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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:87"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&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">E70</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = 1 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/><b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="afinsq_1.html#K9">%&gt;</a></span>;<br/><b>take </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="afinsq_1.html#K9">%&gt;</a></span>;<br/><a NAME="E3:87_1_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="newton.html#K12">!</a>  <a href="hidden.html#R1">=</a> 1 &amp;  <a href="card_fin.html#K6" target="_self">Sum</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K44">addint</a>  <a href="stirl2_1.html#K4">"**"</a> <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="afinsq_1.html#K9">%&gt;</a></span> &amp; 1 <a href="hidden.html#R1">=</a> 1 &amp; <a href="binop_2.html#K44">addint</a>  <a href="stirl2_1.html#K4">"**"</a> <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="afinsq_1.html#K9">%&gt;</a></span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="txt" href="card_fin.html#E1:87_1_1_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="newton.html#T18">NEWTON:18</a>, <a class="ref" href="stirl2_1.html#T44">STIRL2_1:44</a></i>;<br/><b>hence </b><a NAME="E4:87_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="stirl2_1.html#K3">block</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="card_fin.html#K6" target="_self">Sum</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="card_fin.html#E1:87_1_1_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="stirl2_1.html#T26">STIRL2_1:26</a></i>;<br/><b>thus </b><a NAME="E5:87_1_1_1"/>
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="txt" href="card_fin.html#E1:87_1_1_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="afinsq_1.html#T36">AFINSQ_1:36</a></i>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E6:87_1_1_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E7:87_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="card_fin.html#E6:87_1_1_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="afinsq_1.html#T36">AFINSQ_1:36</a>, <a class="ref" href="card_5.html#T1">CARD_5:1</a></i>;<br/><a NAME="E8:87_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E9:87_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K9">choose</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="txt" href="card_fin.html#E1:87_1_1_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="newton.html#T9">NEWTON:9</a>, <a class="ref" href="newton.html#T27">NEWTON:27</a></i>;<br/><b>hence </b><a NAME="E10:87_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</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">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K9">choose</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="afinsq_1.html#T38">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">E70</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="finsop_1.html#K3">--&gt;</a> 0;<br/><a NAME="E2:87_1_1_2"/>
0 is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a> 
 <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/><a NAME="E3:87_1_1_2"/><b>then </b><i><font color="Green">E71</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="finsop_1.html#K3">--&gt;</a> 0</span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> &amp; <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K4">INT</a>  &amp; <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a>  )
 <b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E4:87_1_1_2"/><b>then </b>
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="finsop_1.html#K3">--&gt;</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1 &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="finsop_1.html#K3">--&gt;</a> 0</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K4">INT</a>  )
 <b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="finsop_1.html#K3">--&gt;</a> 0 as   <a href="afinsq_1.html#NM1">XFinSequence</a> <b>by </b><i><a class="ref" href="afinsq_1.html#T7">AFINSQ_1:7</a></i>;<br/><a NAME="E6:87_1_1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K4">INT</a> 
 <b>by </b><i><a class="txt" href="card_fin.html#E3:87_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K4">INT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D8">ORDINAL1:def 8</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="card_fin.html#E3:87_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="ordinal1.html#D8">ORDINAL1:def 8</a></i>;<br/><b>take </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font>;<br/><a NAME="E9:87_1_1_2"/>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> &amp; <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,0</span><a href="domain_1.html#K7">}</a></span> )
 <b>by </b><i><a class="ref" href="enumset1.html#T69">ENUMSET1:69</a>, <a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/><a NAME="E10:87_1_1_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,0</span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E11:87_1_1_2"/><b>then </b>
 <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K10">"</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="card_fin.html#T45" target="_self">Th45</a></i>;<br/><a NAME="E12:87_1_1_2"/><b>then </b>
(  <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_fin.html#K6" target="_self">Sum</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="card_fin.html#T63" target="_self">Th63</a></i>;<br/><b>hence </b><a NAME="E13:87_1_1_2"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="stirl2_1.html#K3">block</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="card_fin.html#K6" target="_self">Sum</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1 )
 <b>by </b><i><a class="txt" href="card_fin.html#E1:87_1_1_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a>, <a class="ref" href="stirl2_1.html#T31">STIRL2_1:31</a></i>;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E14:87_1_1_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">c<sub>7</sub></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">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&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">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:87_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 <b>by </b><i><a class="txt" href="card_fin.html#E1:87_1_1_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="nat_1.html#T39">NAT_1:39</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">c<sub>2</sub></font> <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="newton.html#T16">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">c<sub>2</sub></font> <a href="newton.html#K9">choose</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</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">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:87_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K9">choose</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="card_fin.html#E1:87_1_1_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="newton.html#D3">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">c<sub>2</sub></font> <a href="newton.html#K9">choose</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</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">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K9">choose</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1 )
 <b>by </b><i><a class="txt" href="card_fin.html#E14:87_1_1_2"><i><font color="Green">E72</font></i></a>, <a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/><b>hence </b><a NAME="E17:87_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</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">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K9">choose</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">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">E70</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K4">INT</a> <b> such that </b><br/><a NAME="E3:87_1_1_3"/><i><font color="Green">E71</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1
 <b>and </b><br/><a NAME="E4:87_1_1_3"/><i><font color="Green">E72</font></i>: 
 <a href="card_fin.html#K6" target="_self">Sum</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="funct_2.html#V2">onto</a> </span>}</span> 
 <b>and </b><br/><a NAME="E5:87_1_1_3"/><i><font color="Green">E73</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</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">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</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">card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="card_fin.html#T70" target="_self">Th70</a></i>;<br/><b>take </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> is   <a href="funct_2.html#NM2">Permutation</a> of <font color="Maroon">c<sub>2</sub></font> </span>}</span> ;<br/><a NAME="E6:87_1_1_3"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> is   <a href="funct_2.html#NM2">Permutation</a> of <font color="Maroon">c<sub>2</sub></font> </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="newton.html#K12">!</a> 
 <b>by </b><i><a class="ref" href="card_fin.html#T8" target="_self">Th8</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> is   <a href="funct_2.html#NM2">Permutation</a> of <font color="Maroon">c<sub>2</sub></font> </span>}</span>  as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="card_4.html#T1">CARD_4:1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> : ( <font color="Olive">b<sub>1</sub></font> is <a href="funct_2.html#V2">onto</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="stirl2_1.html#V1">"increasing</a> ) </span>}</span>  as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="stirl2_1.html#T24">STIRL2_1:24</a></i>;<br/><b>set </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="funct_2.html#V2">onto</a> </span>}</span> ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means for <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="partfun1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font>;<br/><a NAME="E9:87_1_1_3"/><i><font color="Green">E74</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>3</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>3</sub></font> is <a href="funct_2.html#V2">onto</a> </span>}</span>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:87_1_1_3_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:87_1_1_3_1"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="txt" href="card_fin.html#E1:87_1_1_3_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E5:87_1_1_3_1"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> is   <a href="funct_2.html#NM2">Permutation</a> of <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="card_fin.html#E3:87_1_1_3_1"><i><font color="Green">E76</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E7:87_1_1_3_1"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> is <a href="funct_2.html#V2">onto</a> &amp; <font color="Maroon">c<sub>13</sub></font> is <a href="stirl2_1.html#V1">"increasing</a> )
 <b>by </b><i><a class="txt" href="card_fin.html#E3:87_1_1_3_1"><i><font color="Green">E76</font></i></a></i>;<br/>
<a NAME="E8:87_1_1_3_1"/>
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E5:87_1_1_3_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="card_fin.html#E7:87_1_1_3_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="card_fin.html#E1:87_1_1_3"><i><font color="Green">E70</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E9:87_1_1_3_1"/><b>then </b><i><font color="Green">E79</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T46">RELAT_1:46</a>, <a class="ref" href="relat_1.html#T47">RELAT_1:47</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="funct_2.html#T3">FUNCT_2:3</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>14</sub></font>
;<br/>

<a NAME="E11:87_1_1_3_1"/>
<font color="Maroon">c<sub>14</sub></font> is <a href="funct_2.html#V2">onto</a>
 
<b>by </b><i><a class="txt" href="card_fin.html#E9:87_1_1_3_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<b>hence </b><a NAME="E12:87_1_1_3_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="funct_2.html#V2">onto</a> </span>}</span> 
 ;<br/>

<b>let </b><font color="Maroon">c<sub>15</sub></font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>16</sub></font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>



<b>assume </b><a NAME="E13:87_1_1_3_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="domain_1.html#K1">]</a></span>
 ;<br/>

<a NAME="E14:87_1_1_3_1"/>
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E3:87_1_1_3_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="card_fin.html#E5:87_1_1_3_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="card_fin.html#E7:87_1_1_3_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="card_fin.html#E13:87_1_1_3_1"><i><font color="Green">E80</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E15:87_1_1_3_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="funct_2.html#V2">onto</a> </span>}</span> <b> such that </b><br/><a NAME="E11:87_1_1_3"/><i><font color="Green">E75</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>(<a class="txt" href="card_fin.html#E9:87_1_1_3"><i><font color="Green">E74</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="stirl2_1.html#T8">STIRL2_1:8</a>, <a class="txt" href="card_fin.html#E1:87_1_1_3"><i><font color="Green">E70</font></i></a></i>;<br/><a NAME="E13:87_1_1_3"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>9</sub></font> is <a href="funct_1.html#V2">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">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/><b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:87_1_1_3_2"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> )
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:87_1_1_3_2"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="txt" href="card_fin.html#E1:87_1_1_3_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E5:87_1_1_3_2"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> is   <a href="funct_2.html#NM2">Permutation</a> of <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="card_fin.html#E3:87_1_1_3_2"><i><font color="Green">E78</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E7:87_1_1_3_2"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> is <a href="funct_2.html#V2">onto</a> &amp; <font color="Maroon">c<sub>17</sub></font> is <a href="stirl2_1.html#V1">"increasing</a> )
 <b>by </b><i><a class="txt" href="card_fin.html#E3:87_1_1_3_2"><i><font color="Green">E78</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:87_1_1_3_2"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="txt" href="card_fin.html#E1:87_1_1_3_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E11:87_1_1_3_2"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> is   <a href="funct_2.html#NM2">Permutation</a> of <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="card_fin.html#E9:87_1_1_3_2"><i><font color="Green">E81</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>21</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E13:87_1_1_3_2"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> is <a href="funct_2.html#V2">onto</a> &amp; <font color="Maroon">c<sub>21</sub></font> is <a href="stirl2_1.html#V1">"increasing</a> )
 <b>by </b><i><a class="txt" href="card_fin.html#E9:87_1_1_3_2"><i><font color="Green">E81</font></i></a></i>;<br/>
<a NAME="E14:87_1_1_3_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="card_fin.html#E1:87_1_1_3_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E15:87_1_1_3_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="funct_2.html#V2">onto</a> </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>22</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font><b> such that </b><br/><a NAME="E17:87_1_1_3_2"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> is <a href="funct_2.html#V2">onto</a> )
 ;<br/>
<a NAME="E18:87_1_1_3_2"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="card_fin.html#E17:87_1_1_3_2"><i><font color="Green">E84</font></i></a>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>20</sub></font> as   <a href="funct_2.html#NM2">Permutation</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>22</sub></font> <b>by </b><i><a class="txt" href="card_fin.html#E5:87_1_1_3_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="card_fin.html#E11:87_1_1_3_2"><i><font color="Green">E82</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>21</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> ;<br/>
<a NAME="E21:87_1_1_3_2"/>
( ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> holds <br/> <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="stirl2_1.html#K1">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="stirl2_1.html#K1">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> ) &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> holds <br/> <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="stirl2_1.html#K1">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="stirl2_1.html#K1">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K6">}</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">E80</font></i></a>, <a class="txt" href="card_fin.html#E13:87_1_1_3_2"><i><font color="Green">E83</font></i></a>, <a class="ref" href="stirl2_1.html#D1">STIRL2_1:def 1</a></i>;<br/>
<a NAME="E22:87_1_1_3_2"/><b>then </b>
( <font color="Maroon">c<sub>25</sub></font> is <a href="stirl2_1.html#V2">"increasing</a> &amp; <font color="Maroon">c<sub>26</sub></font> is <a href="stirl2_1.html#V2">"increasing</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>25</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>26</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="ref" href="stirl2_1.html#D5">STIRL2_1:def 5</a>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a>, <a class="txt" href="card_fin.html#E1:87_1_1_3_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="card_fin.html#E3:87_1_1_3_2"><i><font color="Green">E78</font></i></a>, <a class="txt" href="card_fin.html#E5:87_1_1_3_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="card_fin.html#E7:87_1_1_3_2"><i><font color="Green">E80</font></i></a>, <a class="txt" href="card_fin.html#E9:87_1_1_3_2"><i><font color="Green">E81</font></i></a>, <a class="txt" href="card_fin.html#E11:87_1_1_3_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="card_fin.html#E13:87_1_1_3_2"><i><font color="Green">E83</font></i></a>, <a class="txt" href="card_fin.html#E17:87_1_1_3_2"><i><font color="Green">E84</font></i></a>, <a class="txt" href="card_fin.html#E11:87_1_1_3"><i><font color="Green">E75</font></i></a></i>;<br/>
<a NAME="E23:87_1_1_3_2"/><b>then </b>
( <font color="Maroon">c<sub>23</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>26</sub></font> )
 
<b>by </b><i><a class="ref" href="stirl2_1.html#T75">STIRL2_1:75</a></i>;<br/>
<b>hence </b><a NAME="E24:87_1_1_3_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="card_fin.html#E3:87_1_1_3_2"><i><font color="Green">E78</font></i></a>, <a class="txt" href="card_fin.html#E9:87_1_1_3_2"><i><font color="Green">E81</font></i></a>, <a class="txt" href="card_fin.html#E5:87_1_1_3_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="card_fin.html#E7:87_1_1_3_2"><i><font color="Green">E80</font></i></a>, <a class="txt" href="card_fin.html#E11:87_1_1_3_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="card_fin.html#E13:87_1_1_3_2"><i><font color="Green">E83</font></i></a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E15:87_1_1_3"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> is <a href="funct_2.html#V2">onto</a> &amp; <font color="Maroon">c<sub>12</sub></font> is <a href="stirl2_1.html#V1">"increasing</a> )
 <b>by </b><i><a class="txt" href="card_fin.html#E1:87"><i><font color="Green">E69</font></i></a>, <a class="txt" href="card_fin.html#E1:87_1_1_3"><i><font color="Green">E70</font></i></a>, <a class="ref" href="stirl2_1.html#T23">STIRL2_1:23</a></i>;<br/><a NAME="E16:87_1_1_3"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="funct_2.html#V2">onto</a> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>9</sub></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">c<sub>13</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:87_1_1_3_3"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="funct_2.html#V2">onto</a> </span>}</span> 
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:87_1_1_3_3"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> is <a href="funct_2.html#V2">onto</a> )
 <b>by </b><i><a class="txt" href="card_fin.html#E1:87_1_1_3_3"><i><font color="Green">E78</font></i></a></i>;<br/>
<a NAME="E4:87_1_1_3_3"/>
( <font color="Maroon">c<sub>14</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E3:87_1_1_3_3"><i><font color="Green">E79</font></i></a>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>16</sub></font> being   <a href="funct_2.html#NM2">Permutation</a> of <font color="Maroon">c<sub>11</sub></font><b> such that </b><br/><a NAME="E6:87_1_1_3_3"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> is <a href="stirl2_1.html#V2">"increasing</a> )
 <b>by </b><i><a class="ref" href="stirl2_1.html#T73">STIRL2_1:73</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E8:87_1_1_3_3"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> holds <br/> <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K6">}</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">E80</font></i></a>, <a class="ref" href="stirl2_1.html#D5">STIRL2_1:def 5</a></i>;<br/>
<a NAME="E9:87_1_1_3_3"/><b>then </b>
( <font color="Maroon">c<sub>18</sub></font> is <a href="stirl2_1.html#V1">"increasing</a> &amp; <font color="Maroon">c<sub>18</sub></font> is <a href="funct_2.html#V2">onto</a> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E6:87_1_1_3_3"><i><font color="Green">E80</font></i></a>, <a class="ref" href="stirl2_1.html#D1">STIRL2_1:def 1</a>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E10:87_1_1_3_3"/><b>then </b>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="funct_2.html#V2">onto</a> </span>}</span>  )
 
<b>by </b><i><a class="txt" href="card_fin.html#E15:87_1_1_3"><i><font color="Green">E77</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">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E12:87_1_1_3_3"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E11:87_1_1_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="card_fin.html#E6:87_1_1_3_3"><i><font color="Green">E80</font></i></a></i>;<br/>
<b>hence </b><a NAME="E13:87_1_1_3_3"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="card_fin.html#E3:87_1_1_3_3"><i><font color="Green">E79</font></i></a>, <a class="ref" href="funct_1.html#D5">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">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="funct_2.html#V2">onto</a> </span>}</span>  &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="funct_2.html#V2">onto</a> </span>}</span>  )
 <b>by </b><i><a class="txt" href="card_fin.html#E15:87_1_1_3"><i><font color="Green">E77</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E18:87_1_1_3"/><b>then </b>
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="funct_2.html#V2">onto</a> </span>}</span>  &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">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">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="funct_2.html#V2">onto</a> </span>}</span> ,<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R2">are_equipotent</a>  &amp;  <a href="card_1.html#K4">card</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="card_fin.html#E13:87_1_1_3"><i><font color="Green">E76</font></i></a>, <a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a>, <a class="ref" href="card_2.html#T65">CARD_2:65</a></i>;<br/><a NAME="E20:87_1_1_3"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="funct_2.html#V2">onto</a> </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="newton.html#K12">!</a>  &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="card_1.html#T21">CARD_1:21</a>, <a class="ref" href="card_1.html#T66">CARD_1:66</a>, <a class="ref" href="card_fin.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E21:87_1_1_3"/><b>then </b>
(  <a href="card_fin.html#K6" target="_self">Sum</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K12">!</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>by </b><i><a class="txt" href="card_fin.html#E4:87_1_1_3"><i><font color="Green">E72</font></i></a>, <a class="ref" href="newton.html#T23">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">c<sub>2</sub></font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K12">!</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_fin.html#K6" target="_self">Sum</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K12">!</a> </span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K12">!</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K12">!</a> </span>)</span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a>, <a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>;<br/><a NAME="E23:87_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="stirl2_1.html#K3">block</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_fin.html#K6" target="_self">Sum</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K12">!</a> </span>)</span>
 <b>by </b><i><a class="ref" href="stirl2_1.html#D2">STIRL2_1:def 2</a></i>;<br/><b>hence </b><a NAME="E24:87_1_1_3"/>
<font color="Maroon">c<sub>1</sub></font> <a href="stirl2_1.html#K3">block</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="card_fin.html#K6" target="_self">Sum</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>;<br/><b>thus </b><a NAME="E25:87_1_1_3"/>
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="txt" href="card_fin.html#E3:87_1_1_3"><i><font color="Green">E71</font></i></a>, <a class="ref" href="card_1.html#T66">CARD_1:66</a></i>;<br/><b>let </b><font color="Maroon">c<sub>13</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E26:87_1_1_3"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E27:87_1_1_3"/>
(  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="card_1.html#T66">CARD_1:66</a></i>;<br/><b>hence </b><a NAME="E28:87_1_1_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</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">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K9">choose</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="card_fin.html#E5:87_1_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="card_fin.html#E26:87_1_1_3"><i><font color="Green">E78</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">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:87_1_1_4"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K4">INT</a>  st <br/>( <font color="Maroon">c<sub>1</sub></font> <a href="stirl2_1.html#K3">block</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="card_fin.html#K6" target="_self">Sum</a> <font color="Olive">b<sub>1</sub></font></span>)</span> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1 &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</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">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K9">choose</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K10">-</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="card_fin.html#E1:87"><i><font color="Green">E69</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">b<sub>1</sub></font> being  <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K4">INT</a>  st <br/>( <font color="Maroon">c<sub>1</sub></font> <a href="stirl2_1.html#K3">block</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="card_fin.html#K6" target="_self">Sum</a> <font color="Olive">b<sub>1</sub></font></span>)</span> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1 &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</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">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K9">choose</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K10">-</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> ) )
 ;<br/>


</div>
