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

<span class="kw">let </span><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> implies  ex <font color="Olive" title="b1">F</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  st <br/>( <font color="Maroon" title="c1">n</font> <a href="stirl2_1.html#K3" title="STIRL2_1:func.3">block</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Olive" title="b1">F</font></span>)</span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; (  for <font color="Olive" title="b2">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <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> <font color="Olive" title="b1">F</font> holds <br/><font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Olive" title="b2">m</font></span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> ) ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:70"/><span class="lab"><font color="Green" title="E56">A1</font></span>: 
<font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 ; <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="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  st <br/>( <font color="Maroon" title="c1">n</font> <a href="stirl2_1.html#K3" title="STIRL2_1:func.3">block</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Olive" title="b1">F</font></span>)</span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; (  for <font color="Olive" title="b2">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <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> <font color="Olive" title="b1">F</font> holds <br/><font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Olive" title="b2">m</font></span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> ) )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:70_1_1"/>
( ( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or ( <font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or ( <font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or ( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:70_1_1_1"/><span class="lab"><font color="Green" title="E57">A2</font></span>: 
( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</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">F</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c1">n</font> <a href="stirl2_1.html#K3" title="STIRL2_1:func.3">block</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Olive" title="b1">F</font></span>)</span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; (  for <font color="Olive" title="b2">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <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> <font color="Olive" title="b1">F</font> holds <br/><font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Olive" title="b2">m</font></span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> ) )</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c3">I</font> = 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c4">F</font> = <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">I</font></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c5">F</font> = <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">I</font></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span>; <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="c1">n</font> <a href="stirl2_1.html#K3" title="STIRL2_1:func.3">block</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c5">F</font></span>)</span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &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="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> <font color="Maroon" title="c5">F</font> holds <br/><font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> ) )</span><br/><a NAME="E3:70_1_1_1"/>
<a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">I</font></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T37" title="AFINSQ_2:th.37">AFINSQ_2:37</a></span>;<br/><a NAME="E4:70_1_1_1"/><span class="kw">then </span>
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T50" title="AFINSQ_2:th.50">AFINSQ_2:50</a></span>;<br/><span class="kw">hence </span><a NAME="E5:70_1_1_1"/>
<font color="Maroon" title="c1">n</font> <a href="stirl2_1.html#K3" title="STIRL2_1:func.3">block</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c5">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_1_1"><span class="lab"><font color="Green" title="E57">A2</font></span></a>, <a class="ref" href="newton.html#T12" title="NEWTON:th.12">NEWTON:12</a>, <a class="ref" href="stirl2_1.html#T26" title="STIRL2_1:th.26">STIRL2_1:26</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &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="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> <font color="Maroon" title="c5">F</font> holds <br/><font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> ) )</span><br/><span class="kw">thus </span><a NAME="E6:70_1_1_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_1_1"><span class="lab"><font color="Green" title="E57">A2</font></span></a>, <a class="ref" href="afinsq_1.html#T33" title="AFINSQ_1:th.33">AFINSQ_1:33</a></span>; <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">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <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> <font color="Maroon" title="c5">F</font> holds <br/><font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">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="c5">F</font> implies <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Maroon" title="c6">m</font></span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E7:70_1_1_1"/>
<font color="Maroon" title="c6">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="c5">F</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="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Maroon" title="c6">m</font></span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span><br/><a NAME="E8:70_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A3</font></span>: 
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T33" title="AFINSQ_1:th.33">AFINSQ_1:33</a>, <a class="ref" href="card_1.html#T49" title="CARD_1:th.49">CARD_1:49</a></span>;<br/><a NAME="E9:70_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E10:70_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A4</font></span>: 
<span class="p1">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T4" title="NEWTON:th.4">NEWTON:4</a></span>;<br/><a NAME="E11:70_1_1_1"/><span class="lab"><font color="Green" title="E60">A5</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_1_1"><span class="lab"><font color="Green" title="E57">A2</font></span></a>, <a class="ref" href="newton.html#T4" title="NEWTON:th.4">NEWTON:4</a></span>;<br/><a NAME="E12:70_1_1_1"/><span class="lab"><font color="Green" title="E61">A6</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T19" title="NEWTON:th.19">NEWTON:19</a></span>;<br/><a NAME="E13:70_1_1_1"/>
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:70_1_1_1"><span class="lab"><font color="Green" title="E58">A3</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E14:70_1_1_1"/>
<font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Maroon" title="c6">m</font></span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_1_1"><span class="lab"><font color="Green" title="E57">A2</font></span></a>, <a class="txt" href="#E10:70_1_1_1"><span class="lab"><font color="Green" title="E59">A4</font></span></a>, <a class="txt" href="#E11:70_1_1_1"><span class="lab"><font color="Green" title="E60">A5</font></span></a>, <a class="txt" href="#E12:70_1_1_1"><span class="lab"><font color="Green" title="E61">A6</font></span></a>, <a class="ref" href="afinsq_1.html#T34" title="AFINSQ_1:th.34">AFINSQ_1:34</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:70_1_1_2"/><span class="lab"><font color="Green" title="E57">A7</font></span>: 
( <font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</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">Fi</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  st <br/>( <font color="Maroon" title="c1">n</font> <a href="stirl2_1.html#K3" title="STIRL2_1:func.3">block</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Olive" title="b1">Fi</font></span>)</span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">Fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; (  for <font color="Olive" title="b2">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <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> <font color="Olive" title="b1">Fi</font> holds <br/><font color="Olive" title="b1">Fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Olive" title="b2">m</font></span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> ) )</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c3">F</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>;<br/><a NAME="E2:70_1_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></span>;<br/><a NAME="E3:70_1_1_2"/><span class="kw">then </span>
(  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  )
 <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>, <a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>;<br/><a NAME="E4:70_1_1_2"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">Fi</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c5">Fn</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  ;<br/><span class="kw">take </span><font color="Maroon" title="c6">Fi</font> = <font color="Maroon" title="c4">Fi</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="c1">n</font> <a href="stirl2_1.html#K3" title="STIRL2_1:func.3">block</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c6">Fi</font></span>)</span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">Fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &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="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> <font color="Maroon" title="c6">Fi</font> holds <br/><font color="Maroon" title="c6">Fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> ) )</span><br/><a NAME="E7:70_1_1_2"/>
(  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T29" title="ENUMSET1:th.29">ENUMSET1:29</a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/><a NAME="E8:70_1_1_2"/><span class="kw">then </span>
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c5">Fn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Fn</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T68" title="AFINSQ_2:th.68">AFINSQ_2:68</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="kw">hence </span><a NAME="E9:70_1_1_2"/>
( <font color="Maroon" title="c1">n</font> <a href="stirl2_1.html#K3" title="STIRL2_1:func.3">block</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c6">Fi</font></span>)</span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">Fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_1_2"><span class="lab"><font color="Green" title="E57">A7</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a>, <a class="ref" href="stirl2_1.html#T31" title="STIRL2_1:th.31">STIRL2_1:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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="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> <font color="Maroon" title="c6">Fi</font> holds <br/><font color="Maroon" title="c6">Fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c7">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">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="c6">Fi</font> implies <font color="Maroon" title="c6">Fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E10:70_1_1_2"/><span class="lab"><font color="Green" title="E58">A8</font></span>: 
<font color="Maroon" title="c7">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="c6">Fi</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="c6">Fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:70_1_1_2_1_1"/>
( <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:70_1_1_2_1_1_1"/>
<font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div class="add"><a NAME="E2:70_1_1_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_1_2"><span class="lab"><font color="Green" title="E57">A7</font></span></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a>, <a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></span>;<br/><span class="kw">hence </span><a NAME="E3:70_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:70_1_1_2_1_1_2"/>
<font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div class="add"><a NAME="E2:70_1_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">k</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_1_2"><span class="lab"><font color="Green" title="E57">A7</font></span></a>, <a class="ref" href="newton.html#D3" title="NEWTON:def.3">NEWTON:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:70_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E12:70_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A9</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E13:70_1_1_2"/>
<font color="Maroon" title="c7">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:70_1_1_2"><span class="lab"><font color="Green" title="E58">A8</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E14:70_1_1_2"/>
<font color="Maroon" title="c6">Fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:70_1_1_2"><span class="lab"><font color="Green" title="E59">A9</font></span></a>, <a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:70_1_1_3"/><span class="lab"><font color="Green" title="E57">A10</font></span>: 
( <font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</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">F</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  st <br/>( <font color="Maroon" title="c1">n</font> <a href="stirl2_1.html#K3" title="STIRL2_1:func.3">block</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Olive" title="b1">F</font></span>)</span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; (  for <font color="Olive" title="b2">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <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> <font color="Olive" title="b1">F</font> holds <br/><font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Olive" title="b2">m</font></span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> ) )</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c3">Perm</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c2">k</font> : <font color="Olive" title="b1">p</font> is   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c2">k</font> </span> } </span> ;<br/><a NAME="E2:70_1_1_3"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c2">k</font> : <font color="Olive" title="b1">p</font> is   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c2">k</font> </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K11" title="NEWTON:func.11">!</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_fin.html#T8" target="_self" title="CARD_FIN:th.8">Th8</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">Perm</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c2">k</font> : <font color="Olive" title="b1">p</font> is   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c2">k</font> </span> } </span>  as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c5">Bloc</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font> : ( <font color="Olive" title="b1">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; <font color="Olive" title="b1">f</font> is  <a href="stirl2_1.html#V1" title="STIRL2_1:attr.1">"increasing</a>  ) </span> } </span>  as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="stirl2_1.html#T24" title="STIRL2_1:th.24">STIRL2_1:24</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c6">Onto</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font> : <font color="Olive" title="b1">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  </span> } </span> ;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> for <font color="Olive" title="b1">p</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c2">k</font><br/>  for <font color="Olive" title="b2">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font>  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> holds <br/>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">f</font>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">N</font> = <font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c8">K</font> = <font color="Maroon" title="c2">k</font> 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 <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_1_3"><span class="lab"><font color="Green" title="E57">A10</font></span></a>, <a class="ref" href="stirl2_1.html#T8" title="STIRL2_1:th.8">STIRL2_1:8</a></span>;<br/><a NAME="E6:70_1_1_3"/><span class="lab"><font color="Green" title="E58">A11</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">Perm</font>,<font color="Maroon" title="c5">Bloc</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">Perm</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">Bloc</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T46" title="CARD_2:th.46">CARD_2:46</a></span>;<br/><a NAME="E7:70_1_1_3"/><span class="lab"><font color="Green" title="E59">A12</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">Perm</font>,<font color="Maroon" title="c5">Bloc</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/> ex <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b3">f</font> where <font color="Olive" title="b3">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font> : <font color="Olive" title="b3">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  </span> } </span>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">Perm</font>,<font color="Maroon" title="c5">Bloc</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> implies  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">f</font> where <font color="Olive" title="b2">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font> : <font color="Olive" title="b2">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  </span> } </span>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>] ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:70_1_1_3_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">Perm</font>,<font color="Maroon" title="c5">Bloc</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></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">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">f</font> where <font color="Olive" title="b2">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font> : <font color="Olive" title="b2">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  </span> } </span>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">p9</font>, <font color="Maroon" title="c11">f9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:70_1_1_3_1"/><span class="lab"><font color="Green" title="E60">A13</font></span>: 
<font color="Maroon" title="c10">p9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Perm</font>
 <span class="kw">and </span><br/><a NAME="E4:70_1_1_3_1"/><span class="lab"><font color="Green" title="E61">A14</font></span>: 
<font color="Maroon" title="c11">f9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Bloc</font>
 <span class="kw">and </span><br/><a NAME="E5:70_1_1_3_1"/><span class="lab"><font color="Green" title="E62">A15</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">p9</font>,<font color="Maroon" title="c11">f9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font><span class="kw"> such that </span><br/><a NAME="E7:70_1_1_3_1"/><span class="lab"><font color="Green" title="E63">A16</font></span>: 
<font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f9</font>
 <span class="kw">and </span><br/><a NAME="E8:70_1_1_3_1"/><span class="lab"><font color="Green" title="E64">A17</font></span>: 
( <font color="Maroon" title="c12">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; <font color="Maroon" title="c12">f</font> is  <a href="stirl2_1.html#V1" title="STIRL2_1:attr.1">"increasing</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70_1_1_3_1"><span class="lab"><font color="Green" title="E61">A14</font></span></a></span>;<br/>
<a NAME="E9:70_1_1_3_1"/><span class="lab"><font color="Green" title="E65">A18</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:70_1_1_3_1"><span class="lab"><font color="Green" title="E64">A17</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">p</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c2">k</font><span class="kw"> such that </span><br/><a NAME="E11:70_1_1_3_1"/><span class="lab"><font color="Green" title="E66">A19</font></span>: 
<font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p9</font>
 <span class="kw">and </span><br/><a NAME="E12:70_1_1_3_1"/><span class="lab"><font color="Green" title="E67">A20</font></span>: 
<font color="Maroon" title="c13">p</font> is   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c2">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70_1_1_3_1"><span class="lab"><font color="Green" title="E60">A13</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">pf</font> = <font color="Maroon" title="c13">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c14">pf</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">pf</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font> : <font color="Olive" title="b1">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  </span> } </span>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c14">pf</font>] )</span><br/>

<a NAME="E14:70_1_1_3_1"/><span class="lab"><font color="Green" title="E68">A21</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_1_3"><span class="lab"><font color="Green" title="E57">A10</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E15:70_1_1_3_1"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:70_1_1_3_1"><span class="lab"><font color="Green" title="E67">A20</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E16:70_1_1_3_1"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:70_1_1_3_1"><span class="lab"><font color="Green" title="E65">A18</font></span></a>, <a class="txt" href="#E14:70_1_1_3_1"><span class="lab"><font color="Green" title="E68">A21</font></span></a>, <a class="ref" href="relat_1.html#T28" title="RELAT_1:th.28">RELAT_1:28</a></span>;<br/>
<a NAME="E17:70_1_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">pf</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:70_1_1_3_1"/>
<font color="Maroon" title="c14">pf</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font> : <font color="Olive" title="b1">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  </span> } </span> 
 ; <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>1</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c14">pf</font>]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c15">p1</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c2">k</font>; <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">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font>  st <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c15">p1</font>,<font color="Olive" title="b1">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> holds <br/><font color="Maroon" title="c14">pf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">p1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">f</font></span><br/><span class="kw">let </span><font color="Maroon" title="c16">f1</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</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="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c15">p1</font>,<font color="Maroon" title="c16">f1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> implies <font color="Maroon" title="c14">pf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">p1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c16">f1</font> )</span><br/>



<span class="kw">assume </span><a NAME="E19:70_1_1_3_1"/><span class="lab"><font color="Green" title="E69">A22</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c15">p1</font>,<font color="Maroon" title="c16">f1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 ; <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">pf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">p1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c16">f1</font></span><br/>

<a NAME="E20:70_1_1_3_1"/>
<font color="Maroon" title="c15">p1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c13">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:70_1_1_3_1"><span class="lab"><font color="Green" title="E62">A15</font></span></a>, <a class="txt" href="#E11:70_1_1_3_1"><span class="lab"><font color="Green" title="E66">A19</font></span></a>, <a class="txt" href="#E19:70_1_1_3_1"><span class="lab"><font color="Green" title="E69">A22</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:70_1_1_3_1"/>
<font color="Maroon" title="c14">pf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">p1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c16">f1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:70_1_1_3_1"><span class="lab"><font color="Green" title="E62">A15</font></span></a>, <a class="txt" href="#E7:70_1_1_3_1"><span class="lab"><font color="Green" title="E63">A16</font></span></a>, <a class="txt" href="#E19:70_1_1_3_1"><span class="lab"><font color="Green" title="E69">A22</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="c9">FP</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">Perm</font>,<font color="Maroon" title="c5">Bloc</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font> : <font color="Olive" title="b1">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  </span> } </span> <span class="kw"> such that </span><br/><a NAME="E9:70_1_1_3"/><span class="lab"><font color="Green" title="E60">A23</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">Perm</font>,<font color="Maroon" title="c5">Bloc</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c9">FP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="#E7:70_1_1_3"><span class="lab"><font color="Green" title="E59">A12</font></span></a>);<br/></span><a NAME="E10:70_1_1_3"/><span class="lab"><font color="Green" title="E61">A24</font></span>: 
<font color="Maroon" title="c9">FP</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">x1</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="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</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">b<sub>1</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c9">FP</font> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c9">FP</font> or  not <font color="Maroon" title="c9">FP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">FP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> or <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c11">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c9">FP</font> or  not <font color="Maroon" title="c11">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c9">FP</font> or  not <font color="Maroon" title="c9">FP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">FP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</font> or <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x2</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:70_1_1_3_2"/><span class="lab"><font color="Green" title="E62">A25</font></span>: 
<font color="Maroon" title="c10">x1</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="c9">FP</font>
 <span class="kw">and </span><br/><a NAME="E2:70_1_1_3_2"/><span class="lab"><font color="Green" title="E63">A26</font></span>: 
<font color="Maroon" title="c11">x2</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="c9">FP</font>
 <span class="kw">and </span><br/><a NAME="E3:70_1_1_3_2"/><span class="lab"><font color="Green" title="E64">A27</font></span>: 
<font color="Maroon" title="c9">FP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">FP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</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="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x2</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c12">p19</font>, <font color="Maroon" title="c13">f19</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:70_1_1_3_2"/><span class="lab"><font color="Green" title="E65">A28</font></span>: 
<font color="Maroon" title="c12">p19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Perm</font>
 <span class="kw">and </span><br/><a NAME="E6:70_1_1_3_2"/><span class="lab"><font color="Green" title="E66">A29</font></span>: 
<font color="Maroon" title="c13">f19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Bloc</font>
 <span class="kw">and </span><br/><a NAME="E7:70_1_1_3_2"/><span class="lab"><font color="Green" title="E67">A30</font></span>: 
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">p19</font>,<font color="Maroon" title="c13">f19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_1_3_2"><span class="lab"><font color="Green" title="E62">A25</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">p1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c2">k</font><span class="kw"> such that </span><br/><a NAME="E9:70_1_1_3_2"/><span class="lab"><font color="Green" title="E68">A31</font></span>: 
<font color="Maroon" title="c12">p19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">p1</font>
 <span class="kw">and </span><br/><a NAME="E10:70_1_1_3_2"/><span class="lab"><font color="Green" title="E69">A32</font></span>: 
<font color="Maroon" title="c14">p1</font> is   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c2">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:70_1_1_3_2"><span class="lab"><font color="Green" title="E65">A28</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">p29</font>, <font color="Maroon" title="c16">f29</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E12:70_1_1_3_2"/><span class="lab"><font color="Green" title="E70">A33</font></span>: 
<font color="Maroon" title="c15">p29</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Perm</font>
 <span class="kw">and </span><br/><a NAME="E13:70_1_1_3_2"/><span class="lab"><font color="Green" title="E71">A34</font></span>: 
<font color="Maroon" title="c16">f29</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Bloc</font>
 <span class="kw">and </span><br/><a NAME="E14:70_1_1_3_2"/><span class="lab"><font color="Green" title="E72">A35</font></span>: 
<font color="Maroon" title="c11">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">p29</font>,<font color="Maroon" title="c16">f29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:70_1_1_3_2"><span class="lab"><font color="Green" title="E63">A26</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<a NAME="E15:70_1_1_3_2"/>
<font color="Maroon" title="c9">FP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">FP</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_1_3_2"><span class="lab"><font color="Green" title="E62">A25</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E16:70_1_1_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">FP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font> : <font color="Olive" title="b1">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  </span> } </span> 
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">fp</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c7">N</font>,<font color="Maroon" title="c8">K</font><span class="kw"> such that </span><br/><a NAME="E18:70_1_1_3_2"/><span class="lab"><font color="Green" title="E73">A36</font></span>: 
<font color="Maroon" title="c9">FP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">fp</font>
 <span class="kw">and </span><br/><a NAME="E19:70_1_1_3_2"/><span class="lab"><font color="Green" title="E74">A37</font></span>: 
<font color="Maroon" title="c17">fp</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 ;<br/>
<a NAME="E20:70_1_1_3_2"/><span class="lab"><font color="Green" title="E75">A38</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">fp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:70_1_1_3_2"><span class="lab"><font color="Green" title="E74">A37</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c18">p2</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c2">k</font><span class="kw"> such that </span><br/><a NAME="E22:70_1_1_3_2"/><span class="lab"><font color="Green" title="E76">A39</font></span>: 
<font color="Maroon" title="c15">p29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">p2</font>
 <span class="kw">and </span><br/><a NAME="E23:70_1_1_3_2"/><span class="lab"><font color="Green" title="E77">A40</font></span>: 
<font color="Maroon" title="c18">p2</font> is   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c2">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:70_1_1_3_2"><span class="lab"><font color="Green" title="E70">A33</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c19">f2</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font><span class="kw"> such that </span><br/><a NAME="E25:70_1_1_3_2"/><span class="lab"><font color="Green" title="E78">A41</font></span>: 
<font color="Maroon" title="c16">f29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">f2</font>
 <span class="kw">and </span><br/><a NAME="E26:70_1_1_3_2"/><span class="lab"><font color="Green" title="E79">A42</font></span>: 
( <font color="Maroon" title="c19">f2</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; <font color="Maroon" title="c19">f2</font> is  <a href="stirl2_1.html#V1" title="STIRL2_1:attr.1">"increasing</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:70_1_1_3_2"><span class="lab"><font color="Green" title="E71">A34</font></span></a></span>;<br/>
<a NAME="E27:70_1_1_3_2"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">fp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:70_1_1_3_2"><span class="lab"><font color="Green" title="E74">A37</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">p199</font> = <font color="Maroon" title="c14">p1</font>, <font color="Maroon" title="c21">p299</font> = <font color="Maroon" title="c18">p2</font> as   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">fp</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:70_1_1_3_2"><span class="lab"><font color="Green" title="E69">A32</font></span></a>, <a class="txt" href="#E23:70_1_1_3_2"><span class="lab"><font color="Green" title="E77">A40</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c22">f1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font><span class="kw"> such that </span><br/><a NAME="E30:70_1_1_3_2"/><span class="lab"><font color="Green" title="E80">A43</font></span>: 
<font color="Maroon" title="c13">f19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">f1</font>
 <span class="kw">and </span><br/><a NAME="E31:70_1_1_3_2"/><span class="lab"><font color="Green" title="E81">A44</font></span>: 
( <font color="Maroon" title="c22">f1</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; <font color="Maroon" title="c22">f1</font> is  <a href="stirl2_1.html#V1" title="STIRL2_1:attr.1">"increasing</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:70_1_1_3_2"><span class="lab"><font color="Green" title="E66">A29</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c23">f199</font> = <font color="Maroon" title="c22">f1</font>, <font color="Maroon" title="c24">f299</font> = <font color="Maroon" title="c19">f2</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c7">N</font>,<font color="Maroon" title="c8">K</font> ;<br/>
<a NAME="E33:70_1_1_3_2"/><span class="lab"><font color="Green" title="E82">A45</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c19">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:70_1_1_3_2"><span class="lab"><font color="Green" title="E79">A42</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E34:70_1_1_3_2"/>
 for <font color="Olive" title="b1">l</font>, <font color="Olive" title="b2">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c22">f1</font> &amp; <font color="Olive" title="b2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c22">f1</font> &amp; <font color="Olive" title="b1">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> holds <br/> <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">f1</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">l</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">f1</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:70_1_1_3_2"><span class="lab"><font color="Green" title="E81">A44</font></span></a>, <a class="ref" href="stirl2_1.html#D1" title="STIRL2_1:def.1">STIRL2_1:def 1</a></span>;<br/>
<a NAME="E35:70_1_1_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A46</font></span>: 
<font color="Maroon" title="c23">f199</font> is  <a href="stirl2_1.html#V2" title="STIRL2_1:attr.2">"increasing</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="stirl2_1.html#D3" title="STIRL2_1:def.3">STIRL2_1:def 3</a></span>;<br/>
<a NAME="E36:70_1_1_3_2"/>
 for <font color="Olive" title="b1">l</font>, <font color="Olive" title="b2">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c19">f2</font> &amp; <font color="Olive" title="b2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c19">f2</font> &amp; <font color="Olive" title="b1">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> holds <br/> <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">f2</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">l</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">f2</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:70_1_1_3_2"><span class="lab"><font color="Green" title="E79">A42</font></span></a>, <a class="ref" href="stirl2_1.html#D1" title="STIRL2_1:def.1">STIRL2_1:def 1</a></span>;<br/>
<a NAME="E37:70_1_1_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A47</font></span>: 
<font color="Maroon" title="c24">f299</font> is  <a href="stirl2_1.html#V2" title="STIRL2_1:attr.2">"increasing</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="stirl2_1.html#D3" title="STIRL2_1:def.3">STIRL2_1:def 3</a></span>;<br/>
<a NAME="E38:70_1_1_3_2"/><span class="lab"><font color="Green" title="E85">A48</font></span>: 
<font color="Maroon" title="c17">fp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">p199</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">f199</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:70_1_1_3"><span class="lab"><font color="Green" title="E60">A23</font></span></a>, <a class="txt" href="#E1:70_1_1_3_2"><span class="lab"><font color="Green" title="E62">A25</font></span></a>, <a class="txt" href="#E7:70_1_1_3_2"><span class="lab"><font color="Green" title="E67">A30</font></span></a>, <a class="txt" href="#E9:70_1_1_3_2"><span class="lab"><font color="Green" title="E68">A31</font></span></a>, <a class="txt" href="#E30:70_1_1_3_2"><span class="lab"><font color="Green" title="E80">A43</font></span></a>, <a class="txt" href="#E18:70_1_1_3_2"><span class="lab"><font color="Green" title="E73">A36</font></span></a></span>;<br/>
<a NAME="E39:70_1_1_3_2"/><span class="lab"><font color="Green" title="E86">A49</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c22">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:70_1_1_3_2"><span class="lab"><font color="Green" title="E81">A44</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E40:70_1_1_3_2"/><span class="lab"><font color="Green" title="E87">A50</font></span>: 
<font color="Maroon" title="c17">fp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">p299</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c24">f299</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:70_1_1_3"><span class="lab"><font color="Green" title="E60">A23</font></span></a>, <a class="txt" href="#E2:70_1_1_3_2"><span class="lab"><font color="Green" title="E63">A26</font></span></a>, <a class="txt" href="#E3:70_1_1_3_2"><span class="lab"><font color="Green" title="E64">A27</font></span></a>, <a class="txt" href="#E14:70_1_1_3_2"><span class="lab"><font color="Green" title="E72">A35</font></span></a>, <a class="txt" href="#E22:70_1_1_3_2"><span class="lab"><font color="Green" title="E76">A39</font></span></a>, <a class="txt" href="#E25:70_1_1_3_2"><span class="lab"><font color="Green" title="E78">A41</font></span></a>, <a class="txt" href="#E18:70_1_1_3_2"><span class="lab"><font color="Green" title="E73">A36</font></span></a></span>;<br/>
<a NAME="E41:70_1_1_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">p199</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c21">p299</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:70_1_1_3_2"><span class="lab"><font color="Green" title="E83">A46</font></span></a>, <a class="txt" href="#E37:70_1_1_3_2"><span class="lab"><font color="Green" title="E84">A47</font></span></a>, <a class="txt" href="#E39:70_1_1_3_2"><span class="lab"><font color="Green" title="E86">A49</font></span></a>, <a class="txt" href="#E33:70_1_1_3_2"><span class="lab"><font color="Green" title="E82">A45</font></span></a>, <a class="txt" href="#E38:70_1_1_3_2"><span class="lab"><font color="Green" title="E85">A48</font></span></a>, <a class="txt" href="#E20:70_1_1_3_2"><span class="lab"><font color="Green" title="E75">A38</font></span></a>, <a class="ref" href="stirl2_1.html#T65" title="STIRL2_1:th.65">STIRL2_1:65</a></span>;<br/>
<span class="kw">hence </span><a NAME="E42:70_1_1_3_2"/>
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:70_1_1_3_2"><span class="lab"><font color="Green" title="E67">A30</font></span></a>, <a class="txt" href="#E9:70_1_1_3_2"><span class="lab"><font color="Green" title="E68">A31</font></span></a>, <a class="txt" href="#E30:70_1_1_3_2"><span class="lab"><font color="Green" title="E80">A43</font></span></a>, <a class="txt" href="#E14:70_1_1_3_2"><span class="lab"><font color="Green" title="E72">A35</font></span></a>, <a class="txt" href="#E22:70_1_1_3_2"><span class="lab"><font color="Green" title="E76">A39</font></span></a>, <a class="txt" href="#E25:70_1_1_3_2"><span class="lab"><font color="Green" title="E78">A41</font></span></a>, <a class="txt" href="#E35:70_1_1_3_2"><span class="lab"><font color="Green" title="E83">A46</font></span></a>, <a class="txt" href="#E37:70_1_1_3_2"><span class="lab"><font color="Green" title="E84">A47</font></span></a>, <a class="txt" href="#E39:70_1_1_3_2"><span class="lab"><font color="Green" title="E86">A49</font></span></a>, <a class="txt" href="#E33:70_1_1_3_2"><span class="lab"><font color="Green" title="E82">A45</font></span></a>, <a class="txt" href="#E38:70_1_1_3_2"><span class="lab"><font color="Green" title="E85">A48</font></span></a>, <a class="txt" href="#E40:70_1_1_3_2"><span class="lab"><font color="Green" title="E87">A50</font></span></a>, <a class="txt" href="#E20:70_1_1_3_2"><span class="lab"><font color="Green" title="E75">A38</font></span></a>, <a class="ref" href="stirl2_1.html#T65" title="STIRL2_1:th.65">STIRL2_1:65</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="c10">h</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font><span class="kw"> such that </span><br/><a NAME="E12:70_1_1_3"/><span class="lab"><font color="Green" title="E62">A51</font></span>: 
( <font color="Maroon" title="c10">h</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; <font color="Maroon" title="c10">h</font> is  <a href="stirl2_1.html#V1" title="STIRL2_1:attr.1">"increasing</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="txt" href="#E1:70_1_1_3"><span class="lab"><font color="Green" title="E57">A10</font></span></a>, <a class="ref" href="stirl2_1.html#T23" title="STIRL2_1:th.23">STIRL2_1:23</a></span>;<br/><a NAME="E13:70_1_1_3"/>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font> : <font color="Olive" title="b1">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">FP</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_3_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">x</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="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font> : <font color="Olive" title="b1">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  </span> } </span>  or <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">FP</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:70_1_1_3_3"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font> : <font color="Olive" title="b1">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  </span> } </span> 
 ; <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="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">FP</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font><span class="kw"> such that </span><br/><a NAME="E3:70_1_1_3_3"/><span class="lab"><font color="Green" title="E63">A52</font></span>: 
<font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font>
 <span class="kw">and </span><br/><a NAME="E4:70_1_1_3_3"/><span class="lab"><font color="Green" title="E64">A53</font></span>: 
<font color="Maroon" title="c12">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 ;<br/>
<a NAME="E5:70_1_1_3_3"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70_1_1_3_3"><span class="lab"><font color="Green" title="E64">A53</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">I</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c7">N</font>,<font color="Maroon" title="c8">K</font>, <font color="Maroon" title="c14">P</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c8">K</font><span class="kw"> such that </span><br/><a NAME="E7:70_1_1_3_3"/><span class="lab"><font color="Green" title="E65">A54</font></span>: 
<font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c13">I</font>
 <span class="kw">and </span><br/><a NAME="E8:70_1_1_3_3"/><span class="lab"><font color="Green" title="E66">A55</font></span>: 
<font color="Maroon" title="c8">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c13">I</font>
 <span class="kw">and </span><br/><a NAME="E9:70_1_1_3_3"/><span class="lab"><font color="Green" title="E67">A56</font></span>: 
<font color="Maroon" title="c13">I</font> is  <a href="stirl2_1.html#V2" title="STIRL2_1:attr.2">"increasing</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="stirl2_1.html#T63" title="STIRL2_1:th.63">STIRL2_1:63</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">p</font> = <font color="Maroon" title="c14">P</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">i</font> = <font color="Maroon" title="c13">I</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font> ;<br/>
<a NAME="E11:70_1_1_3_3"/>
 for <font color="Olive" title="b1">l</font>, <font color="Olive" title="b2">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c13">I</font> &amp; <font color="Olive" title="b2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c13">I</font> &amp; <font color="Olive" title="b1">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> holds <br/> <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">I</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">l</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">I</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:70_1_1_3_3"><span class="lab"><font color="Green" title="E67">A56</font></span></a>, <a class="ref" href="stirl2_1.html#D3" title="STIRL2_1:def.3">STIRL2_1:def 3</a></span>;<br/>
<a NAME="E12:70_1_1_3_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A57</font></span>: 
<font color="Maroon" title="c16">i</font> is  <a href="stirl2_1.html#V1" title="STIRL2_1:attr.1">"increasing</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="stirl2_1.html#D1" title="STIRL2_1:def.1">STIRL2_1:def 1</a></span>;<br/>
<a NAME="E13:70_1_1_3_3"/>
<font color="Maroon" title="c16">i</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:70_1_1_3_3"><span class="lab"><font color="Green" title="E66">A55</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E14:70_1_1_3_3"/><span class="kw">then </span>
( <font color="Maroon" title="c14">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Perm</font> &amp; <font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Bloc</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:70_1_1_3_3"><span class="lab"><font color="Green" title="E68">A57</font></span></a></span>;<br/>
<a NAME="E15:70_1_1_3_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A58</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">P</font>,<font color="Maroon" title="c16">i</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">Perm</font>,<font color="Maroon" title="c5">Bloc</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<a NAME="E16:70_1_1_3_3"/>
<font color="Maroon" title="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font> : <font color="Olive" title="b1">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:70_1_1_3"><span class="lab"><font color="Green" title="E62">A51</font></span></a></span>;<br/>
<a NAME="E17:70_1_1_3_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A59</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">P</font>,<font color="Maroon" title="c16">i</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</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="c9">FP</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:70_1_1_3_3"><span class="lab"><font color="Green" title="E69">A58</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E18:70_1_1_3_3"/>
<font color="Maroon" title="c9">FP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">P</font>,<font color="Maroon" title="c16">i</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:70_1_1_3"><span class="lab"><font color="Green" title="E60">A23</font></span></a>, <a class="txt" href="#E7:70_1_1_3_3"><span class="lab"><font color="Green" title="E65">A54</font></span></a>, <a class="txt" href="#E15:70_1_1_3_3"><span class="lab"><font color="Green" title="E69">A58</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:70_1_1_3_3"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">FP</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70_1_1_3_3"><span class="lab"><font color="Green" title="E63">A52</font></span></a>, <a class="txt" href="#E17:70_1_1_3_3"><span class="lab"><font color="Green" title="E70">A59</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</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><a NAME="E14:70_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A60</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">FP</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font> : <font color="Olive" title="b1">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/><a NAME="E15:70_1_1_3"/>
<font color="Maroon" title="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font> : <font color="Olive" title="b1">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:70_1_1_3"><span class="lab"><font color="Green" title="E62">A51</font></span></a></span>;<br/><a NAME="E16:70_1_1_3"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">FP</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="c4">Perm</font>,<font color="Maroon" title="c5">Bloc</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E17:70_1_1_3"/><span class="kw">then </span>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font> : <font color="Olive" title="b1">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  </span> } </span> ,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">Perm</font>,<font color="Maroon" title="c5">Bloc</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:70_1_1_3"><span class="lab"><font color="Green" title="E61">A24</font></span></a>, <a class="txt" href="#E14:70_1_1_3"><span class="lab"><font color="Green" title="E63">A60</font></span></a>, <a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a></span>;<br/><a NAME="E18:70_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A61</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font> : <font color="Olive" title="b1">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">Perm</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">Bloc</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:70_1_1_3"><span class="lab"><font color="Green" title="E58">A11</font></span></a>, <a class="ref" href="card_1.html#T5" title="CARD_1:th.5">CARD_1:5</a></span>;<br/><a NAME="E19:70_1_1_3"/>
<font color="Maroon" title="c2">k</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T17" title="NEWTON:th.17">NEWTON:17</a></span>;<br/><a NAME="E20:70_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A62</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">Bloc</font></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">Bloc</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a>, <a class="ref" href="xcmplx_1.html#T74" title="XCMPLX_1:th.74">XCMPLX_1:74</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c11">F</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of <span class="kw"> such that </span><br/><a NAME="E22:70_1_1_3"/><span class="lab"><font color="Green" title="E66">A63</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">and </span><br/><a NAME="E23:70_1_1_3"/><span class="lab"><font color="Green" title="E67">A64</font></span>: 
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c11">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font> : <font color="Olive" title="b1">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  </span> } </span> 
 <span class="kw">and </span><br/><a NAME="E24:70_1_1_3"/><span class="lab"><font color="Green" title="E68">A65</font></span>: 
 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="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> <font color="Maroon" title="c11">F</font> holds <br/><font color="Maroon" title="c11">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_1_3"><span class="lab"><font color="Green" title="E57">A10</font></span></a>, <a class="ref" href="card_fin.html#T59" target="_self" title="CARD_FIN:th.59">Th70</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c12">F</font> = <font color="Maroon" title="c11">F</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="c1">n</font> <a href="stirl2_1.html#K3" title="STIRL2_1:func.3">block</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c12">F</font></span>)</span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &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="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> <font color="Maroon" title="c12">F</font> holds <br/><font color="Maroon" title="c12">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> ) )</span><br/><a NAME="E25:70_1_1_3"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">Perm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K11" title="NEWTON:func.11">!</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_fin.html#T8" target="_self" title="CARD_FIN:th.8">Th8</a></span>;<br/><a NAME="E26:70_1_1_3"/><span class="kw">then </span>
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c12">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">Bloc</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:70_1_1_3"><span class="lab"><font color="Green" title="E67">A64</font></span></a>, <a class="txt" href="#E18:70_1_1_3"><span class="lab"><font color="Green" title="E64">A61</font></span></a>, <a class="ref" href="card_1.html#D2" title="CARD_1:def.2">CARD_1:def 2</a></span>;<br/><a NAME="E27:70_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c1">n</font> <a href="stirl2_1.html#K3" title="STIRL2_1:func.3">block</a> <font color="Maroon" title="c2">k</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="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c12">F</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> 1</span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:70_1_1_3"><span class="lab"><font color="Green" title="E65">A62</font></span></a>, <a class="ref" href="stirl2_1.html#D2" title="STIRL2_1:def.2">STIRL2_1:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E28:70_1_1_3"/>
<font color="Maroon" title="c1">n</font> <a href="stirl2_1.html#K3" title="STIRL2_1:func.3">block</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c12">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T74" title="XCMPLX_1:th.74">XCMPLX_1:74</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &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="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> <font color="Maroon" title="c12">F</font> holds <br/><font color="Maroon" title="c12">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> ) )</span><br/><span class="kw">thus </span><a NAME="E29:70_1_1_3"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:70_1_1_3"><span class="lab"><font color="Green" title="E66">A63</font></span></a>, <a class="ref" href="card_1.html#D2" title="CARD_1:def.2">CARD_1:def 2</a></span>; <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">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <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> <font color="Maroon" title="c12">F</font> holds <br/><font color="Maroon" title="c12">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span><br/><a NAME="E30:70_1_1_3"/><span class="lab"><font color="Green" title="E69">A66</font></span>: 
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D2" title="CARD_1:def.2">CARD_1:def 2</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c13">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" 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="c12">F</font> implies <font color="Maroon" title="c12">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Maroon" title="c13">m</font></span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E31:70_1_1_3"/>
<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="c12">F</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="c12">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Maroon" title="c13">m</font></span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span><br/><span class="kw">hence </span><a NAME="E32:70_1_1_3"/>
<font color="Maroon" title="c12">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Maroon" title="c13">m</font></span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:70_1_1_3"><span class="lab"><font color="Green" title="E68">A65</font></span></a>, <a class="txt" href="#E30:70_1_1_3"><span class="lab"><font color="Green" title="E69">A66</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:70_1_1_4"/>
( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</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">F</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  st <br/>( <font color="Maroon" title="c1">n</font> <a href="stirl2_1.html#K3" title="STIRL2_1:func.3">block</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Olive" title="b1">F</font></span>)</span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; (  for <font color="Olive" title="b2">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <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> <font color="Olive" title="b1">F</font> holds <br/><font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Olive" title="b2">m</font></span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> ) )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:70_1_1_4"/>
 ex <font color="Olive" title="b1">F</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  st <br/>( <font color="Maroon" title="c1">n</font> <a href="stirl2_1.html#K3" title="STIRL2_1:func.3">block</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Olive" title="b1">F</font></span>)</span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; (  for <font color="Olive" title="b2">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <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> <font color="Olive" title="b1">F</font> holds <br/><font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Olive" title="b2">m</font></span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E56">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:70"/>
 ex <font color="Olive" title="b1">F</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  st <br/>( <font color="Maroon" title="c1">n</font> <a href="stirl2_1.html#K3" title="STIRL2_1:func.3">block</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Olive" title="b1">F</font></span>)</span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; (  for <font color="Olive" title="b2">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <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> <font color="Olive" title="b1">F</font> holds <br/><font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Olive" title="b2">m</font></span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</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>
