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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:15"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>2</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="E3:15"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K13">FinSETS</a>  )
 <b>by </b><i><a class="ref" href="neckla_2.html#D4" target="_self">Def4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  <b>by </b><i><a class="txt" href="neckla_2.html#E3:15"><i><font color="Green">E11</font></i></a></i>;<br/>
<div><i><font color="Green">E12</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="struct_0.html#V3">empty</a>
 ;<br/><b>set </b><font color="Maroon">c<sub>4</sub></font> = <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K1">}</a></span>;<br/><b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a> ;<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K1">}</a></span> as   <a href="subset_1.html#NM2">Subset</a> of <a href="neckla_2.html#K3" target="_self">fin_RelStr</a>  ;<br/><div><i><font color="Green">E14</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be  <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>assume </b><a NAME="E1:15_1_1"/><i><font color="Green">E15</font></i>: 
( not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</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>7</sub></font> is <a href="realset1.html#V1">trivial</a> &amp; 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>  )
 ;<br/><a NAME="E2:15_1_1"/><b>then </b><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a> 
 <b>by </b><i><a class="ref" href="neckla_2.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E3:15_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="neckla_2.html#E1:15_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="neckla_2.html#E1:15_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><a NAME="E4:15_1_1"/><b>then </b>
not <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E5:15_1_1"/>
<font color="Maroon">c<sub>7</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#E2:15_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</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="E1:15_1_2"/><i><font color="Green">E15</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</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>8</sub></font>
 <b>and </b><br/><a NAME="E2:15_1_2"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E3:15_1_2"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E4:15_1_2"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  )
 <b>by </b><i><a class="txt" href="neckla_2.html#E2:15_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="neckla_2.html#E3:15_1_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E5:15_1_2"/><b>then </b><i><font color="Green">E18</font></i>: 
(  <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> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  &amp;  <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  )
 <b>by </b><i><a class="txt" href="neckla_2.html#E1:15_1_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="neckla_2.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E6:15_1_2"/><i><font color="Green">E19</font></i>: 
not <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="neckla_2.html#E2:15_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E7:15_1_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_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="E1:15_1_2_1_1"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> or the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>3</sub></font> )
 </span><b>by </b><i><a class="txt" href="neckla_2.html#E6:15_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:15_1_2_1_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:15_1_2_1_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</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#E1:15_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:15_1_2_1_1_2"/><i><font color="Green">E20</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>9</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>10</sub></font> = the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>7</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>11</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font>;<br/><a NAME="E2:15_1_2_1_1_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="neckla_2.html#E1:15_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><a NAME="E3:15_1_2_1_1_2"/><b>then </b>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>7</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#E1:15_1_2_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="relset_1.html#T26">RELSET_1:26</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:15_1_2_1_1_2"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E6:15_1_2_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <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="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="neckla_2.html#E5:15_1_2_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="relset_1.html#T6">RELSET_1:6</a></i>;<br/><b>hence </b><a NAME="E7:15_1_2_1_1_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><a NAME="E9:15_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E10:15_1_2"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><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></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="neckla_2.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E11:15_1_2"/><b>then </b>
 <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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="neckla_2.html#E1:15_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><a NAME="E12:15_1_2"/><b>then </b>
not  <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> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E13:15_1_2"/>
 <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> <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#E5:15_1_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E14:15_1_2"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="neckla_2.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E15:15_1_2"/><b>then </b>
 <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="neckla_2.html#E1:15_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><a NAME="E16:15_1_2"/><b>then </b>
not  <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E17:15_1_2"/>
 <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</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#E5:15_1_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><a NAME="E5:15_1"/><b>then </b><i><font color="Green">E15</font></i>: 
 <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="neckla_2.html#E3:15_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="neckla_2.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E6:15_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E7:15_1"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_2.html#E1:15"><i><font color="Green">E10</font></i></a>, <a class="txt" href="neckla_2.html#E5:15_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:15"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="txt" href="neckla_2.html#E3:15"><i><font color="Green">E11</font></i></a>, <a class="ref" href="neckla_2.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E7:15"/>
<font color="Maroon">c<sub>1</sub></font> is  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V6">finite</a> <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a> 
 <b>by </b><i><a class="txt" href="neckla_2.html#E5:15"><i><font color="Green">E12</font></i></a>, <a class="ref" href="group_1.html#D14">GROUP_1:def 14</a></i>;<br/>


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