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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 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="necklace.html#V1">symmetric</a> <a href="necklace.html#V3">irreflexive</a>  <a href="orders_2.html#L1">RelStr</a>  st <font color="Olive">b<sub>1</sub></font> is <a href="neckla_2.html#V1">N-free</a> &amp;  <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &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/> <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Olive">b<sub>1</sub></font> #) <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a> ;<br/>
<a NAME="E1:62"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1"><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="E1:62_1"/><i><font color="Green">E38</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="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be  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="necklace.html#V1">symmetric</a> <a href="necklace.html#V3">irreflexive</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>assume </b><a NAME="E2:62_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="neckla_2.html#V1">N-free</a> &amp;  <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</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>  )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:62_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="realset2.html#V3">trivial</a> or ( not <font color="Maroon">c<sub>2</sub></font> is <a href="neckla_3.html#V1" target="_self">path-connected</a> &amp; not <font color="Maroon">c<sub>2</sub></font> is <a href="realset2.html#V3">trivial</a> ) or ( not  <a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>2</sub></font> is <a href="neckla_3.html#V1" target="_self">path-connected</a> &amp; not <font color="Maroon">c<sub>2</sub></font> is <a href="realset2.html#V3">trivial</a> ) or ( not <font color="Maroon">c<sub>2</sub></font> is <a href="realset2.html#V3">trivial</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="neckla_3.html#V1" target="_self">path-connected</a> &amp;  <a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>2</sub></font> is <a href="neckla_3.html#V1" target="_self">path-connected</a> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_1"><b>suppose </b></a><a NAME="E1:62_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="realset2.html#V3">trivial</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  non <a href="struct_0.html#V3">empty</a> <a href="realset2.html#V3">trivial</a>  <a href="orders_2.html#L1">RelStr</a>  ;<br/><a NAME="E3:62_1_1_1"/>
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>4</sub></font> #) <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a> 
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:62_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="neckla_2.html#D5">NECKLA_2:def 5</a></i>;<br/><b>hence </b><a NAME="E4:62_1_1_1"/>
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> #) <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_2"><b>suppose </b></a><a NAME="E1:62_1_1_2"/>
( not <font color="Maroon">c<sub>2</sub></font> is <a href="neckla_3.html#V1" target="_self">path-connected</a> &amp; not <font color="Maroon">c<sub>2</sub></font> is <a href="realset2.html#V3">trivial</a> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> being  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V1">strict</a> <a href="necklace.html#V1">symmetric</a> <a href="necklace.html#V3">irreflexive</a>  <a href="orders_2.html#L1">RelStr</a> <b> such that </b><br/><a NAME="E3:62_1_1_2"/><i><font color="Green">E40</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.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="E4:62_1_1_2"/><i><font color="Green">E41</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1">union_of</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="neckla_3.html#T31" target="_self">Th31</a></i>;<br/><b>set </b><font color="Maroon">c<sub>6</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>7</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> #);<br/><b>set </b><font color="Maroon">c<sub>9</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> #);<br/><a NAME="E5:62_1_1_2"/><i><font color="Green">E42</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> #) <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E4:62_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="neckla_2.html#D2">NECKLA_2:def 2</a></i>;<br/><a NAME="E6:62_1_1_2"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> #)
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:62_1_1_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="neckla_3.html#E36"><i><font color="Green">Lemma31</font></i></a></i>;<br/><a NAME="E7:62_1_1_2"/><b>then </b><i><font color="Green">E44</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="card_2.html#T68">CARD_2:68</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</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="neckla_3.html#E6:62_1_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="card_2.html#T68">CARD_2:68</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>4</sub></font> as  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="necklace.html#V1">symmetric</a> <a href="necklace.html#V3">irreflexive</a>  <a href="orders_2.html#L1">RelStr</a>  <b>by </b><i><a class="txt" href="neckla_3.html#E7:62_1_1_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="group_1.html#D14">GROUP_1:def 14</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> #) as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/><a NAME="E11:62_1_1_2"/><i><font color="Green">E45</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E6:62_1_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="card_2.html#T58">CARD_2:58</a></i>;<br/><a NAME="E12:62_1_1_2"/>
<font color="Maroon">c<sub>11</sub></font> is  <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:62_1_1_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="neckla_3.html#E4:62_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="neckla_3.html#T10" target="_self">Th10</a></i>;<br/><a NAME="E13:62_1_1_2"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>11</sub></font> is <a href="neckla_2.html#V1">N-free</a>
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:62_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="neckla_3.html#T23" target="_self">Th23</a></i>;<br/><a NAME="E14:62_1_1_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K13">FinSETS</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:62_1_1_2_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E5:62_1_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<b>hence </b><a NAME="E2:62_1_1_2_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>11</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_3.html#E2:62_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="classes1.html#T6">CLASSES1:6</a>, <a class="ref" href="classes2.html#D2">CLASSES2:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E15:62_1_1_2"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a> 
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:62_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="neckla_3.html#E2:62_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="neckla_3.html#E11:62_1_1_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="neckla_3.html#E13:62_1_1_2"><i><font color="Green">E46</font></i></a></i>;<br/><a NAME="E16:62_1_1_2"/><i><font color="Green">E48</font></i>: 
 <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:62_1_1_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="neckla_3.html#E5:62_1_1_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="neckla_3.html#E36"><i><font color="Green">Lemma31</font></i></a></i>;<br/><a NAME="E17:62_1_1_2"/><b>then </b><i><font color="Green">E49</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="card_2.html#T68">CARD_2:68</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</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="neckla_3.html#E16:62_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="card_2.html#T68">CARD_2:68</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>5</sub></font> as  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="necklace.html#V1">symmetric</a> <a href="necklace.html#V3">irreflexive</a>  <a href="orders_2.html#L1">RelStr</a>  <b>by </b><i><a class="txt" href="neckla_3.html#E17:62_1_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="group_1.html#D14">GROUP_1:def 14</a></i>;<br/><a NAME="E20:62_1_1_2"/><i><font color="Green">E50</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E16:62_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="card_2.html#T58">CARD_2:58</a></i>;<br/><a NAME="E21:62_1_1_2"/>
<font color="Maroon">c<sub>14</sub></font> is  <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:62_1_1_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="neckla_3.html#E4:62_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="neckla_3.html#T10" target="_self">Th10</a></i>;<br/><a NAME="E22:62_1_1_2"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>14</sub></font> is <a href="neckla_2.html#V1">N-free</a>
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:62_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="neckla_3.html#T23" target="_self">Th23</a></i>;<br/><a NAME="E23:62_1_1_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K13">FinSETS</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:62_1_1_2_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E5:62_1_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<b>hence </b><a NAME="E2:62_1_1_2_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>14</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_3.html#E2:62_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="classes1.html#T6">CLASSES1:6</a>, <a class="ref" href="classes2.html#D2">CLASSES2:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E24:62_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a> 
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:62_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="neckla_3.html#E2:62_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="neckla_3.html#E20:62_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="neckla_3.html#E22:62_1_1_2"><i><font color="Green">E51</font></i></a></i>;<br/><b>hence </b><a NAME="E25:62_1_1_2"/>
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> #) <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a> 
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:62_1_1_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="neckla_3.html#E4:62_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="neckla_3.html#E15:62_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="neckla_2.html#D5">NECKLA_2:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_3"><b>suppose </b></a><a NAME="E1:62_1_1_3"/>
( not  <a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>2</sub></font> is <a href="neckla_3.html#V1" target="_self">path-connected</a> &amp; not <font color="Maroon">c<sub>2</sub></font> is <a href="realset2.html#V3">trivial</a> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> being  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V1">strict</a> <a href="necklace.html#V1">symmetric</a> <a href="necklace.html#V3">irreflexive</a>  <a href="orders_2.html#L1">RelStr</a> <b> such that </b><br/><a NAME="E3:62_1_1_3"/><i><font color="Green">E40</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.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="E4:62_1_1_3"/><i><font color="Green">E41</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="neckla_3.html#T32" target="_self">Th32</a></i>;<br/><b>set </b><font color="Maroon">c<sub>6</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>7</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> #);<br/><b>set </b><font color="Maroon">c<sub>9</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</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>4</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>11</sub></font> = the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>5</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/><b>set </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/><b>set </b><font color="Maroon">c<sub>14</sub></font> = the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E5:62_1_1_3"/><i><font color="Green">E42</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> #) <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E4:62_1_1_3"><i><font color="Green">E41</font></i></a>, <a class="ref" href="neckla_2.html#D3">NECKLA_2:def 3</a></i>;<br/><a NAME="E6:62_1_1_3"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> #)
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:62_1_1_3"><i><font color="Green">E40</font></i></a>, <a class="txt" href="neckla_3.html#E36"><i><font color="Green">Lemma31</font></i></a></i>;<br/><a NAME="E7:62_1_1_3"/><b>then </b><i><font color="Green">E44</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="card_2.html#T68">CARD_2:68</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</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="neckla_3.html#E6:62_1_1_3"><i><font color="Green">E43</font></i></a>, <a class="ref" href="card_2.html#T68">CARD_2:68</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>4</sub></font> as  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="necklace.html#V1">symmetric</a> <a href="necklace.html#V3">irreflexive</a>  <a href="orders_2.html#L1">RelStr</a>  <b>by </b><i><a class="txt" href="neckla_3.html#E7:62_1_1_3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="group_1.html#D14">GROUP_1:def 14</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> #) as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/><a NAME="E11:62_1_1_3"/><i><font color="Green">E45</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E6:62_1_1_3"><i><font color="Green">E43</font></i></a>, <a class="ref" href="card_2.html#T58">CARD_2:58</a></i>;<br/><a NAME="E12:62_1_1_3"/>
<font color="Maroon">c<sub>16</sub></font> is  <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:62_1_1_3"><i><font color="Green">E40</font></i></a>, <a class="txt" href="neckla_3.html#E4:62_1_1_3"><i><font color="Green">E41</font></i></a>, <a class="ref" href="neckla_3.html#T10" target="_self">Th10</a></i>;<br/><a NAME="E13:62_1_1_3"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>16</sub></font> is <a href="neckla_2.html#V1">N-free</a>
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:62_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="neckla_3.html#T23" target="_self">Th23</a></i>;<br/><a NAME="E14:62_1_1_3"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K13">FinSETS</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:62_1_1_3_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E5:62_1_1_3"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<b>hence </b><a NAME="E2:62_1_1_3_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>16</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_3.html#E2:62_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="classes1.html#T6">CLASSES1:6</a>, <a class="ref" href="classes2.html#D2">CLASSES2:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E15:62_1_1_3"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a> 
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:62_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="neckla_3.html#E2:62_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="neckla_3.html#E11:62_1_1_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="neckla_3.html#E13:62_1_1_3"><i><font color="Green">E46</font></i></a></i>;<br/><a NAME="E16:62_1_1_3"/><i><font color="Green">E48</font></i>: 
 <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:62_1_1_3"><i><font color="Green">E40</font></i></a>, <a class="txt" href="neckla_3.html#E5:62_1_1_3"><i><font color="Green">E42</font></i></a>, <a class="txt" href="neckla_3.html#E36"><i><font color="Green">Lemma31</font></i></a></i>;<br/><a NAME="E17:62_1_1_3"/><b>then </b><i><font color="Green">E49</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="card_2.html#T68">CARD_2:68</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</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="neckla_3.html#E16:62_1_1_3"><i><font color="Green">E48</font></i></a>, <a class="ref" href="card_2.html#T68">CARD_2:68</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>5</sub></font> as  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="necklace.html#V1">symmetric</a> <a href="necklace.html#V3">irreflexive</a>  <a href="orders_2.html#L1">RelStr</a>  <b>by </b><i><a class="txt" href="neckla_3.html#E17:62_1_1_3"><i><font color="Green">E49</font></i></a>, <a class="ref" href="group_1.html#D14">GROUP_1:def 14</a></i>;<br/><a NAME="E20:62_1_1_3"/><i><font color="Green">E50</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E16:62_1_1_3"><i><font color="Green">E48</font></i></a>, <a class="ref" href="card_2.html#T58">CARD_2:58</a></i>;<br/><a NAME="E21:62_1_1_3"/>
<font color="Maroon">c<sub>19</sub></font> is  <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:62_1_1_3"><i><font color="Green">E40</font></i></a>, <a class="txt" href="neckla_3.html#E4:62_1_1_3"><i><font color="Green">E41</font></i></a>, <a class="ref" href="neckla_3.html#T10" target="_self">Th10</a></i>;<br/><a NAME="E22:62_1_1_3"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>19</sub></font> is <a href="neckla_2.html#V1">N-free</a>
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:62_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="neckla_3.html#T23" target="_self">Th23</a></i>;<br/><a NAME="E23:62_1_1_3"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K13">FinSETS</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_3_2"><b>proof </b></a><div class="add">

<a NAME="E1:62_1_1_3_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E5:62_1_1_3"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<b>hence </b><a NAME="E2:62_1_1_3_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>19</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_3.html#E2:62_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="classes1.html#T6">CLASSES1:6</a>, <a class="ref" href="classes2.html#D2">CLASSES2:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E24:62_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a> 
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:62_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="neckla_3.html#E2:62_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="neckla_3.html#E20:62_1_1_3"><i><font color="Green">E50</font></i></a>, <a class="txt" href="neckla_3.html#E22:62_1_1_3"><i><font color="Green">E51</font></i></a></i>;<br/><b>hence </b><a NAME="E25:62_1_1_3"/>
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> #) <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a> 
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:62_1_1_3"><i><font color="Green">E40</font></i></a>, <a class="txt" href="neckla_3.html#E4:62_1_1_3"><i><font color="Green">E41</font></i></a>, <a class="txt" href="neckla_3.html#E15:62_1_1_3"><i><font color="Green">E47</font></i></a>, <a class="ref" href="neckla_2.html#D5">NECKLA_2:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_4"><b>suppose </b></a><a NAME="E1:62_1_1_4"/><i><font color="Green">E40</font></i>: 
( not <font color="Maroon">c<sub>2</sub></font> is <a href="realset2.html#V3">trivial</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="neckla_3.html#V1" target="_self">path-connected</a> &amp;  <a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>2</sub></font> is <a href="neckla_3.html#V1" target="_self">path-connected</a> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:62_1_1_4"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="neckla_3.html#E3:62_1_1_4"><i><font color="Green">E41</font></i></a></i>;<br/><b>set </b><font color="Maroon">c<sub>6</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span>;<br/><b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/><a NAME="E6:62_1_1_4"/><i><font color="Green">E42</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>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>2</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/><b>set </b><font color="Maroon">c<sub>10</sub></font> =  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>9</sub></font>;<br/><a NAME="E8:62_1_1_4"/>
not  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>9</sub></font> is <a href="struct_0.html#V3">empty</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:62_1_1_4_1"/>
 <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>9</sub></font> is <a href="struct_0.html#V3">empty</a>
 ;<br/>

<a NAME="E2:62_1_1_4_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<a NAME="E3:62_1_1_4_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<a NAME="E4:62_1_1_4_1"/><b>then </b><i><font color="Green">E43</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T66">ZFMISC_1:66</a></i>;<br/>
<a NAME="E5:62_1_1_4_1"/>
not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> is <a href="realset1.html#V1">trivial</a>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E1:62_1_1_4"><i><font color="Green">E40</font></i></a>, <a class="ref" href="realset2.html#D5">REALSET2:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:62_1_1_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E4:62_1_1_4_1"><i><font color="Green">E43</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> =  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>9</sub></font> as  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V6">finite</a> <a href="necklace.html#V1">symmetric</a> <a href="necklace.html#V3">irreflexive</a>  <a href="orders_2.html#L1">RelStr</a>  ;<br/><a NAME="E10:62_1_1_4"/><i><font color="Green">E43</font></i>: 
 <a href="card_1.html#K4">card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_4_2"><b>proof </b></a><div class="add">

<a NAME="E1:62_1_1_4_2"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="card_2.html#T63">CARD_2:63</a></i>;<br/>
<a NAME="E2:62_1_1_4_2"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="txt" href="neckla_3.html#E2:62_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="card_2.html#T60">CARD_2:60</a></i>;<br/>
<a NAME="E3:62_1_1_4_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<b>hence </b><a NAME="E4:62_1_1_4_2"/>
 <a href="card_1.html#K4">card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:62_1_1_4_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>


</div><b>end;</b></div><a NAME="E11:62_1_1_4"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>11</sub></font> is <a href="neckla_2.html#V1">N-free</a>
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:62_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="neckla_3.html#T23" target="_self">Th23</a></i>;<br/><a NAME="E12:62_1_1_4"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K13">FinSETS</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_4_3"><b>proof </b></a><div class="add">

<a NAME="E1:62_1_1_4_3"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E6:62_1_1_4"><i><font color="Green">E42</font></i></a>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<b>hence </b><a NAME="E2:62_1_1_4_3"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>11</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_3.html#E2:62_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="classes1.html#T6">CLASSES1:6</a>, <a class="ref" href="classes2.html#D2">CLASSES2:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E13:62_1_1_4"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a> 
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:62_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="neckla_3.html#E10:62_1_1_4"><i><font color="Green">E43</font></i></a>, <a class="txt" href="neckla_3.html#E11:62_1_1_4"><i><font color="Green">E44</font></i></a></i>;<br/><b>thus </b><a NAME="E14:62_1_1_4"/>
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> #) <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_4_4"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_4_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:62_1_1_4_4_1"/>
( <font color="Maroon">c<sub>11</sub></font> is  <a href="realset2.html#V3">trivial</a>  <a href="orders_2.html#L1">RelStr</a>  or ex <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 <br/>( 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#K4">fin_RelStr_sp</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a>  &amp; ( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1">union_of</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> or <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K2">sum_of</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> ) ) )
 </span><b>by </b><i><a class="txt" href="neckla_3.html#E13:62_1_1_4"><i><font color="Green">E45</font></i></a>, <a class="ref" href="neckla_2.html#T6">NECKLA_2:6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_4_4_1_1"><b>suppose </b></a><a NAME="E1:62_1_1_4_4_1_1"/>
<font color="Maroon">c<sub>11</sub></font> is  <a href="realset2.html#V3">trivial</a>  <a href="orders_2.html#L1">RelStr</a> 
 ;<br/><div class="add"><a NAME="E2:62_1_1_4_4_1_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>11</sub></font> is <a href="realset1.html#V1">trivial</a>
 <b>by </b><i><a class="ref" href="realset2.html#D5">REALSET2:def 5</a></i>;<br/><a NAME="E3:62_1_1_4_4_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>9</sub></font> is <a href="realset1.html#V1">trivial</a>
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E4:62_1_1_4_4_1_1"/>
not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>11</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><a NAME="E5:62_1_1_4_4_1_1"/><b>then </b>
not <font color="Maroon">c<sub>9</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</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="E7:62_1_1_4_4_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:62_1_1_4_4_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="realset1.html#D4">REALSET1:def 4</a></i>;<br/><a NAME="E8:62_1_1_4_4_1_1"/><i><font color="Green">E48</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E7:62_1_1_4_4_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>;<br/><a NAME="E9:62_1_1_4_4_1_1"/><i><font color="Green">E49</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_4_4_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:62_1_1_4_4_1_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_4_4_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:62_1_1_4_4_1_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E2:62_1_1_4_4_1_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:62_1_1_4_4_1_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


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

<b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:62_1_1_4_4_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>hence </b><a NAME="E3:62_1_1_4_4_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:62_1_1_4_4_1_1"/><b>then </b><i><font color="Green">E50</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E8:62_1_1_4_4_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="enumset1.html#T41">ENUMSET1:41</a></i>;<br/><a NAME="E11:62_1_1_4_4_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:62_1_1_4"><i><font color="Green">E40</font></i></a>, <a class="txt" href="neckla_3.html#E8:62_1_1_4_4_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="neckla_3.html#E9:62_1_1_4_4_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="realset2.html#D5">REALSET2:def 5</a></i>;<br/><a NAME="E12:62_1_1_4_4_1_1"/><b>then </b>
 <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 2
 <b>by </b><i><a class="txt" href="neckla_3.html#E10:62_1_1_4_4_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="card_2.html#T76">CARD_2:76</a></i>;<br/><b>hence </b><a NAME="E13:62_1_1_4_4_1_1"/>
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> #) <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a> 
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:62_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="neckla_3.html#T34" target="_self">Th34</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_4_4_1_2"><b>suppose </b></a><a NAME="E1:62_1_1_4_4_1_2"/>
ex <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 <br/>( 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#K4">fin_RelStr_sp</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a>  &amp; ( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1">union_of</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> or <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K2">sum_of</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> ) )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</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="E3:62_1_1_4_4_1_2"/><i><font color="Green">E46</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>12</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>13</sub></font>
 <b>and </b><br/><a NAME="E4:62_1_1_4_4_1_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a>  &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a>  &amp; ( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1">union_of</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> or <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> ) )
 ;<br/><b>thus </b><a NAME="E5:62_1_1_4_4_1_2"/>
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> #) <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_4_4_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_4_4_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="E1:62_1_1_4_4_1_2_1_1"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1">union_of</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> or <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> )
 </span><b>by </b><i><a class="txt" href="neckla_3.html#E4:62_1_1_4_4_1_2"><i><font color="Green">E47</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_4_4_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:62_1_1_4_4_1_2_1_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1">union_of</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 ;<br/><div class="add"><a NAME="E2:62_1_1_4_4_1_2_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> is    <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:62_1_1_4_4_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="neckla_3.html#E4:62_1_1_4_4_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="neckla_3.html#T10" target="_self">Th10</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>12</sub></font> as  non <a href="struct_0.html#V3">empty</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="neckla_3.html#E4:62_1_1_4_4_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="neckla_2.html#T4">NECKLA_2:4</a>, <a class="ref" href="yellow_6.html#T16">YELLOW_6:16</a></i>;<br/><a NAME="E4:62_1_1_4_4_1_2_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> is    <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:62_1_1_4_4_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="neckla_3.html#E4:62_1_1_4_4_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="neckla_3.html#T10" target="_self">Th10</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>13</sub></font> as  non <a href="struct_0.html#V3">empty</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="neckla_3.html#E4:62_1_1_4_4_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="neckla_2.html#T4">NECKLA_2:4</a>, <a class="ref" href="yellow_6.html#T16">YELLOW_6:16</a></i>;<br/><a NAME="E6:62_1_1_4_4_1_2_1_1_1"/>
 <a href="yellow_0.html#K5">subrelstr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1">union_of</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:62_1_1_4_4_1_2_1_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="pre_topc.html#D3">PRE_TOPC:def 3</a></i>;<br/><a NAME="E7:62_1_1_4_4_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="necklace.html#R2">embeds</a>  <a href="necklace.html#K4">Necklace</a> 4
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:62_1_1_4"><i><font color="Green">E40</font></i></a>, <a class="txt" href="neckla_3.html#E3:62_1_1_4_4_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="neckla_3.html#T40" target="_self">Th40</a></i>;<br/><b>hence </b><a NAME="E8:62_1_1_4_4_1_2_1_1_1"/>
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> #) <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a> 
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:62_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="neckla_2.html#D1">NECKLA_2:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_4_4_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:62_1_1_4_4_1_2_1_1_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 ;<br/><div class="add"><a NAME="E2:62_1_1_4_4_1_2_1_1_2"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1">union_of</a> <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:62_1_1_4_4_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="neckla_3.html#T18" target="_self">Th18</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>14</sub></font> =  <a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>2</sub></font> as  non <a href="struct_0.html#V3">empty</a> <a href="necklace.html#V1">symmetric</a> <a href="necklace.html#V3">irreflexive</a>  <a href="orders_2.html#L1">RelStr</a>  ;<br/><a NAME="E4:62_1_1_4_4_1_2_1_1_2"/><i><font color="Green">E49</font></i>: 
not  <a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>12</sub></font> is <a href="struct_0.html#V3">empty</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_4_4_1_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:62_1_1_4_4_1_2_1_1_2_1"/>
 <a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>12</sub></font> is <a href="struct_0.html#V3">empty</a>
 ;<br/>

<a NAME="E2:62_1_1_4_4_1_2_1_1_2_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<a NAME="E3:62_1_1_4_4_1_2_1_1_2_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>12</sub></font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="necklace.html#D9">NECKLACE:def 9</a></i>;<br/>
<a NAME="E4:62_1_1_4_4_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> is <a href="struct_0.html#V3">empty</a>
 
<b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:62_1_1_4_4_1_2_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E4:62_1_1_4_4_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="neckla_2.html#T4">NECKLA_2:4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:62_1_1_4_4_1_2_1_1_2"/><i><font color="Green">E50</font></i>: 
not  <a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>13</sub></font> is <a href="struct_0.html#V3">empty</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_4_4_1_2_1_1_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:62_1_1_4_4_1_2_1_1_2_2"/>
 <a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>13</sub></font> is <a href="struct_0.html#V3">empty</a>
 ;<br/>

<a NAME="E2:62_1_1_4_4_1_2_1_1_2_2"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<a NAME="E3:62_1_1_4_4_1_2_1_1_2_2"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>13</sub></font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="necklace.html#D9">NECKLACE:def 9</a></i>;<br/>
<a NAME="E4:62_1_1_4_4_1_2_1_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> is <a href="struct_0.html#V3">empty</a>
 
<b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:62_1_1_4_4_1_2_1_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E4:62_1_1_4_4_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="neckla_2.html#T4">NECKLA_2:4</a></i>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> =  <a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>12</sub></font> as  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a>  <b>by </b><i><a class="txt" href="neckla_3.html#E4:62_1_1_4_4_1_2_1_1_2"><i><font color="Green">E49</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>16</sub></font> =  <a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>13</sub></font> as  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a>  <b>by </b><i><a class="txt" href="neckla_3.html#E5:62_1_1_4_4_1_2_1_1_2"><i><font color="Green">E50</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>14</sub></font> <b>by </b><i><a class="ref" href="necklace.html#D9">NECKLACE:def 9</a></i>;<br/><a NAME="E9:62_1_1_4_4_1_2_1_1_2"/><i><font color="Green">E51</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>12</sub></font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="ref" href="necklace.html#D9">NECKLACE:def 9</a></i>;<br/><a NAME="E10:62_1_1_4_4_1_2_1_1_2"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> is <a href="neckla_3.html#V1" target="_self">path-connected</a> &amp;  <a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>14</sub></font> is <a href="neckla_3.html#V1" target="_self">path-connected</a> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:62_1_1_4"><i><font color="Green">E40</font></i></a>, <a class="ref" href="neckla_3.html#T16" target="_self">Th16</a></i>;<br/><a NAME="E11:62_1_1_4_4_1_2_1_1_2"/><i><font color="Green">E53</font></i>: 
not <font color="Maroon">c<sub>14</sub></font> is <a href="realset2.html#V3">trivial</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_4_4_1_2_1_1_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:62_1_1_4_4_1_2_1_1_2_3"/>
<font color="Maroon">c<sub>14</sub></font> is <a href="realset2.html#V3">trivial</a>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>18</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:62_1_1_4_4_1_2_1_1_2_3"/><i><font color="Green">E54</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>18</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="group_6.html#D2">GROUP_6:def 2</a></i>;<br/>
<a NAME="E4:62_1_1_4_4_1_2_1_1_2_3"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>18</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E3:62_1_1_4_4_1_2_1_1_2_3"><i><font color="Green">E54</font></i></a>, <a class="ref" href="necklace.html#D9">NECKLACE:def 9</a></i>;<br/>
<b>hence </b><a NAME="E5:62_1_1_4_4_1_2_1_1_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:62_1_1_4"><i><font color="Green">E40</font></i></a>, <a class="ref" href="group_6.html#D2">GROUP_6:def 2</a></i>;<br/>


</div><b>end;</b></div> <a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>11</sub></font> = 
 <a href="necklace.html#K3">ComplRelStr</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D3">PRE_TOPC:def 3</a></i>
<br/>.= 
 <a href="yellow_0.html#K5">subrelstr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>17</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="neckla_3.html#T20" target="_self">Th20</a></i>
;<br/><a NAME="E13:62_1_1_4_4_1_2_1_1_2"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="necklace.html#R2">embeds</a>  <a href="necklace.html#K4">Necklace</a> 4
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:62_1_1_4_4_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="neckla_3.html#E2:62_1_1_4_4_1_2_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="neckla_3.html#E9:62_1_1_4_4_1_2_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="neckla_3.html#E10:62_1_1_4_4_1_2_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="neckla_3.html#E11:62_1_1_4_4_1_2_1_1_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="neckla_3.html#T40" target="_self">Th40</a></i>;<br/><a NAME="E14:62_1_1_4_4_1_2_1_1_2"/>
<font color="Maroon">c<sub>14</sub></font> is <a href="neckla_2.html#V1">N-free</a>
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:62_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="neckla_3.html#T25" target="_self">Th25</a></i>;<br/><b>hence </b><a NAME="E15:62_1_1_4_4_1_2_1_1_2"/>
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> #) <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a> 
 <b>by </b><i><a class="txt" href="neckla_3.html#E13:62_1_1_4_4_1_2_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="neckla_2.html#D1">NECKLA_2:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

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

</div><b>end;</b></div>
<a NAME="E2:62"/><i><font color="Green">E38</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#S4">NAT_1:sch 4</a>(<a class="txt" href="neckla_3.html#E1:62"><i><font color="Green">E37</font></i></a>);<br/></i>
<b>let </b><font color="Maroon">c<sub>1</sub></font> be  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="necklace.html#V1">symmetric</a> <a href="necklace.html#V3">irreflexive</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>assume </b><a NAME="E3:62"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="neckla_2.html#V1">N-free</a>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:62"/><i><font color="Green">E40</font></i>: 
 <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="card_4.html#T4">CARD_4:4</a></i>;<br/>
<a NAME="E6:62"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="card_1.html#T66">CARD_1:66</a></i>;<br/>
<b>hence </b><a NAME="E7:62"/>
(  not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K13">FinSETS</a>  or  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> #) <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a>  )
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:62"><i><font color="Green">E38</font></i></a>, <a class="txt" href="neckla_3.html#E3:62"><i><font color="Green">E39</font></i></a>, <a class="txt" href="neckla_3.html#E5:62"><i><font color="Green">E40</font></i></a></i>;<br/>


</div>
