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<div class="add">

<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:87"/><i><font color="Green" title="E69">A1</font></i>: 
<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>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:87_1_1"/>
( ( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) or ( <font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) or ( <font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c2">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 ) or ( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c2">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_1_1"><b>suppose </b></a><a NAME="E1:87_1_1_1"/><i><font color="Green" title="E70">A2</font></i>: 
( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 ;<br/><div class="add"><b>reconsider </b><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>  <b>by </b><i><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/><b>set </b><font color="Maroon" title="c4">F</font> = <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><font color="Maroon" title="c3">I</font></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span>;<br/><b>take </b><font color="Maroon" title="c5">F</font> = <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><font color="Maroon" title="c3">I</font></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span>;<br/><a NAME="E3:87_1_1_1"/>
( <font color="Maroon" title="c1">n</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="card_fin.html#K6" title="CARD_FIN:func.6">Sum</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><font color="Maroon" title="c3">I</font></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> &amp; 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><font color="Maroon" title="c3">I</font></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="card_fin.html#E1:87_1_1_1"><i><font color="Green" title="E70">A2</font></i></a>, <a class="ref" href="newton.html#T18" title="NEWTON:th.18">NEWTON:18</a>, <a class="ref" href="stirl2_1.html#T44" title="STIRL2_1:th.44">STIRL2_1:44</a></i>;<br/><b>hence </b><a NAME="E4:87_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#K12" title="NEWTON:func.12">!</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="card_fin.html#K6" title="CARD_FIN:func.6">Sum</a> <font color="Maroon" title="c5">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="card_fin.html#E1:87_1_1_1"><i><font color="Green" title="E70">A2</font></i></a>, <a class="ref" href="stirl2_1.html#T26" title="STIRL2_1:th.26">STIRL2_1:26</a></i>;<br/><b>thus </b><a NAME="E5:87_1_1_1"/>
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">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
 <b>by </b><i><a class="txt" href="card_fin.html#E1:87_1_1_1"><i><font color="Green" title="E70">A2</font></i></a>, <a class="ref" href="afinsq_1.html#T36" title="AFINSQ_1:th.36">AFINSQ_1:36</a></i>;<br/><b>let </b><font color="Maroon" title="c6">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E6:87_1_1_1"/><i><font color="Green" title="E71">A3</font></i>: 
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c5">F</font>
 ;<br/><a NAME="E7:87_1_1_1"/>
<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">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="card_fin.html#E6:87_1_1_1"><i><font color="Green" title="E71">A3</font></i></a>, <a class="ref" href="afinsq_1.html#T36" title="AFINSQ_1:th.36">AFINSQ_1:36</a>, <a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a></i>;<br/><a NAME="E8:87_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E9:87_1_1_1"/><b>then </b>
( <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">k</font> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="card_fin.html#E1:87_1_1_1"><i><font color="Green" title="E70">A2</font></i></a>, <a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a>, <a class="ref" href="newton.html#T27" title="NEWTON:th.27">NEWTON:27</a></i>;<br/><b>hence </b><a NAME="E10:87_1_1_1"/>
<font color="Maroon" title="c5">F</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</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#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <font color="Maroon" title="c6">m</font></span>)</span></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="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="afinsq_1.html#T38" title="AFINSQ_1:th.38">AFINSQ_1:38</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_1_2"><b>suppose </b></a><a NAME="E1:87_1_1_2"/><i><font color="Green" title="E70">A4</font></i>: 
( <font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 ;<br/><div class="add"><b>set </b><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="finsop_1.html#K3" title="FINSOP_1:func.3">--&gt;</a> 0;<br/><a NAME="E2:87_1_1_2"/>
0 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> 
 <b>by </b><i><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/><a NAME="E3:87_1_1_2"/><b>then </b><i><font color="Green" title="E71">A5</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">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="finsop_1.html#K3" title="FINSOP_1:func.3">--&gt;</a> 0</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">0</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">0</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>  &amp; <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</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#K5" title="NUMBERS:func.5">NAT</a>  )
 <b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><a NAME="E4:87_1_1_2"/><b>then </b>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="finsop_1.html#K3" title="FINSOP_1:func.3">--&gt;</a> 0</span>)</span> <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;  <a href="relset_1.html#K5" title="RELSET_1:func.5">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="finsop_1.html#K3" title="FINSOP_1:func.3">--&gt;</a> 0</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  )
 <b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c4">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="finsop_1.html#K3" title="FINSOP_1:func.3">--&gt;</a> 0 as   <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</a> <b>by </b><i><a class="ref" href="afinsq_1.html#T7" title="AFINSQ_1:th.7">AFINSQ_1:7</a></i>;<br/><a NAME="E6:87_1_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 <b>by </b><i><a class="txt" href="card_fin.html#E3:87_1_1_2"><i><font color="Green" title="E71">A5</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c5">Fi</font> = <font color="Maroon" title="c4">F</font> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D8" title="ORDINAL1:def.8">ORDINAL1:def 8</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">Fn</font> = <font color="Maroon" title="c4">F</font> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="card_fin.html#E3:87_1_1_2"><i><font color="Green" title="E71">A5</font></i></a>, <a class="ref" href="ordinal1.html#D8" title="ORDINAL1:def.8">ORDINAL1:def 8</a></i>;<br/><b>take </b><font color="Maroon" title="c7">Fi</font> = <font color="Maroon" title="c5">Fi</font>;<br/><a NAME="E9:87_1_1_2"/>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font> <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">0</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">0</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">0,0</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )
 <b>by </b><i><a class="ref" href="enumset1.html#T69" title="ENUMSET1:th.69">ENUMSET1:69</a>, <a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/><a NAME="E10:87_1_1_2"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font> <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">0,0</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E11:87_1_1_2"/><b>then </b>
 <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c6">Fn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">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">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="card_fin.html#T45" target="_self" title="CARD_FIN:th.45">Th45</a></i>;<br/><a NAME="E12:87_1_1_2"/><b>then </b>
(  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c6">Fn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp;  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c6">Fn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_fin.html#K6" title="CARD_FIN:func.6">Sum</a> <font color="Maroon" title="c7">Fi</font> )
 <b>by </b><i><a class="ref" href="card_fin.html#T63" target="_self" title="CARD_FIN:th.63">Th63</a></i>;<br/><b>hence </b><a NAME="E13:87_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#K12" title="NEWTON:func.12">!</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="card_fin.html#K6" title="CARD_FIN:func.6">Sum</a> <font color="Maroon" title="c7">Fi</font></span>)</span> &amp;  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">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 )
 <b>by </b><i><a class="txt" href="card_fin.html#E1:87_1_1_2"><i><font color="Green" title="E70">A4</font></i></a>, <a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a>, <a class="ref" href="stirl2_1.html#T31" title="STIRL2_1:th.31">STIRL2_1:31</a></i>;<br/><b>let </b><font color="Maroon" title="c8">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E14:87_1_1_2"/><i><font color="Green" title="E72">A6</font></i>: 
<font color="Maroon" title="c8">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">Fi</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:87_1_1_2_1_1"/>
( <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:87_1_1_2_1_1_1"/>
<font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:87_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c2">k</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 <b>by </b><i><a class="txt" href="card_fin.html#E1:87_1_1_2"><i><font color="Green" title="E70">A4</font></i></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><a NAME="E3:87_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="newton.html#T16" title="NEWTON:th.16">NEWTON:16</a></i>;<br/><b>hence </b><a NAME="E4:87_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <font color="Maroon" title="c8">m</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="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:87_1_1_2_1_1_2"/>
<font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:87_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">k</font> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="card_fin.html#E1:87_1_1_2"><i><font color="Green" title="E70">A4</font></i></a>, <a class="ref" href="newton.html#D3" title="NEWTON:def.3">NEWTON:def 3</a></i>;<br/><b>hence </b><a NAME="E3:87_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <font color="Maroon" title="c8">m</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="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E16:87_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c8">m</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#K9" title="NEWTON:func.9">choose</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</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#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c8">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 )
 <b>by </b><i><a class="txt" href="card_fin.html#E14:87_1_1_2"><i><font color="Green" title="E72">A6</font></i></a>, <a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/><b>hence </b><a NAME="E17:87_1_1_2"/>
<font color="Maroon" title="c7">Fi</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <font color="Maroon" title="c8">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#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <font color="Maroon" title="c8">m</font></span>)</span></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="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_1_3"><b>suppose </b></a><a NAME="E1:87_1_1_3"/><i><font color="Green" title="E70">A7</font></i>: 
( <font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c2">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c3">F</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <b> such that </b><br/><a NAME="E3:87_1_1_3"/><i><font color="Green" title="E71">A8</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <b>and </b><br/><a NAME="E4:87_1_1_3"/><i><font color="Green" title="E72">A9</font></i>: 
 <a href="card_fin.html#K6" title="CARD_FIN:func.6">Sum</a> <font color="Maroon" title="c3">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> 
 <b>and </b><br/><a NAME="E5:87_1_1_3"/><i><font color="Green" title="E73">A10</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c3">F</font> holds <br/><font color="Maroon" title="c3">F</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</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#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="card_fin.html#T70" target="_self" title="CARD_FIN:th.70">Th70</a></i>;<br/><b>take </b><font color="Maroon" title="c4">F</font> = <font color="Maroon" title="c3">F</font>;<br/><b>set </b><font color="Maroon" title="c5">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="E6:87_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#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K12" title="NEWTON:func.12">!</a> 
 <b>by </b><i><a class="ref" href="card_fin.html#T8" target="_self" title="CARD_FIN:th.8">Th8</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c6">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>  <b>by </b><i><a class="ref" href="card_4.html#T1" title="CARD_4:th.1">CARD_4:1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">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>  <b>by </b><i><a class="ref" href="stirl2_1.html#T24" title="STIRL2_1:th.24">STIRL2_1:24</a></i>;<br/><b>set </b><font color="Maroon" title="c8">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/><b>defpred </b><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> ] <b>means </b>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/><a NAME="E9:87_1_1_3"/><i><font color="Green" title="E74">A11</font></i>: 
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="c6">Perm</font>,<font color="Maroon" title="c7">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="87_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:87_1_1_3_1"/><i><font color="Green" title="E75">A12</font></i>: 
<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="c6">Perm</font>,<font color="Maroon" title="c7">Bloc</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/>

<b>consider </b><font color="Maroon" title="c10">p'</font>, <font color="Maroon" title="c11">f'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:87_1_1_3_1"/><i><font color="Green" title="E76">A13</font></i>: 
( <font color="Maroon" title="c10">p'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Perm</font> &amp; <font color="Maroon" title="c11">f'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Bloc</font> &amp; <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">p'</font>,<font color="Maroon" title="c11">f'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="card_fin.html#E1:87_1_1_3_1"><i><font color="Green" title="E75">A12</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">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><b> such that </b><br/><a NAME="E5:87_1_1_3_1"/><i><font color="Green" title="E77">A14</font></i>: 
( <font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p'</font> &amp; <font color="Maroon" title="c12">p</font> is   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c2">k</font> )
 <b>by </b><i><a class="txt" href="card_fin.html#E3:87_1_1_3_1"><i><font color="Green" title="E76">A13</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">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><b> such that </b><br/><a NAME="E7:87_1_1_3_1"/><i><font color="Green" title="E78">A15</font></i>: 
( <font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f'</font> &amp; <font color="Maroon" title="c13">f</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> &amp; <font color="Maroon" title="c13">f</font> is <a href="stirl2_1.html#V1" title="STIRL2_1:attr.1">"increasing</a> )
 <b>by </b><i><a class="txt" href="card_fin.html#E3:87_1_1_3_1"><i><font color="Green" title="E76">A13</font></i></a></i>;<br/>
<a NAME="E8:87_1_1_3_1"/>
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E5:87_1_1_3_1"><i><font color="Green" title="E77">A14</font></i></a>, <a class="txt" href="card_fin.html#E7:87_1_1_3_1"><i><font color="Green" title="E78">A15</font></i></a>, <a class="txt" href="card_fin.html#E1:87_1_1_3"><i><font color="Green" title="E70">A7</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E9:87_1_1_3_1"/><b>then </b><i><font color="Green" title="E79">A16</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a>, <a class="ref" href="relat_1.html#T47" title="RELAT_1:th.47">RELAT_1:47</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">pf</font> = <font color="Maroon" title="c12">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c13">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> <b>by </b><i><a class="ref" href="funct_2.html#T3" title="FUNCT_2:th.3">FUNCT_2:3</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c14">pf</font>
;<br/>

<a NAME="E11:87_1_1_3_1"/>
<font color="Maroon" title="c14">pf</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>
 
<b>by </b><i><a class="txt" href="card_fin.html#E9:87_1_1_3_1"><i><font color="Green" title="E79">A16</font></i></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<b>hence </b><a NAME="E12:87_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> 
 ;<br/>

<b>let </b><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>;<br/><b>let </b><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>;<br/>



<b>assume </b><a NAME="E13:87_1_1_3_1"/><i><font color="Green" title="E80">A17</font></i>: 
<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>
 ;<br/>

<a NAME="E14:87_1_1_3_1"/>
( <font color="Maroon" title="c15">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">p</font> &amp; <font color="Maroon" title="c16">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f</font> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E3:87_1_1_3_1"><i><font color="Green" title="E76">A13</font></i></a>, <a class="txt" href="card_fin.html#E5:87_1_1_3_1"><i><font color="Green" title="E77">A14</font></i></a>, <a class="txt" href="card_fin.html#E7:87_1_1_3_1"><i><font color="Green" title="E78">A15</font></i></a>, <a class="txt" href="card_fin.html#E13:87_1_1_3_1"><i><font color="Green" title="E80">A17</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E15:87_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>
 ;<br/>


</div><b>end;</b></div><b>consider </b><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="c6">Perm</font>,<font color="Maroon" title="c7">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> <b> such that </b><br/><a NAME="E11:87_1_1_3"/><i><font color="Green" title="E75">A18</font></i>: 
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="c6">Perm</font>,<font color="Maroon" title="c7">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>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="card_fin.html#E9:87_1_1_3"><i><font color="Green" title="E74">A11</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon" title="c10">N</font> = <font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c11">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>  <b>by </b><i><a class="ref" href="stirl2_1.html#T8" title="STIRL2_1:th.8">STIRL2_1:8</a>, <a class="txt" href="card_fin.html#E1:87_1_1_3"><i><font color="Green" title="E70">A7</font></i></a></i>;<br/><a NAME="E13:87_1_1_3"/><i><font color="Green" title="E76">A19</font></i>: 
<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="87_1_1_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">x1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a><br/><b>let </b><font color="Maroon" title="c13">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:87_1_1_3_2"/><i><font color="Green" title="E77">A20</font></i>: 
( <font color="Maroon" title="c12">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">FP</font> &amp; <font color="Maroon" title="c13">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">FP</font> &amp; <font color="Maroon" title="c9">FP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">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="c13">x2</font> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c14">p1'</font>, <font color="Maroon" title="c15">f1'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:87_1_1_3_2"/><i><font color="Green" title="E78">A21</font></i>: 
( <font color="Maroon" title="c14">p1'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Perm</font> &amp; <font color="Maroon" title="c15">f1'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Bloc</font> &amp; <font color="Maroon" title="c12">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="c14">p1'</font>,<font color="Maroon" title="c15">f1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="card_fin.html#E1:87_1_1_3_2"><i><font color="Green" title="E77">A20</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">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><b> such that </b><br/><a NAME="E5:87_1_1_3_2"/><i><font color="Green" title="E79">A22</font></i>: 
( <font color="Maroon" title="c14">p1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">p1</font> &amp; <font color="Maroon" title="c16">p1</font> is   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c2">k</font> )
 <b>by </b><i><a class="txt" href="card_fin.html#E3:87_1_1_3_2"><i><font color="Green" title="E78">A21</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">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><b> such that </b><br/><a NAME="E7:87_1_1_3_2"/><i><font color="Green" title="E80">A23</font></i>: 
( <font color="Maroon" title="c15">f1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f1</font> &amp; <font color="Maroon" title="c17">f1</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> &amp; <font color="Maroon" title="c17">f1</font> is <a href="stirl2_1.html#V1" title="STIRL2_1:attr.1">"increasing</a> )
 <b>by </b><i><a class="txt" href="card_fin.html#E3:87_1_1_3_2"><i><font color="Green" title="E78">A21</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">p2'</font>, <font color="Maroon" title="c19">f2'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:87_1_1_3_2"/><i><font color="Green" title="E81">A24</font></i>: 
( <font color="Maroon" title="c18">p2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Perm</font> &amp; <font color="Maroon" title="c19">f2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Bloc</font> &amp; <font color="Maroon" title="c13">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="c18">p2'</font>,<font color="Maroon" title="c19">f2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="card_fin.html#E1:87_1_1_3_2"><i><font color="Green" title="E77">A20</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c20">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><b> such that </b><br/><a NAME="E11:87_1_1_3_2"/><i><font color="Green" title="E82">A25</font></i>: 
( <font color="Maroon" title="c18">p2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">p2</font> &amp; <font color="Maroon" title="c20">p2</font> is   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c2">k</font> )
 <b>by </b><i><a class="txt" href="card_fin.html#E9:87_1_1_3_2"><i><font color="Green" title="E81">A24</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c21">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><b> such that </b><br/><a NAME="E13:87_1_1_3_2"/><i><font color="Green" title="E83">A26</font></i>: 
( <font color="Maroon" title="c19">f2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">f2</font> &amp; <font color="Maroon" title="c21">f2</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> &amp; <font color="Maroon" title="c21">f2</font> is <a href="stirl2_1.html#V1" title="STIRL2_1:attr.1">"increasing</a> )
 <b>by </b><i><a class="txt" href="card_fin.html#E9:87_1_1_3_2"><i><font color="Green" title="E81">A24</font></i></a></i>;<br/>
<a NAME="E14:87_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="c12">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">FP</font>
 
<b>by </b><i><a class="txt" href="card_fin.html#E1:87_1_1_3_2"><i><font color="Green" title="E77">A20</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E15:87_1_1_3_2"/><b>then </b>
<font color="Maroon" title="c9">FP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">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/>
<b>then consider </b><font color="Maroon" title="c22">fp</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c10">N</font>,<font color="Maroon" title="c11">K</font><b> such that </b><br/><a NAME="E17:87_1_1_3_2"/><i><font color="Green" title="E84">A27</font></i>: 
( <font color="Maroon" title="c9">FP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">fp</font> &amp; <font color="Maroon" title="c22">fp</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> )
 ;<br/>
<a NAME="E18:87_1_1_3_2"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c22">fp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">K</font>
 
<b>by </b><i><a class="txt" href="card_fin.html#E17:87_1_1_3_2"><i><font color="Green" title="E84">A27</font></i></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c23">p1''</font> = <font color="Maroon" title="c16">p1</font>, <font color="Maroon" title="c24">p2''</font> = <font color="Maroon" title="c20">p2</font> as   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c22">fp</font> <b>by </b><i><a class="txt" href="card_fin.html#E5:87_1_1_3_2"><i><font color="Green" title="E79">A22</font></i></a>, <a class="txt" href="card_fin.html#E11:87_1_1_3_2"><i><font color="Green" title="E82">A25</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c25">f1''</font> = <font color="Maroon" title="c17">f1</font>, <font color="Maroon" title="c26">f2''</font> = <font color="Maroon" title="c21">f2</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c10">N</font>,<font color="Maroon" title="c11">K</font> ;<br/>
<a NAME="E21:87_1_1_3_2"/>
( ( for <font color="Olive" title="b1">l</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c21">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#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c21">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="henmodel.html#K1" title="HENMODEL:func.1">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">f2</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">l</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="henmodel.html#K1" title="HENMODEL:func.1">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">f2</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b2">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> ) &amp; ( for <font color="Olive" title="b1">l</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">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#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">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="henmodel.html#K1" title="HENMODEL:func.1">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">f1</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">l</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="henmodel.html#K1" title="HENMODEL:func.1">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">f1</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b2">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> ) )
 
<b>by </b><i><a class="txt" href="card_fin.html#E7:87_1_1_3_2"><i><font color="Green" title="E80">A23</font></i></a>, <a class="txt" href="card_fin.html#E13:87_1_1_3_2"><i><font color="Green" title="E83">A26</font></i></a>, <a class="ref" href="stirl2_1.html#D1" title="STIRL2_1:def.1">STIRL2_1:def 1</a></i>;<br/>
<a NAME="E22:87_1_1_3_2"/><b>then </b>
( <font color="Maroon" title="c25">f1''</font> is <a href="stirl2_1.html#V2" title="STIRL2_1:attr.2">"increasing</a> &amp; <font color="Maroon" title="c26">f2''</font> is <a href="stirl2_1.html#V2" title="STIRL2_1:attr.2">"increasing</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">K</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c21">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">K</font> &amp; <font color="Maroon" title="c22">fp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p1''</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c25">f1''</font> &amp; <font color="Maroon" title="c22">fp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">p2''</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c26">f2''</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c22">fp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">K</font> )
 
<b>by </b><i><a class="ref" href="stirl2_1.html#D5" title="STIRL2_1:def.5">STIRL2_1:def 5</a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a>, <a class="txt" href="card_fin.html#E1:87_1_1_3_2"><i><font color="Green" title="E77">A20</font></i></a>, <a class="txt" href="card_fin.html#E3:87_1_1_3_2"><i><font color="Green" title="E78">A21</font></i></a>, <a class="txt" href="card_fin.html#E5:87_1_1_3_2"><i><font color="Green" title="E79">A22</font></i></a>, <a class="txt" href="card_fin.html#E7:87_1_1_3_2"><i><font color="Green" title="E80">A23</font></i></a>, <a class="txt" href="card_fin.html#E9:87_1_1_3_2"><i><font color="Green" title="E81">A24</font></i></a>, <a class="txt" href="card_fin.html#E11:87_1_1_3_2"><i><font color="Green" title="E82">A25</font></i></a>, <a class="txt" href="card_fin.html#E13:87_1_1_3_2"><i><font color="Green" title="E83">A26</font></i></a>, <a class="txt" href="card_fin.html#E17:87_1_1_3_2"><i><font color="Green" title="E84">A27</font></i></a>, <a class="txt" href="card_fin.html#E11:87_1_1_3"><i><font color="Green" title="E75">A18</font></i></a></i>;<br/>
<a NAME="E23:87_1_1_3_2"/><b>then </b>
( <font color="Maroon" title="c23">p1''</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c24">p2''</font> &amp; <font color="Maroon" title="c25">f1''</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c26">f2''</font> )
 
<b>by </b><i><a class="ref" href="stirl2_1.html#T75" title="STIRL2_1:th.75">STIRL2_1:75</a></i>;<br/>
<b>hence </b><a NAME="E24:87_1_1_3_2"/>
<font color="Maroon" title="c12">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x2</font>
 <b>by </b><i><a class="txt" href="card_fin.html#E3:87_1_1_3_2"><i><font color="Green" title="E78">A21</font></i></a>, <a class="txt" href="card_fin.html#E9:87_1_1_3_2"><i><font color="Green" title="E81">A24</font></i></a>, <a class="txt" href="card_fin.html#E5:87_1_1_3_2"><i><font color="Green" title="E79">A22</font></i></a>, <a class="txt" href="card_fin.html#E7:87_1_1_3_2"><i><font color="Green" title="E80">A23</font></i></a>, <a class="txt" href="card_fin.html#E11:87_1_1_3_2"><i><font color="Green" title="E82">A25</font></i></a>, <a class="txt" href="card_fin.html#E13:87_1_1_3_2"><i><font color="Green" title="E83">A26</font></i></a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon" title="c12">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><b> such that </b><br/><a NAME="E15:87_1_1_3"/><i><font color="Green" title="E77">A28</font></i>: 
( <font color="Maroon" title="c12">h</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> &amp; <font color="Maroon" title="c12">h</font> is <a href="stirl2_1.html#V1" title="STIRL2_1:attr.1">"increasing</a> )
 <b>by </b><i><a class="txt" href="card_fin.html#E1:87"><i><font color="Green" title="E69">A1</font></i></a>, <a class="txt" href="card_fin.html#E1:87_1_1_3"><i><font color="Green" title="E70">A7</font></i></a>, <a class="ref" href="stirl2_1.html#T23" title="STIRL2_1:th.23">STIRL2_1:23</a></i>;<br/><a NAME="E16:87_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#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">FP</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_1_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:87_1_1_3_3"/><i><font color="Green" title="E78">A29</font></i>: 
<font color="Maroon" title="c13">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> 
 ;<br/>

<b>consider </b><font color="Maroon" title="c14">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font><b> such that </b><br/><a NAME="E3:87_1_1_3_3"/><i><font color="Green" title="E79">A30</font></i>: 
( <font color="Maroon" title="c14">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font> &amp; <font color="Maroon" title="c14">f</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> )
 <b>by </b><i><a class="txt" href="card_fin.html#E1:87_1_1_3_3"><i><font color="Green" title="E78">A29</font></i></a></i>;<br/>
<a NAME="E4:87_1_1_3_3"/>
( <font color="Maroon" title="c14">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c10">N</font>,<font color="Maroon" title="c11">K</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">K</font> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E3:87_1_1_3_3"><i><font color="Green" title="E79">A30</font></i></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">I</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c10">N</font>,<font color="Maroon" title="c11">K</font>, <font color="Maroon" title="c16">P</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c11">K</font><b> such that </b><br/><a NAME="E6:87_1_1_3_3"/><i><font color="Green" title="E80">A31</font></i>: 
( <font color="Maroon" title="c14">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c15">I</font> &amp; <font color="Maroon" title="c11">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c15">I</font> &amp; <font color="Maroon" title="c15">I</font> is <a href="stirl2_1.html#V2" title="STIRL2_1:attr.2">"increasing</a> )
 <b>by </b><i><a class="ref" href="stirl2_1.html#T73" title="STIRL2_1:th.73">STIRL2_1:73</a></i>;<br/>
<b>set </b><font color="Maroon" title="c17">p</font> = <font color="Maroon" title="c16">P</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">i</font> = <font color="Maroon" title="c15">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="E8:87_1_1_3_3"/>
for <font color="Olive" title="b1">l</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c15">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#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c15">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="henmodel.html#K1" title="HENMODEL:func.1">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">I</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">l</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="henmodel.html#K1" title="HENMODEL:func.1">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">I</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b2">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="card_fin.html#E6:87_1_1_3_3"><i><font color="Green" title="E80">A31</font></i></a>, <a class="ref" href="stirl2_1.html#D5" title="STIRL2_1:def.5">STIRL2_1:def 5</a></i>;<br/>
<a NAME="E9:87_1_1_3_3"/><b>then </b>
( <font color="Maroon" title="c18">i</font> is <a href="stirl2_1.html#V1" title="STIRL2_1:attr.1">"increasing</a> &amp; <font color="Maroon" title="c18">i</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E6:87_1_1_3_3"><i><font color="Green" title="E80">A31</font></i></a>, <a class="ref" href="stirl2_1.html#D1" title="STIRL2_1:def.1">STIRL2_1:def 1</a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E10:87_1_1_3_3"/><b>then </b>
( <font color="Maroon" title="c16">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Perm</font> &amp; <font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Bloc</font> &amp; <font color="Maroon" title="c12">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>  )
 
<b>by </b><i><a class="txt" href="card_fin.html#E15:87_1_1_3"><i><font color="Green" title="E77">A28</font></i></a></i>;<br/>
<a NAME="E11:87_1_1_3_3"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c18">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="c6">Perm</font>,<font color="Maroon" title="c7">Bloc</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c6">Perm</font>,<font color="Maroon" title="c7">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>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">FP</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E12:87_1_1_3_3"/><b>then </b>
( <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="c16">P</font>,<font color="Maroon" title="c18">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="c14">f</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c18">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#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">FP</font> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E11:87_1_1_3"><i><font color="Green" title="E75">A18</font></i></a>, <a class="txt" href="card_fin.html#E6:87_1_1_3_3"><i><font color="Green" title="E80">A31</font></i></a></i>;<br/>
<b>hence </b><a NAME="E13:87_1_1_3_3"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">FP</font>
 <b>by </b><i><a class="txt" href="card_fin.html#E3:87_1_1_3_3"><i><font color="Green" title="E79">A30</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E17:87_1_1_3"/><b>then </b>
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">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>  &amp; <font color="Maroon" title="c12">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>  )
 <b>by </b><i><a class="txt" href="card_fin.html#E15:87_1_1_3"><i><font color="Green" title="E77">A28</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E18:87_1_1_3"/><b>then </b>
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">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>  &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">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="c6">Perm</font>,<font color="Maroon" title="c7">Bloc</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E19:87_1_1_3"/><b>then </b>
( <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="c6">Perm</font>,<font color="Maroon" title="c7">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>  &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c6">Perm</font>,<font color="Maroon" title="c7">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#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">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#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c7">Bloc</font></span>)</span> )
 <b>by </b><i><a class="txt" href="card_fin.html#E13:87_1_1_3"><i><font color="Green" title="E76">A19</font></i></a>, <a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a>, <a class="ref" href="card_2.html#T65" title="CARD_2:th.65">CARD_2:65</a></i>;<br/><a NAME="E20:87_1_1_3"/><b>then </b>
(  <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#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">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#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c7">Bloc</font></span>)</span> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">Perm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K12" title="NEWTON:func.12">!</a>  &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">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> )
 <b>by </b><i><a class="ref" href="card_1.html#T21" title="CARD_1:th.21">CARD_1:21</a>, <a class="ref" href="card_1.html#T66" title="CARD_1:th.66">CARD_1:66</a>, <a class="ref" href="card_fin.html#T8" target="_self" title="CARD_FIN:th.8">Th8</a></i>;<br/><a NAME="E21:87_1_1_3"/><b>then </b>
(  <a href="card_fin.html#K6" title="CARD_FIN:func.6">Sum</a> <font color="Maroon" title="c4">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#K12" title="NEWTON:func.12">!</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#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c7">Bloc</font></span>)</span> &amp; <font color="Maroon" title="c2">k</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 <b>by </b><i><a class="txt" href="card_fin.html#E4:87_1_1_3"><i><font color="Green" title="E72">A9</font></i></a>, <a class="ref" href="newton.html#T23" title="NEWTON:th.23">NEWTON:23</a></i>;<br/><a NAME="E22:87_1_1_3"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K12" title="NEWTON:func.12">!</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#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c7">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#K12" title="NEWTON:func.12">!</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_fin.html#K6" title="CARD_FIN:func.6">Sum</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K12" title="NEWTON:func.12">!</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#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c7">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#K12" title="NEWTON:func.12">!</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c7">Bloc</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K12" title="NEWTON:func.12">!</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#K12" title="NEWTON:func.12">!</a> </span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K12" title="NEWTON:func.12">!</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#K12" title="NEWTON:func.12">!</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a>, <a class="ref" href="xcmplx_1.html#T75" title="XCMPLX_1:th.75">XCMPLX_1:75</a></i>;<br/><a NAME="E23:87_1_1_3"/><b>then </b>
<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="card_fin.html#K6" title="CARD_FIN:func.6">Sum</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span>
 <b>by </b><i><a class="ref" href="stirl2_1.html#D2" title="STIRL2_1:def.2">STIRL2_1:def 2</a></i>;<br/><b>hence </b><a NAME="E24:87_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#K12" title="NEWTON:func.12">!</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="card_fin.html#K6" title="CARD_FIN:func.6">Sum</a> <font color="Maroon" title="c4">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T75" title="XCMPLX_1:th.75">XCMPLX_1:75</a></i>;<br/><b>thus </b><a NAME="E25:87_1_1_3"/>
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">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
 <b>by </b><i><a class="txt" href="card_fin.html#E3:87_1_1_3"><i><font color="Green" title="E71">A8</font></i></a>, <a class="ref" href="card_1.html#T66" title="CARD_1:th.66">CARD_1:66</a></i>;<br/><b>let </b><font color="Maroon" title="c13">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E26:87_1_1_3"/><i><font color="Green" title="E78">A32</font></i>: 
<font color="Maroon" title="c13">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">F</font>
 ;<br/><a NAME="E27:87_1_1_3"/>
(  <a href="card_1.html#K4" title="CARD_1:func.4">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#K4" title="CARD_1:func.4">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> )
 <b>by </b><i><a class="ref" href="card_1.html#T66" title="CARD_1:th.66">CARD_1:66</a></i>;<br/><b>hence </b><a NAME="E28:87_1_1_3"/>
<font color="Maroon" title="c4">F</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</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#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <font color="Maroon" title="c13">m</font></span>)</span></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="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="card_fin.html#E5:87_1_1_3"><i><font color="Green" title="E73">A10</font></i></a>, <a class="txt" href="card_fin.html#E26:87_1_1_3"><i><font color="Green" title="E78">A32</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_1_4"><b>suppose </b></a><a NAME="E1:87_1_1_4"/>
( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c2">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:87_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  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</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#K12" title="NEWTON:func.12">!</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="card_fin.html#K6" title="CARD_FIN:func.6">Sum</a> <font color="Olive" title="b1">F</font></span>)</span> &amp;  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Olive" title="b1">F</font> holds <br/><font color="Olive" title="b1">F</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</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#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <font color="Olive" title="b2">m</font></span>)</span></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="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="card_fin.html#E1:87"><i><font color="Green" title="E69">A1</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:87"/>
 ex <font color="Olive" title="b1">F</font> being  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</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#K12" title="NEWTON:func.12">!</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="card_fin.html#K6" title="CARD_FIN:func.6">Sum</a> <font color="Olive" title="b1">F</font></span>)</span> &amp;  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Olive" title="b1">F</font> holds <br/><font color="Olive" title="b1">F</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</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#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <font color="Olive" title="b2">m</font></span>)</span></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="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> ) )
 ;<br/>


</div>
