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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means ( ( for <font color="Olive">b<sub>1</sub></font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st not the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> is <a href="realset1.html#V1">trivial</a> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K13">FinSETS</a>  holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> ) &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> holds <br/>(  <a href="neckla_2.html#K1" target="_self">union_of</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> &amp;  <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> ) ) );<br/>
<b>consider </b><font color="Maroon">c<sub>1</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:13_1_1"/><i><font color="Green">E8</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <a href="neckla_2.html#K3" target="_self">fin_RelStr</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E3:13_1_1"/>
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>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <a href="neckla_2.html#K3" target="_self">fin_RelStr</a> 
 
<b>by </b><i><a class="txt" href="neckla_2.html#E2:13_1_1"><i><font color="Green">E8</font></i></a></i>;<br/>
<a NAME="E4:13_1_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <a href="neckla_2.html#K3" target="_self">fin_RelStr</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <a href="neckla_2.html#K3" target="_self">fin_RelStr</a>  ;<br/>
<b>take </b><font color="Maroon">c<sub>3</sub></font> =  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>2</sub></font>;<br/>

<a NAME="E6:13_1_1"/><i><font color="Green">E9</font></i>: 
 <a href="neckla_2.html#K3" target="_self">fin_RelStr</a>  <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <a href="neckla_2.html#K3" target="_self">fin_RelStr</a> 
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D1">ZFMISC_1:def 1</a></i>;<br/>
<a NAME="E7:13_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="neckla_2.html#K3" target="_self">fin_RelStr</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="neckla_2.html#K3" target="_self">fin_RelStr</a> ;<br/>
<a NAME="E1:13_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K3" target="_self">fin_RelStr</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K3" target="_self">fin_RelStr</a>  holds <br/>(  <a href="neckla_2.html#K1" target="_self">union_of</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K3" target="_self">fin_RelStr</a>  &amp;  <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K3" target="_self">fin_RelStr</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be  <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:13_1_1_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E2:13_1_1_1_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K3" target="_self">fin_RelStr</a> 
 <b>and </b><br/><a NAME="E3:13_1_1_1_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K3" target="_self">fin_RelStr</a> 
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>7</sub></font> being  <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a> <b> such that </b><br/><a NAME="E5:13_1_1_1_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E6:13_1_1_1_1"/><i><font color="Green">E13</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K13">FinSETS</a> 
 <b>by </b><i><a class="txt" href="neckla_2.html#E2:13_1_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="neckla_2.html#D4" target="_self">Def4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being  <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a> <b> such that </b><br/><a NAME="E8:13_1_1_1_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E9:13_1_1_1_1"/><i><font color="Green">E15</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K13">FinSETS</a> 
 <b>by </b><i><a class="txt" href="neckla_2.html#E3:13_1_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="neckla_2.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E10:13_1_1_1_1"/><i><font color="Green">E16</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K13">FinSETS</a> 
 
<b>by </b><i><a class="txt" href="neckla_2.html#E5:13_1_1_1_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="neckla_2.html#E6:13_1_1_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="neckla_2.html#E8:13_1_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="neckla_2.html#E9:13_1_1_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="classes2.html#T66">CLASSES2:66</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> =  <a href="neckla_2.html#K1" target="_self">union_of</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> as  <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  ;<br/>
<a NAME="E12:13_1_1_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K13">FinSETS</a> 
 
<b>by </b><i><a class="txt" href="neckla_2.html#E5:13_1_1_1_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="neckla_2.html#E8:13_1_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="neckla_2.html#E10:13_1_1_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="neckla_2.html#D2" target="_self">Def2</a></i>;<br/>
<b>hence </b><a NAME="E13:13_1_1_1_1"/>
 <a href="neckla_2.html#K1" target="_self">union_of</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K3" target="_self">fin_RelStr</a> 
 <b>by </b><i><a class="txt" href="neckla_2.html#E5:13_1_1_1_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="neckla_2.html#E8:13_1_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="neckla_2.html#D4" target="_self">Def4</a></i>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> =  <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> as  <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  ;<br/>
<a NAME="E15:13_1_1_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K13">FinSETS</a> 
 
<b>by </b><i><a class="txt" href="neckla_2.html#E10:13_1_1_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="neckla_2.html#D3" target="_self">Def3</a></i>;<br/>
<b>hence </b><a NAME="E16:13_1_1_1_1"/>
 <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K3" target="_self">fin_RelStr</a> 
 <b>by </b><i><a class="ref" href="neckla_2.html#D4" target="_self">Def4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:13_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="neckla_2.html#K3" target="_self">fin_RelStr</a> ]
 <b>by </b><i><a class="ref" href="neckla_2.html#D4" target="_self">Def4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:13_1_1"/><b>then </b><i><font color="Green">E10</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="neckla_2.html#E2:13_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="neckla_2.html#E6:13_1_1"><i><font color="Green">E9</font></i></a></i>;<br/>
<a NAME="E9:13_1_1"/><b>then </b><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/>
<a NAME="E10:13_1_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>] &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <a href="neckla_2.html#K3" target="_self">fin_RelStr</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:13_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:13_1_1_2_1"/><i><font color="Green">E12</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st not the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> is <a href="realset1.html#V1">trivial</a> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K13">FinSETS</a>  holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be  <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:13_1_1_2_1_1"/><i><font color="Green">E13</font></i>: 
( not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> is <a href="realset1.html#V1">trivial</a> )
 <b>and </b><br/><a NAME="E2:13_1_1_2_1_1"/><i><font color="Green">E14</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K13">FinSETS</a> 
 ;<br/>

<a NAME="E3:13_1_1_2_1_1"/>
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>2</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="neckla_2.html#E2:13_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="neckla_2.html#E1:13_1_1_2_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="neckla_2.html#E2:13_1_1_2_1_1"><i><font color="Green">E14</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:13_1_1_2_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="neckla_2.html#E8:13_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="neckla_2.html#E9:13_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:13_1_1_2_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/>(  <a href="neckla_2.html#K1" target="_self">union_of</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be  <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:13_1_1_2_1_2"/><i><font color="Green">E13</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E2:13_1_1_2_1_2"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E3:13_1_1_2_1_2"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E4:13_1_1_2_1_2"/><i><font color="Green">E16</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> <font color="Maroon">c<sub>2</sub></font> holds <br/> <a href="neckla_2.html#K1" target="_self">union_of</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_2_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:13_1_1_2_1_2_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:13_1_1_2_1_2_1"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>4</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="neckla_2.html#E9:13_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="neckla_2.html#E2:13_1_1_2_1_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E3:13_1_1_2_1_2_1"/>
<font color="Maroon">c<sub>5</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="neckla_2.html#E9:13_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="neckla_2.html#E3:13_1_1_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="neckla_2.html#E1:13_1_1_2_1_2_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:13_1_1_2_1_2_1"/>
 <a href="neckla_2.html#K1" target="_self">union_of</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</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="neckla_2.html#E2:13_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="neckla_2.html#E1:13_1_1_2_1_2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="neckla_2.html#E1:13_1_1_2_1_2_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="neckla_2.html#E2:13_1_1_2_1_2_1"><i><font color="Green">E18</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:13_1_1_2_1_2"/>
 <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_2_1_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:13_1_1_2_1_2_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>2</sub></font> holds <br/> <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_2_1_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:13_1_1_2_1_2_2_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:13_1_1_2_1_2_2_1"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>4</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="neckla_2.html#E9:13_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="neckla_2.html#E2:13_1_1_2_1_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E3:13_1_1_2_1_2_2_1"/>
<font color="Maroon">c<sub>5</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="neckla_2.html#E9:13_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="neckla_2.html#E3:13_1_1_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="neckla_2.html#E1:13_1_1_2_1_2_2_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:13_1_1_2_1_2_2_1"/>
 <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</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="neckla_2.html#E2:13_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="neckla_2.html#E1:13_1_1_2_1_2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="neckla_2.html#E1:13_1_1_2_1_2_2_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="neckla_2.html#E2:13_1_1_2_1_2_2_1"><i><font color="Green">E18</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:13_1_1_2_1_2_2"/>
 <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="neckla_2.html#E8:13_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="neckla_2.html#E9:13_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:13_1_1_2_1_2"/>
(  <a href="neckla_2.html#K1" target="_self">union_of</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="neckla_2.html#E8:13_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="neckla_2.html#E9:13_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="neckla_2.html#E4:13_1_1_2_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:13_1_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>]
 <b>by </b><i><a class="txt" href="neckla_2.html#E1:13_1_1_2_1"><i><font color="Green">E12</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="neckla_2.html#K3" target="_self">fin_RelStr</a> ;<br/>

<b>assume </b><a NAME="E2:13_1_1_2"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 ;<br/>

<a NAME="E3:13_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</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="E1:13_1_1_2_2"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:13_1_1_2_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="neckla_2.html#E2:13_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="neckla_2.html#E2:13_1_1_2"><i><font color="Green">E12</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:13_1_1_2_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="neckla_2.html#E9:13_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="neckla_2.html#E1:13_1_1_2_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:13_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:13_1_1"/>
( ( for <font color="Olive">b<sub>1</sub></font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st not the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> is <a href="realset1.html#V1">trivial</a> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K13">FinSETS</a>  holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/>(  <a href="neckla_2.html#K1" target="_self">union_of</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <a href="neckla_2.html#K3" target="_self">fin_RelStr</a>  st ( for <font color="Olive">b<sub>2</sub></font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st not the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>2</sub></font> is <a href="realset1.html#V1">trivial</a> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K13">FinSETS</a>  holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>2</sub></font> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/>(  <a href="neckla_2.html#K1" target="_self">union_of</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) ) holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/>


</div>
