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<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means for <font color="Olive">b<sub>1</sub></font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="eqrel_1.html#M1">a_partition</a> of <font color="Olive">b<sub>1</sub></font> st  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Olive">b<sub>2</sub></font> st <font color="Olive">b<sub>3</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> holds <br/>for <font color="Olive">b<sub>4</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>3</sub></font> &amp; ( for <font color="Olive">b<sub>5</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">b<sub>4</sub></font> holds <br/><font color="Olive">b<sub>4</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">b<sub>5</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>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>5</sub></font></span>)</span> ) holds <br/> <a href="card_1.html#K1">Card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K8">Sum</a> <font color="Olive">b<sub>4</sub></font>;<br/>
<a NAME="E1:6"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:6_1"/><i><font color="Green">E8</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><a NAME="E2:6_1"/><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E3:6_1"/><i><font color="Green">E10</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> &amp; ( 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="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) )
 ;<br/>

<a NAME="E4:6_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="weddwitt.html#E1:6_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="graph_5.html#T5">GRAPH_5:5</a></i>;<br/>
<a NAME="E5:6_1"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="weddwitt.html#E1:6_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="graph_5.html#T5">GRAPH_5:5</a>, <a class="ref" href="zfmisc_1.html#T2">ZFMISC_1:2</a></i>;<br/>
<a NAME="E6:6_1"/><b>then </b><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/>
<a NAME="E7:6_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="weddwitt.html#E2:6_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="weddwitt.html#E4:6_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="finseq_1.html#T27">FINSEQ_1:27</a></i>;<br/>
<b>hence </b><a NAME="E8:6_1"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K8">Sum</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E3:6_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="weddwitt.html#E6:6_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="card_1.html#T78">CARD_1:78</a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a>, <a class="ref" href="rvsum_1.html#T102">RVSUM_1:102</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:6"/><i><font color="Green">E8</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:6_2"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>]
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>



<b>assume </b><a NAME="E2:6_2"/><i><font color="Green">E10</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E3:6_2"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E4:6_2"/><i><font color="Green">E12</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E5:6_2"/><i><font color="Green">E13</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E6:6_2"/><i><font color="Green">E14</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="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 ;<br/>

<a NAME="E7:6_2"/><i><font color="Green">E15</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="weddwitt.html#E2:6_2"><i><font color="Green">E10</font></i></a>, <a class="txt" href="weddwitt.html#E3:6_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="weddwitt.html#E4:6_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="finseq_4.html#T77">FINSEQ_4:77</a></i>;<br/>
<a NAME="E8:6_2"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="weddwitt.html#E2:6_2"><i><font color="Green">E10</font></i></a>, <a class="ref" href="graph_5.html#T5">GRAPH_5:5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="weddwitt.html#E2:6_2"><i><font color="Green">E10</font></i></a>, <a class="ref" href="graph_5.html#T5">GRAPH_5:5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>6</sub></font> ;<br/>
<a NAME="E11:6_2"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="weddwitt.html#E8:6_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="weddwitt.html#E8:6_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>6</sub></font> <b>by </b><i><a class="ref" href="finseq_1.html#T23">FINSEQ_1:23</a></i>;<br/>
<a NAME="E14:6_2"/><i><font color="Green">E18</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E7:6_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="finseq_3.html#T59">FINSEQ_3:59</a></i>;<br/>
<a NAME="E15:6_2"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E7:6_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="finseq_3.html#T61">FINSEQ_3:61</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> =  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>9</sub></font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E17:6_2"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T95">ZFMISC_1:95</a></i>;<br/>
<a NAME="E18:6_2"/><b>then </b><i><font color="Green">E20</font></i>: 
 <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>10</sub></font> <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E20:6_2"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>11</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E3:6_2"><i><font color="Green">E11</font></i></a>, <a class="ref" href="funct_1.html#T84">FUNCT_1:84</a></i>;<br/>
<a NAME="E21:6_2"/><b>then </b><i><font color="Green">E22</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E14:6_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="finseq_4.html#T77">FINSEQ_4:77</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> =  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>10</sub></font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="weddwitt.html#E18:6_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<div><i><font color="Green">E23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_2_1_1"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_2_1_1_1"><b>suppose </b></a><a NAME="E1:6_2_1_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">c<sub>12</sub></font>;<br/><a NAME="E3:6_2_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="weddwitt.html#E1:6_2_1_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a>, <a class="ref" href="zfmisc_1.html#T2">ZFMISC_1:2</a></i>;<br/><b>hence </b><a NAME="E4:6_2_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E1:6_2_1_1_1"><i><font color="Green">E24</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_2_1_1_2"><b>suppose </b></a><a NAME="E1:6_2_1_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:6_2_1_1_2"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E4:6_2_1_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E3:6_2_1_1_2"><i><font color="Green">E24</font></i></a></i>;<br/><a NAME="E5:6_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="weddwitt.html#E11:6_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:6_2_1_1_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E8:6_2_1_1_2"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>12</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>12</sub></font> holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">b<sub>2</sub></font> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>12</sub></font>;<br/>

<b>assume </b><a NAME="E1:6_2_1_1_2_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<a NAME="E2:6_2_1_1_2_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E1:6_2_1_1_2_1"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E3:6_2_1_1_2_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>12</sub></font> holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> or <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> or <font color="Maroon">c<sub>15</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">b<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>12</sub></font>;<br/>

<b>assume </b><a NAME="E1:6_2_1_1_2_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<a NAME="E2:6_2_1_1_2_1_1"/>
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="weddwitt.html#E1:6_2_1_1_2_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="weddwitt.html#E1:6_2_1_1_2_1_1"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:6_2_1_1_2_1_1"/>
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> or <font color="Maroon">c<sub>15</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="weddwitt.html#E11:6_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:6_2_1_1_2_1"/>
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>12</sub></font> holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> or <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> or <font color="Maroon">c<sub>15</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">b<sub>1</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="weddwitt.html#E11:6_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="weddwitt.html#E2:6_2_1_1_2_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:6_2_1_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T100">ZFMISC_1:100</a></i>;<br/><a NAME="E10:6_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>12</sub></font>
 ;<br/><b>hence </b><a NAME="E11:6_2_1_1_2"/>
<font color="Maroon">c<sub>10</sub></font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E3:6_2_1_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="weddwitt.html#E7:6_2_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="weddwitt.html#E8:6_2_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#T23">FINSEQ_1:23</a></i>;<br/>
<a NAME="E25:6_2"/><i><font color="Green">E24</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E5:6_2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="weddwitt.html#E7:6_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="weddwitt.html#E14:6_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="finseq_3.html#T59">FINSEQ_3:59</a></i>;<br/>
<a NAME="E26:6_2"/>
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="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></font> holds <br/><font color="Maroon">c<sub>13</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:6_2_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>

<a NAME="E2:6_2_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E3:6_2_2"/><b>then </b>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E5:6_2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="weddwitt.html#E7:6_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a></i>;<br/>
<a NAME="E4:6_2_2"/><b>then </b><i><font color="Green">E27</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E7:6_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="weddwitt.html#E2:6_2_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a></i>;<br/>
<a NAME="E5:6_2_2"/><i><font color="Green">E28</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T89">RELAT_1:89</a></i>;<br/>
<a NAME="E6:6_2_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E1:6_2_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E7:6_2_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E6:6_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="weddwitt.html#E1:6_2_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="weddwitt.html#E5:6_2_2"><i><font color="Green">E28</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:6_2_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="weddwitt.html#E1:6_2_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="weddwitt.html#E4:6_2_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E27:6_2"/><b>then </b><i><font color="Green">E25</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K8">Sum</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E1:6_2"><i><font color="Green">E9</font></i></a>, <a class="txt" href="weddwitt.html#E20:6_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="weddwitt.html#E21:6_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="weddwitt.html#E23:6_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="weddwitt.html#E25:6_2"><i><font color="Green">E24</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as    <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E29:6_2"/>
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> <font color="Maroon">c<sub>10</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>14</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_2_3"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<a NAME="E2:6_2_3"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E1:6_2_3"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E3:6_2_3"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/>
<a NAME="E4:6_2_3"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:6_2_3"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E7:6_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="weddwitt.html#E3:6_2_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E6:6_2_3"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>14</sub></font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E4:6_2"><i><font color="Green">E12</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:6_2_3"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="txt" href="weddwitt.html#E1:6_2_3"><i><font color="Green">E26</font></i></a>, <a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:6_2_3_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/><a NAME="E2:6_2_3_1"/><i><font color="Green">E33</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E14:6_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:6_2_3_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E14:6_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="weddwitt.html#E8:6_2_3"><i><font color="Green">E31</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E4:6_2_3_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E8:6_2_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="weddwitt.html#E2:6_2_3_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:6_2_3_1"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:6_2_3_1"/><i><font color="Green">E36</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="weddwitt.html#E7:6_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E7:6_2_3_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E8:6_2_3_1"/><b>then </b><i><font color="Green">E37</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/><a NAME="E9:6_2_3_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E6:6_2_3_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/><a NAME="E10:6_2_3_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E8:6_2_3"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><b>hence </b><a NAME="E11:6_2_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="weddwitt.html#E3:6_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="weddwitt.html#E1:6_2_3_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="weddwitt.html#E3:6_2_3_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="weddwitt.html#E5:6_2_3_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="weddwitt.html#E6:6_2_3_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="weddwitt.html#E8:6_2_3_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="weddwitt.html#E9:6_2_3_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:6_2_3"/>
<font color="Maroon">c<sub>15</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E2:6_2_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="weddwitt.html#E5:6_2_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="weddwitt.html#E6:6_2_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="weddwitt.html#E8:6_2_3"><i><font color="Green">E31</font></i></a>, <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E30:6_2"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T98">ZFMISC_1:98</a></i>;<br/>
<a NAME="E31:6_2"/><i><font color="Green">E27</font></i>: 
 <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/>
<a NAME="E32:6_2"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/>
<a NAME="E33:6_2"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E34:6_2"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E7:6_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="weddwitt.html#E32:6_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="weddwitt.html#E31:6_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="finset_1.html#T25">FINSET_1:25</a></i>;<br/>
<i><font color="Green">E29</font></i>: <font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="weddwitt.html#E4:6_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="weddwitt.html#E15:6_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
;<br/>
<i><font color="Green">E30</font></i>: <font color="Maroon">c<sub>8</sub></font> = 
 <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="weddwitt.html#E36:6_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="zfmisc_1.html#T96">ZFMISC_1:96</a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T31">ZFMISC_1:31</a></i>
;<br/>
<a NAME="E38:6_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/>
<a NAME="E39:6_2"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E5:6_2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="weddwitt.html#E7:6_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E40:6_2"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>13</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>8</sub></font> = 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Sum</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="weddwitt.html#E27:6_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="weddwitt.html#E30:6_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="weddwitt.html#E37:6_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="card_2.html#T53">CARD_2:53</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Sum</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wsierp_1.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="weddwitt.html#E6:6_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="weddwitt.html#E39:6_2"><i><font color="Green">E31</font></i></a></i>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wsierp_1.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T104">RVSUM_1:104</a></i>
<br/>.= 
 <a href="wsierp_1.html#K8">Sum</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="weddwitt.html#E5:6_2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="weddwitt.html#E7:6_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="finseq_3.html#T61">FINSEQ_3:61</a></i>
;<br/>
<b>hence </b><a NAME="E43:6_2"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K8">Sum</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:6"/><i><font color="Green">E9</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="weddwitt.html#E1:6"><i><font color="Green">E7</font></i></a>, <a class="txt" href="weddwitt.html#E2:6"><i><font color="Green">E8</font></i></a>);<br/></i>
<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E4:6"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<b>hence </b><a NAME="E5:6"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">b<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> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> ) holds <br/> <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K8">Sum</a> <font color="Olive">b<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E3:6"><i><font color="Green">E9</font></i></a></i>;<br/>


</div>
