ( (  not <font color="Maroon" title="c3">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="c4">p</font> or  not <font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Olive" title="b1">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="c13">FF</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</font></span>)</span> or  not <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Olive" title="b1">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="c13">FF</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">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="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Olive" title="b1">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="c2">f2</font> ) implies <font color="Maroon" title="c13">FF</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">FF</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</font> ) ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E19:88_1_1"/>
( <font color="Maroon" title="c13">FF</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">FF</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i</font></span>)</span> &amp; ( <font color="Maroon" title="c3">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="c4">p</font> &amp;  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">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="c1">f1</font> implies <font color="Maroon" title="c13">FF</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="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">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="c1">f1</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="c4">p</font>,<font color="Maroon" title="c3">i</font>)</span>)</span></span>)</span> ) &amp; ( (  not <font color="Maroon" title="c3">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="c4">p</font> or  not  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">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="c1">f1</font> ) implies <font color="Maroon" title="c13">FF</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>  ) &amp; (  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 <br/> ( ( <font color="Maroon" title="c3">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="c4">p</font> &amp; <font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Olive" title="b1">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="c13">FF</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Olive" title="b1">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="c13">FF</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">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="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Olive" title="b1">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="c2">f2</font> implies <font color="Maroon" title="c13">FF</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">FF</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">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="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<span class="p3">(<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>)</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="c2">f2</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="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Olive" title="b1">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="c13">FF</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">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="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Olive" title="b1">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> ) &amp; ( (  not <font color="Maroon" title="c3">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="c4">p</font> or  not <font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Olive" title="b1">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="c13">FF</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</font></span>)</span> or  not <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Olive" title="b1">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="c13">FF</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">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="c4">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c3">i</font>,<font color="Olive" title="b1">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="c2">f2</font> ) implies <font color="Maroon" title="c13">FF</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">FF</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</font> ) ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:88_1_1"><span class="lab"><font color="Green" title="E51">A2</font></span></a>, <a class="txt" href="#E14:88_1_1"><span class="lab"><font color="Green" title="E52">A3</font></span></a>, <a class="txt" href="#E17:88_1_1"><span class="lab"><font color="Green" title="E54">A13</font></span></a>, <a class="txt" href="#E18:88_1_1"><span class="lab"><font color="Green" title="E55">A14</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/>


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