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<b>let </b><font color="Maroon" title="c1">A</font> be   <a href="aofa_000.html#NM1" title="AOFA_000:NM.1">preIfWhileAlgebra</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">S</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">T</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Olive" title="b1">S</font><br/> for <font color="Olive" title="b3">f</font> being   <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">S</font>,<font color="Olive" title="b2">T</font><br/> for <font color="Olive" title="b4">I</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font><br/> for <font color="Olive" title="b5">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">S</font> holds <br/> ( not <font color="Olive" title="b3">f</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <font color="Olive" title="b4">I</font>,<font color="Olive" title="b5">s</font> iff <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Olive" title="b3">f</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Olive" title="b4">I</font></span>)</span> <a href="aofa_000.html#K5" title="AOFA_000:func.5">orbit</a> <font color="Olive" title="b5">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">T</font> )</span><br/><b>let </b><font color="Maroon" title="c2">S</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">T</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font><br/> for <font color="Olive" title="b2">f</font> being   <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">T</font><br/> for <font color="Olive" title="b3">I</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font><br/> for <font color="Olive" title="b4">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> holds <br/> ( not <font color="Olive" title="b2">f</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <font color="Olive" title="b3">I</font>,<font color="Olive" title="b4">s</font> iff <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Olive" title="b2">f</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Olive" title="b3">I</font></span>)</span> <a href="aofa_000.html#K5" title="AOFA_000:func.5">orbit</a> <font color="Olive" title="b4">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">T</font> )</span><br/><b>let </b><font color="Maroon" title="c3">T</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being   <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font><br/> for <font color="Olive" title="b2">I</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font><br/> for <font color="Olive" title="b3">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> holds <br/> ( not <font color="Olive" title="b1">f</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <font color="Olive" title="b2">I</font>,<font color="Olive" title="b3">s</font> iff <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Olive" title="b2">I</font></span>)</span> <a href="aofa_000.html#K5" title="AOFA_000:func.5">orbit</a> <font color="Olive" title="b3">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">T</font> )</span><br/><b>let </b><font color="Maroon" title="c4">f</font> be    <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">I</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font><br/> for <font color="Olive" title="b2">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> holds <br/> ( not <font color="Maroon" title="c4">f</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <font color="Olive" title="b1">I</font>,<font color="Olive" title="b2">s</font> iff <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Olive" title="b1">I</font></span>)</span> <a href="aofa_000.html#K5" title="AOFA_000:func.5">orbit</a> <font color="Olive" title="b2">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">T</font> )</span><br/><b>let </b><font color="Maroon" title="c5">I</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> holds <br/> ( not <font color="Maroon" title="c4">f</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <font color="Maroon" title="c5">I</font>,<font color="Olive" title="b1">s</font> iff <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="aofa_000.html#K5" title="AOFA_000:func.5">orbit</a> <font color="Olive" title="b1">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">T</font> )</span><br/><b>let </b><font color="Maroon" title="c6">s</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c4">f</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c6">s</font> iff <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="aofa_000.html#K5" title="AOFA_000:func.5">orbit</a> <font color="Maroon" title="c6">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">T</font> )</span><br/>











<b>set </b><font color="Maroon" title="c7">g</font> = <span class="p1">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c5">I</font>;<br/>
<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="aofa_000.html#K5" title="AOFA_000:func.5">orbit</a> <font color="Maroon" title="c6">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">T</font> implies  not <font color="Maroon" title="c4">f</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c6">s</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:145_1"/><i><font color="Green" title="E92">A1</font></i>: 
not <font color="Maroon" title="c4">f</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c6">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="aofa_000.html#K5" title="AOFA_000:func.5">orbit</a> <font color="Maroon" title="c6">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">T</font></span><br/><b>thus </b><a NAME="E2:145_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="aofa_000.html#K5" title="AOFA_000:func.5">orbit</a> <font color="Maroon" title="c6">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">T</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="145_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">x</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="aofa_000.html#K5" title="AOFA_000:func.5">orbit</a> <font color="Maroon" title="c6">s</font> or  not <font color="Maroon" title="c8">x</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c3">T</font> )</span><br/>

<b>assume </b><a NAME="E1:145_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="aofa_000.html#K5" title="AOFA_000:func.5">orbit</a> <font color="Maroon" title="c6">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c8">x</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c3">T</font></span><br/>

<b>then consider </b><font color="Maroon" title="c9">n</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> <b> such that </b><br/><a NAME="E3:145_1_1"/><i><font color="Green" title="E93">A2</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c5">I</font></span>)</span>,<font color="Maroon" title="c9">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">s</font> &amp; <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c5">I</font></span>)</span>,<font color="Maroon" title="c9">n</font></span>)</span> )
 ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b><span class="p1">(<span class="default"><a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c5">I</font></span>)</span>,$1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c3">T</font>;<br/>
<b>assume </b><a NAME="E4:145_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c3">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E5:145_1_1"/><b>then </b><i><font color="Green" title="E94">A3</font></i>: 
 ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>by </b><i><a class="txt" href="aofa_000.html#E3:145_1_1"><i><font color="Green" title="E93">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E7:145_1_1"/><i><font color="Green" title="E95">A4</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">n</font>] &amp; ( for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>] holds <br/><font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="aofa_000.html#E5:145_1_1"><i><font color="Green" title="E94">A3</font></i></a>);<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> = <span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c5">I</font></span>)</span>,<span class="p2">(<span class="default">$1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">s</font>;<br/>
<b>consider </b><font color="Maroon" title="c11">r</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E9:145_1_1"/><i><font color="Green" title="E96">A5</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; ( for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</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="c11">r</font> holds <br/><font color="Maroon" title="c11">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S2" title="FINSEQ_1:sch.2">FINSEQ_1:sch 2</a>();<br/></i>
<a NAME="E10:145_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="145_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">a</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">r</font> or  not <font color="Maroon" title="c12">a</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c2">S</font> )</span><br/>

<b>assume </b><a NAME="E1:145_1_1_1"/>
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c12">a</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c2">S</font></span><br/>

<b>then consider </b><font color="Maroon" title="c13">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:145_1_1_1"/><i><font color="Green" title="E97">A6</font></i>: 
( <font color="Maroon" title="c13">b</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="c11">r</font> &amp; <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">b</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">b</font> = <font color="Maroon" title="c13">b</font> 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>  <b>by </b><i><a class="txt" href="aofa_000.html#E3:145_1_1_1"><i><font color="Green" title="E97">A6</font></i></a></i>;<br/>
<a NAME="E5:145_1_1_1"/>
(  <a href="abian.html#K1" title="ABIAN:func.1">iter</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c5">I</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">b</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c11">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c5">I</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">b</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">s</font> )
 
<b>by </b><i><a class="txt" href="aofa_000.html#E9:145_1_1"><i><font color="Green" title="E96">A5</font></i></a>, <a class="txt" href="aofa_000.html#E3:145_1_1_1"><i><font color="Green" title="E97">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:145_1_1_1"/>
not <font color="Maroon" title="c12">a</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="aofa_000.html#E3:145_1_1_1"><i><font color="Green" title="E97">A6</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c12">r</font> = <font color="Maroon" title="c11">r</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="aofa_000.html#E9:145_1_1"><i><font color="Green" title="E96">A5</font></i></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E12:145_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E13:145_1_1"/><b>then </b>
( 1 <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="c12">r</font> &amp; <font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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="c12">r</font> )
 
<b>by </b><i><a class="txt" href="aofa_000.html#E9:145_1_1"><i><font color="Green" title="E96">A5</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E14:145_1_1"/><b>then </b>
( 1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n</font> &amp;  <a href="abian.html#K1" title="ABIAN:func.1">iter</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c5">I</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c12">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(1) &amp; <font color="Maroon" title="c12">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) )
 
<b>by </b><i><a class="txt" href="aofa_000.html#E9:145_1_1"><i><font color="Green" title="E96">A5</font></i></a>, <a class="ref" href="funct_7.html#T86" title="FUNCT_7:th.86">FUNCT_7:86</a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a>, <a class="ref" href="binarith.html#T51" title="BINARITH:th.51">BINARITH:51</a></i>;<br/>
<a NAME="E15:145_1_1"/><b>then </b><i><font color="Green" title="E97">A7</font></i>: 
( <font color="Maroon" title="c12">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font> &amp; <font color="Maroon" title="c12">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">r</font></span>)</span> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c3">T</font> )
 
<b>by </b><i><a class="txt" href="aofa_000.html#E7:145_1_1"><i><font color="Green" title="E95">A4</font></i></a>, <a class="ref" href="funct_1.html#T34" title="FUNCT_1:th.34">FUNCT_1:34</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="145_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">i</font> &amp; <font color="Maroon" title="c13">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">r</font> implies ( <font color="Maroon" title="c12">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T</font> &amp; <font color="Maroon" title="c12">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font></span>)</span>,<font color="Maroon" title="c5">I</font> ) )</span><br/><b>assume </b><a NAME="E1:145_1_1_2"/><i><font color="Green" title="E98">A8</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">i</font> &amp; <font color="Maroon" title="c13">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">r</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T</font> &amp; <font color="Maroon" title="c12">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font></span>)</span>,<font color="Maroon" title="c5">I</font> )</span><br/><b>consider </b><font color="Maroon" title="c14">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:145_1_1_2"/><i><font color="Green" title="E99">A9</font></i>: 
<font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">j</font>
 <b>by </b><i><a class="txt" href="aofa_000.html#E1:145_1_1_2"><i><font color="Green" title="E98">A8</font></i></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">j</font> = <font color="Maroon" title="c14">j</font> 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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E5:145_1_1_2"/><i><font color="Green" title="E100">A10</font></i>: 
( <font color="Maroon" title="c13">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i</font> )
 <b>by </b><i><a class="txt" href="aofa_000.html#E3:145_1_1_2"><i><font color="Green" title="E99">A9</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/><a NAME="E6:145_1_1_2"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">S</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E7:145_1_1_2"/><b>then </b><i><font color="Green" title="E101">A11</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c5">I</font></span>)</span>,<font color="Maroon" title="c15">j</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c5">I</font></span>)</span>,<font color="Maroon" title="c15">j</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">s</font></span>)</span>,<font color="Maroon" title="c5">I</font> &amp;  <a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c5">I</font></span>)</span>,<font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c5">I</font></span>)</span>,<font color="Maroon" title="c15">j</font></span>)</span> )
 <b>by </b><i><a class="txt" href="aofa_000.html#E3:145_1_1_2"><i><font color="Green" title="E99">A9</font></i></a>, <a class="ref" href="funct_7.html#T73" title="FUNCT_7:th.73">FUNCT_7:73</a>, <a class="ref" href="funct_5.html#T41" title="FUNCT_5:th.41">FUNCT_5:41</a></i>;<br/><a NAME="E8:145_1_1_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="aofa_000.html#E9:145_1_1"><i><font color="Green" title="E96">A5</font></i></a>, <a class="txt" href="aofa_000.html#E1:145_1_1_2"><i><font color="Green" title="E98">A8</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E9:145_1_1_2"/><b>then </b>
( <font color="Maroon" title="c13">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">n</font> &amp; <font color="Maroon" title="c13">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="c12">r</font> &amp; <font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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="c12">r</font> )
 <b>by </b><i><a class="txt" href="aofa_000.html#E9:145_1_1"><i><font color="Green" title="E96">A5</font></i></a>, <a class="txt" href="aofa_000.html#E1:145_1_1_2"><i><font color="Green" title="E98">A8</font></i></a>, <a class="txt" href="aofa_000.html#E3:145_1_1_2"><i><font color="Green" title="E99">A9</font></i></a>, <a class="txt" href="aofa_000.html#E5:145_1_1_2"><i><font color="Green" title="E100">A10</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E10:145_1_1_2"/><b>then </b>
( <font color="Maroon" title="c12">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c5">I</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">s</font> &amp; <font color="Maroon" title="c12">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c5">I</font></span>)</span>,<font color="Maroon" title="c13">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s</font> &amp;  not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1] )
 <b>by </b><i><a class="txt" href="aofa_000.html#E7:145_1_1"><i><font color="Green" title="E95">A4</font></i></a>, <a class="txt" href="aofa_000.html#E9:145_1_1"><i><font color="Green" title="E96">A5</font></i></a>, <a class="txt" href="aofa_000.html#E5:145_1_1_2"><i><font color="Green" title="E100">A10</font></i></a></i>;<br/><b>hence </b><a NAME="E11:145_1_1_2"/>
( <font color="Maroon" title="c12">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T</font> &amp; <font color="Maroon" title="c12">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font></span>)</span>,<font color="Maroon" title="c5">I</font> )
 <b>by </b><i><a class="txt" href="aofa_000.html#E5:145_1_1_2"><i><font color="Green" title="E100">A10</font></i></a>, <a class="txt" href="aofa_000.html#E7:145_1_1_2"><i><font color="Green" title="E101">A11</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E17:145_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aofa_000.html#E1:145_1"><i><font color="Green" title="E92">A1</font></i></a>, <a class="txt" href="aofa_000.html#E15:145_1_1"><i><font color="Green" title="E97">A7</font></i></a>, <a class="ref" href="aofa_000.html#D33" target="_self" title="AOFA_000:def.33">Def33</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>assume </b><a NAME="E2:145"/><i><font color="Green" title="E92">A12</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="aofa_000.html#K5" title="AOFA_000:func.5">orbit</a> <font color="Maroon" title="c6">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c4">f</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c6">s</font></span><br/>

<b>given </b><font color="Maroon" title="c8">r</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">S</font><b> such that </b><a NAME="E4:145"/><i><font color="Green" title="E93">A13</font></i>: 
( <font color="Maroon" title="c8">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font> &amp; <font color="Maroon" title="c8">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c3">T</font> )
 <b>and </b><br/><a NAME="E5:145"/><i><font color="Green" title="E94">A14</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">r</font> holds <br/>( <font color="Maroon" title="c8">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T</font> &amp; <font color="Maroon" title="c8">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<font color="Maroon" title="c5">I</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="aofa_000.html#D33" target="_self" title="AOFA_000:def.33">AOFA_000:def 33</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( $1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">r</font> implies <font color="Maroon" title="c8">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="aofa_000.html#K5" title="AOFA_000:func.5">orbit</a> <font color="Maroon" title="c6">s</font> );<br/>
<a NAME="E6:145"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:145"/><b>then </b><i><font color="Green" title="E95">A15</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="txt" href="aofa_000.html#E4:145"><i><font color="Green" title="E93">A13</font></i></a>, <a class="ref" href="aofa_000.html#T6" target="_self" title="AOFA_000:th.6">Th6</a></i>;<br/>
<a NAME="E8:145"/><i><font color="Green" title="E96">A16</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="145_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">i</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:145_2"/><i><font color="Green" title="E97">A17</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">i</font>] &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">r</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="aofa_000.html#K5" title="AOFA_000:func.5">orbit</a> <font color="Maroon" title="c6">s</font></span><br/>

<a NAME="E2:145_2"/><i><font color="Green" title="E98">A18</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">r</font> )
 
<b>by </b><i><a class="txt" href="aofa_000.html#E1:145_2"><i><font color="Green" title="E97">A17</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E3:145_2"/><b>then </b>
<font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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="c8">r</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E4:145_2"/><b>then </b>
( <font color="Maroon" title="c8">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">r</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:145_2"/><b>then </b><i><font color="Green" title="E99">A19</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c5">I</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">n</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> <b> such that </b><br/><a NAME="E7:145_2"/><i><font color="Green" title="E100">A20</font></i>: 
( <font color="Maroon" title="c8">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c5">I</font></span>)</span>,<font color="Maroon" title="c10">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">s</font> &amp; <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c5">I</font></span>)</span>,<font color="Maroon" title="c10">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="aofa_000.html#E1:145_2"><i><font color="Green" title="E97">A17</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<i><font color="Green" title="E101">A21</font></i>: <font color="Maroon" title="c8">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<font color="Maroon" title="c4">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c5">I</font>
<b>by </b><i><a class="txt" href="aofa_000.html#E2:145_2"><i><font color="Green" title="E98">A18</font></i></a>, <a class="txt" href="aofa_000.html#E5:145"><i><font color="Green" title="E94">A14</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="aofa_000.html#E5:145_2"><i><font color="Green" title="E99">A19</font></i></a>, <a class="ref" href="funct_5.html#T29" title="FUNCT_5:th.29">FUNCT_5:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c5">I</font></span>)</span>,<font color="Maroon" title="c10">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">s</font>
<b>by </b><i><a class="txt" href="aofa_000.html#E7:145_2"><i><font color="Green" title="E100">A20</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c5">I</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">s</font>
<b>by </b><i><a class="ref" href="funct_7.html#T73" title="FUNCT_7:th.73">FUNCT_7:73</a></i>
;<br/>
<a NAME="E9:145_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c5">I</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">n</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> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E10:145_2"/>
<font color="Maroon" title="c8">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="aofa_000.html#K5" title="AOFA_000:func.5">orbit</a> <font color="Maroon" title="c6">s</font>
 <b>by </b><i><a class="txt" href="aofa_000.html#E8:145_2"><i><font color="Green" title="E101">A21</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c9">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E10:145"/><i><font color="Green" title="E97">A22</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/>
<a NAME="E11:145"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="aofa_000.html#E7:145"><i><font color="Green" title="E95">A15</font></i></a>, <a class="txt" href="aofa_000.html#E8:145"><i><font color="Green" title="E96">A16</font></i></a>);<br/></i>
<a NAME="E12:145"/><b>then </b>
<font color="Maroon" title="c8">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="aofa_000.html#K5" title="AOFA_000:func.5">orbit</a> <font color="Maroon" title="c6">s</font>
 
<b>by </b><i><a class="txt" href="aofa_000.html#E10:145"><i><font color="Green" title="E97">A22</font></i></a></i>;<br/>
<b>hence </b><a NAME="E13:145"/>
contradiction
 <b>by </b><i><a class="txt" href="aofa_000.html#E10:145"><i><font color="Green" title="E97">A22</font></i></a>, <a class="txt" href="aofa_000.html#E4:145"><i><font color="Green" title="E93">A13</font></i></a>, <a class="txt" href="aofa_000.html#E2:145"><i><font color="Green" title="E92">A12</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
