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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>for <font color="Olive" title="b1">G</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a> <a href="necklace.html#V3" title="NECKLACE:attr.3">irreflexive</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>   st <font color="Olive" title="b1">G</font> is <a href="neckla_2.html#V1" title="NECKLA_2:attr.1">N-free</a> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes2.html#K13" title="CLASSES2:func.13">FinSETS</a>  holds <br/> <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">G</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Olive" title="b1">G</font> #) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a> ;<br/>
<a NAME="E1:62"/><i><font color="Green" title="E36">A1</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st ( for <font color="Olive" title="b2">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">n</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">k</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</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" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:62_1"/><i><font color="Green" title="E37">A2</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 ;<br/>

<b>let </b><font color="Maroon" title="c2">G</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a> <a href="necklace.html#V3" title="NECKLACE:attr.3">irreflexive</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/>

<b>set </b><font color="Maroon" title="c3">CG</font> =  <a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c2">G</font>;<br/>
<b>assume </b><a NAME="E2:62_1"/><i><font color="Green" title="E38">A3</font></i>: 
( <font color="Maroon" title="c2">G</font> is <a href="neckla_2.html#V1" title="NECKLA_2:attr.1">N-free</a> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes2.html#K13" title="CLASSES2:func.13">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" title="c2">G</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> or ( not <font color="Maroon" title="c2">G</font> is <a href="neckla_3.html#V1" title="NECKLA_3:attr.1">path-connected</a> &amp; not <font color="Maroon" title="c2">G</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> ) or ( not  <a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c2">G</font> is <a href="neckla_3.html#V1" title="NECKLA_3:attr.1">path-connected</a> &amp; not <font color="Maroon" title="c2">G</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> ) or ( not <font color="Maroon" title="c2">G</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> &amp; <font color="Maroon" title="c2">G</font> is <a href="neckla_3.html#V1" title="NECKLA_3:attr.1">path-connected</a> &amp;  <a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c2">G</font> is <a href="neckla_3.html#V1" title="NECKLA_3:attr.1">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" title="c2">G</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c4">G</font> = <font color="Maroon" title="c2">G</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ;<br/><a NAME="E3:62_1_1_1"/>
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">G</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">G</font> #) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a> 
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:62_1"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="neckla_2.html#D5" title="NECKLA_2:def.5">NECKLA_2:def 5</a></i>;<br/><b>hence </b><a NAME="E4:62_1_1_1"/>
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">G</font> #) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">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" title="c2">G</font> is <a href="neckla_3.html#V1" title="NECKLA_3:attr.1">path-connected</a> &amp; not <font color="Maroon" title="c2">G</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c4">G1</font>, <font color="Maroon" title="c5">G2</font> being  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a> <a href="necklace.html#V3" title="NECKLACE:attr.3">irreflexive</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> <b> such that </b><br/><a NAME="E3:62_1_1_2"/><i><font color="Green" title="E39">A4</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">G1</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">G2</font>
 <b>and </b><br/><a NAME="E4:62_1_1_2"/><i><font color="Green" title="E40">A5</font></i>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">G</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Maroon" title="c4">G1</font>,<font color="Maroon" title="c5">G2</font>
 <b>by </b><i><a class="ref" href="neckla_3.html#T31" target="_self" title="NECKLA_3:th.31">Th31</a></i>;<br/><b>set </b><font color="Maroon" title="c6">cG1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">G1</font>;<br/><b>set </b><font color="Maroon" title="c7">cG2</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">G2</font>;<br/><b>set </b><font color="Maroon" title="c8">R</font> =  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">G</font> #);<br/><b>set </b><font color="Maroon" title="c9">cR</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">G</font> #);<br/><a NAME="E5:62_1_1_2"/><i><font color="Green" title="E41">A6</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">G</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">G1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">G2</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E4:62_1_1_2"><i><font color="Green" title="E40">A5</font></i></a>, <a class="ref" href="neckla_2.html#D2" title="NECKLA_2:def.2">NECKLA_2:def 2</a></i>;<br/><a NAME="E6:62_1_1_2"/><b>then </b><i><font color="Green" title="E42">A7</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">G1</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">G</font> #)
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:62_1_1_2"><i><font color="Green" title="E39">A4</font></i></a>, <a class="txt" href="neckla_3.html#E36"><i><font color="Green" title="E30">Lm1</font></i></a></i>;<br/><a NAME="E7:62_1_1_2"/><b>then </b><i><font color="Green" title="E43">A8</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">G1</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="ref" href="card_2.html#T68" title="CARD_2:th.68">CARD_2:68</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">cG1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">G1</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="neckla_3.html#E6:62_1_1_2"><i><font color="Green" title="E42">A7</font></i></a>, <a class="ref" href="card_2.html#T68" title="CARD_2:th.68">CARD_2:68</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">G1</font> = <font color="Maroon" title="c4">G1</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a> <a href="necklace.html#V3" title="NECKLACE:attr.3">irreflexive</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  <b>by </b><i><a class="txt" href="neckla_3.html#E7:62_1_1_2"><i><font color="Green" title="E43">A8</font></i></a>, <a class="ref" href="struct_0.html#D11" title="STRUCT_0:def.11">STRUCT_0:def 11</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">cR</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">G</font> #) as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><a NAME="E11:62_1_1_2"/><i><font color="Green" title="E44">A9</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">cG1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">cR</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E6:62_1_1_2"><i><font color="Green" title="E42">A7</font></i></a>, <a class="ref" href="card_2.html#T58" title="CARD_2:th.58">CARD_2:58</a></i>;<br/><a NAME="E12:62_1_1_2"/>
<font color="Maroon" title="c11">G1</font> is  <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:62_1_1_2"><i><font color="Green" title="E39">A4</font></i></a>, <a class="txt" href="neckla_3.html#E4:62_1_1_2"><i><font color="Green" title="E40">A5</font></i></a>, <a class="ref" href="neckla_3.html#T10" target="_self" title="NECKLA_3:th.10">Th10</a></i>;<br/><a NAME="E13:62_1_1_2"/><b>then </b><i><font color="Green" title="E45">A10</font></i>: 
<font color="Maroon" title="c11">G1</font> is <a href="neckla_2.html#V1" title="NECKLA_2:attr.1">N-free</a>
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:62_1"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="neckla_3.html#T23" target="_self" title="NECKLA_3:th.23">Th23</a></i>;<br/><a NAME="E14:62_1_1_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">G1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes2.html#K13" title="CLASSES2:func.13">FinSETS</a> 
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:62_1"><i><font color="Green" title="E38">A3</font></i></a>, <a class="txt" href="neckla_3.html#E5:62_1_1_2"><i><font color="Green" title="E41">A6</font></i></a>, <a class="ref" href="classes1.html#T6" title="CLASSES1:th.6">CLASSES1:6</a>, <a class="ref" href="classes2.html#D2" title="CLASSES2:def.2">CLASSES2:def 2</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E15:62_1_1_2"/><b>then </b><i><font color="Green" title="E46">A11</font></i>: 
<font color="Maroon" title="c11">G1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a> 
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:62_1"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="neckla_3.html#E2:62_1"><i><font color="Green" title="E38">A3</font></i></a>, <a class="txt" href="neckla_3.html#E11:62_1_1_2"><i><font color="Green" title="E44">A9</font></i></a>, <a class="txt" href="neckla_3.html#E13:62_1_1_2"><i><font color="Green" title="E45">A10</font></i></a></i>;<br/><a NAME="E16:62_1_1_2"/><i><font color="Green" title="E47">A12</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">G2</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c12">cR</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:62_1_1_2"><i><font color="Green" title="E39">A4</font></i></a>, <a class="txt" href="neckla_3.html#E5:62_1_1_2"><i><font color="Green" title="E41">A6</font></i></a>, <a class="txt" href="neckla_3.html#E36"><i><font color="Green" title="E30">Lm1</font></i></a></i>;<br/><a NAME="E17:62_1_1_2"/><b>then </b><i><font color="Green" title="E48">A13</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">G2</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="ref" href="card_2.html#T68" title="CARD_2:th.68">CARD_2:68</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">cG2</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">G2</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="neckla_3.html#E16:62_1_1_2"><i><font color="Green" title="E47">A12</font></i></a>, <a class="ref" href="card_2.html#T68" title="CARD_2:th.68">CARD_2:68</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">G2</font> = <font color="Maroon" title="c5">G2</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a> <a href="necklace.html#V3" title="NECKLACE:attr.3">irreflexive</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  <b>by </b><i><a class="txt" href="neckla_3.html#E17:62_1_1_2"><i><font color="Green" title="E48">A13</font></i></a>, <a class="ref" href="struct_0.html#D11" title="STRUCT_0:def.11">STRUCT_0:def 11</a></i>;<br/><a NAME="E20:62_1_1_2"/><i><font color="Green" title="E49">A14</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">cG2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">cR</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E16:62_1_1_2"><i><font color="Green" title="E47">A12</font></i></a>, <a class="ref" href="card_2.html#T58" title="CARD_2:th.58">CARD_2:58</a></i>;<br/><a NAME="E21:62_1_1_2"/>
<font color="Maroon" title="c14">G2</font> is  <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:62_1_1_2"><i><font color="Green" title="E39">A4</font></i></a>, <a class="txt" href="neckla_3.html#E4:62_1_1_2"><i><font color="Green" title="E40">A5</font></i></a>, <a class="ref" href="neckla_3.html#T10" target="_self" title="NECKLA_3:th.10">Th10</a></i>;<br/><a NAME="E22:62_1_1_2"/><b>then </b><i><font color="Green" title="E50">A15</font></i>: 
<font color="Maroon" title="c14">G2</font> is <a href="neckla_2.html#V1" title="NECKLA_2:attr.1">N-free</a>
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:62_1"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="neckla_3.html#T23" target="_self" title="NECKLA_3:th.23">Th23</a></i>;<br/><a NAME="E23:62_1_1_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c14">G2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes2.html#K13" title="CLASSES2:func.13">FinSETS</a> 
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:62_1"><i><font color="Green" title="E38">A3</font></i></a>, <a class="txt" href="neckla_3.html#E5:62_1_1_2"><i><font color="Green" title="E41">A6</font></i></a>, <a class="ref" href="classes1.html#T6" title="CLASSES1:th.6">CLASSES1:6</a>, <a class="ref" href="classes2.html#D2" title="CLASSES2:def.2">CLASSES2:def 2</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E24:62_1_1_2"/><b>then </b>
<font color="Maroon" title="c14">G2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a> 
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:62_1"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="neckla_3.html#E2:62_1"><i><font color="Green" title="E38">A3</font></i></a>, <a class="txt" href="neckla_3.html#E20:62_1_1_2"><i><font color="Green" title="E49">A14</font></i></a>, <a class="txt" href="neckla_3.html#E22:62_1_1_2"><i><font color="Green" title="E50">A15</font></i></a></i>;<br/><b>hence </b><a NAME="E25:62_1_1_2"/>
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">G</font> #) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a> 
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:62_1_1_2"><i><font color="Green" title="E39">A4</font></i></a>, <a class="txt" href="neckla_3.html#E4:62_1_1_2"><i><font color="Green" title="E40">A5</font></i></a>, <a class="txt" href="neckla_3.html#E15:62_1_1_2"><i><font color="Green" title="E46">A11</font></i></a>, <a class="ref" href="neckla_2.html#D5" title="NECKLA_2:def.5">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" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c2">G</font> is <a href="neckla_3.html#V1" title="NECKLA_3:attr.1">path-connected</a> &amp; not <font color="Maroon" title="c2">G</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c4">G1</font>, <font color="Maroon" title="c5">G2</font> being  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a> <a href="necklace.html#V3" title="NECKLACE:attr.3">irreflexive</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> <b> such that </b><br/><a NAME="E3:62_1_1_3"/><i><font color="Green" title="E39">A16</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">G1</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">G2</font>
 <b>and </b><br/><a NAME="E4:62_1_1_3"/><i><font color="Green" title="E40">A17</font></i>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">G</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Maroon" title="c4">G1</font>,<font color="Maroon" title="c5">G2</font>
 <b>by </b><i><a class="ref" href="neckla_3.html#T32" target="_self" title="NECKLA_3:th.32">Th32</a></i>;<br/><b>set </b><font color="Maroon" title="c6">cG1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">G1</font>;<br/><b>set </b><font color="Maroon" title="c7">cG2</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">G2</font>;<br/><b>set </b><font color="Maroon" title="c8">R</font> =  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">G</font> #);<br/><b>set </b><font color="Maroon" title="c9">cR</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">G</font> #);<br/><a NAME="E5:62_1_1_3"/><i><font color="Green" title="E41">A18</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">G</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">G1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">G2</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E4:62_1_1_3"><i><font color="Green" title="E40">A17</font></i></a>, <a class="ref" href="neckla_2.html#D3" title="NECKLA_2:def.3">NECKLA_2:def 3</a></i>;<br/><a NAME="E6:62_1_1_3"/><b>then </b><i><font color="Green" title="E42">A19</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">G1</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">G</font> #)
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:62_1_1_3"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="neckla_3.html#E36"><i><font color="Green" title="E30">Lm1</font></i></a></i>;<br/><a NAME="E7:62_1_1_3"/><b>then </b><i><font color="Green" title="E43">A20</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">G1</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="ref" href="card_2.html#T68" title="CARD_2:th.68">CARD_2:68</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">cG1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">G1</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="neckla_3.html#E6:62_1_1_3"><i><font color="Green" title="E42">A19</font></i></a>, <a class="ref" href="card_2.html#T68" title="CARD_2:th.68">CARD_2:68</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">G1</font> = <font color="Maroon" title="c4">G1</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a> <a href="necklace.html#V3" title="NECKLACE:attr.3">irreflexive</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  <b>by </b><i><a class="txt" href="neckla_3.html#E7:62_1_1_3"><i><font color="Green" title="E43">A20</font></i></a>, <a class="ref" href="struct_0.html#D11" title="STRUCT_0:def.11">STRUCT_0:def 11</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">cR</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">G</font> #) as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><a NAME="E11:62_1_1_3"/><i><font color="Green" title="E44">A21</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">cG1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">cR</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E6:62_1_1_3"><i><font color="Green" title="E42">A19</font></i></a>, <a class="ref" href="card_2.html#T58" title="CARD_2:th.58">CARD_2:58</a></i>;<br/><a NAME="E12:62_1_1_3"/>
<font color="Maroon" title="c11">G1</font> is  <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:62_1_1_3"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="neckla_3.html#E4:62_1_1_3"><i><font color="Green" title="E40">A17</font></i></a>, <a class="ref" href="neckla_3.html#T10" target="_self" title="NECKLA_3:th.10">Th10</a></i>;<br/><a NAME="E13:62_1_1_3"/><b>then </b><i><font color="Green" title="E45">A22</font></i>: 
<font color="Maroon" title="c11">G1</font> is <a href="neckla_2.html#V1" title="NECKLA_2:attr.1">N-free</a>
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:62_1"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="neckla_3.html#T23" target="_self" title="NECKLA_3:th.23">Th23</a></i>;<br/><a NAME="E14:62_1_1_3"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">G1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes2.html#K13" title="CLASSES2:func.13">FinSETS</a> 
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:62_1"><i><font color="Green" title="E38">A3</font></i></a>, <a class="txt" href="neckla_3.html#E5:62_1_1_3"><i><font color="Green" title="E41">A18</font></i></a>, <a class="ref" href="classes1.html#T6" title="CLASSES1:th.6">CLASSES1:6</a>, <a class="ref" href="classes2.html#D2" title="CLASSES2:def.2">CLASSES2:def 2</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E15:62_1_1_3"/><b>then </b><i><font color="Green" title="E46">A23</font></i>: 
<font color="Maroon" title="c11">G1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a> 
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:62_1"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="neckla_3.html#E2:62_1"><i><font color="Green" title="E38">A3</font></i></a>, <a class="txt" href="neckla_3.html#E11:62_1_1_3"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="neckla_3.html#E13:62_1_1_3"><i><font color="Green" title="E45">A22</font></i></a></i>;<br/><a NAME="E16:62_1_1_3"/><i><font color="Green" title="E47">A24</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">G2</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c12">cR</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:62_1_1_3"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="neckla_3.html#E5:62_1_1_3"><i><font color="Green" title="E41">A18</font></i></a>, <a class="txt" href="neckla_3.html#E36"><i><font color="Green" title="E30">Lm1</font></i></a></i>;<br/><a NAME="E17:62_1_1_3"/><b>then </b><i><font color="Green" title="E48">A25</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">G2</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="ref" href="card_2.html#T68" title="CARD_2:th.68">CARD_2:68</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">cG2</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">G2</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="neckla_3.html#E16:62_1_1_3"><i><font color="Green" title="E47">A24</font></i></a>, <a class="ref" href="card_2.html#T68" title="CARD_2:th.68">CARD_2:68</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">G2</font> = <font color="Maroon" title="c5">G2</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a> <a href="necklace.html#V3" title="NECKLACE:attr.3">irreflexive</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  <b>by </b><i><a class="txt" href="neckla_3.html#E17:62_1_1_3"><i><font color="Green" title="E48">A25</font></i></a>, <a class="ref" href="struct_0.html#D11" title="STRUCT_0:def.11">STRUCT_0:def 11</a></i>;<br/><a NAME="E20:62_1_1_3"/><i><font color="Green" title="E49">A26</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">cG2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">cR</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E16:62_1_1_3"><i><font color="Green" title="E47">A24</font></i></a>, <a class="ref" href="card_2.html#T58" title="CARD_2:th.58">CARD_2:58</a></i>;<br/><a NAME="E21:62_1_1_3"/>
<font color="Maroon" title="c14">G2</font> is  <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:62_1_1_3"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="neckla_3.html#E4:62_1_1_3"><i><font color="Green" title="E40">A17</font></i></a>, <a class="ref" href="neckla_3.html#T10" target="_self" title="NECKLA_3:th.10">Th10</a></i>;<br/><a NAME="E22:62_1_1_3"/><b>then </b><i><font color="Green" title="E50">A27</font></i>: 
<font color="Maroon" title="c14">G2</font> is <a href="neckla_2.html#V1" title="NECKLA_2:attr.1">N-free</a>
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:62_1"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="neckla_3.html#T23" target="_self" title="NECKLA_3:th.23">Th23</a></i>;<br/><a NAME="E23:62_1_1_3"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c14">G2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes2.html#K13" title="CLASSES2:func.13">FinSETS</a> 
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:62_1"><i><font color="Green" title="E38">A3</font></i></a>, <a class="txt" href="neckla_3.html#E5:62_1_1_3"><i><font color="Green" title="E41">A18</font></i></a>, <a class="ref" href="classes1.html#T6" title="CLASSES1:th.6">CLASSES1:6</a>, <a class="ref" href="classes2.html#D2" title="CLASSES2:def.2">CLASSES2:def 2</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E24:62_1_1_3"/><b>then </b>
<font color="Maroon" title="c14">G2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a> 
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:62_1"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="neckla_3.html#E2:62_1"><i><font color="Green" title="E38">A3</font></i></a>, <a class="txt" href="neckla_3.html#E20:62_1_1_3"><i><font color="Green" title="E49">A26</font></i></a>, <a class="txt" href="neckla_3.html#E22:62_1_1_3"><i><font color="Green" title="E50">A27</font></i></a></i>;<br/><b>hence </b><a NAME="E25:62_1_1_3"/>
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">G</font> #) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a> 
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:62_1_1_3"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="neckla_3.html#E4:62_1_1_3"><i><font color="Green" title="E40">A17</font></i></a>, <a class="txt" href="neckla_3.html#E15:62_1_1_3"><i><font color="Green" title="E46">A23</font></i></a>, <a class="ref" href="neckla_2.html#D5" title="NECKLA_2:def.5">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" title="E39">A28</font></i>: 
( not <font color="Maroon" title="c2">G</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> &amp; <font color="Maroon" title="c2">G</font> is <a href="neckla_3.html#V1" title="NECKLA_3:attr.1">path-connected</a> &amp;  <a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c2">G</font> is <a href="neckla_3.html#V1" title="NECKLA_3:attr.1">path-connected</a> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:62_1_1_4"/><i><font color="Green" title="E40">A29</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="txt" href="neckla_3.html#E3:62_1_1_4"><i><font color="Green" title="E40">A29</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c6">A</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/><a NAME="E5:62_1_1_4"/><i><font color="Green" title="E41">A30</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c7">A</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">G</font> ;<br/><b>set </b><font color="Maroon" title="c8">R</font> =  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c7">A</font>;<br/><a NAME="E7:62_1_1_4"/>
not  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c7">A</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">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" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c7">A</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>
 ;<br/>

<a NAME="E2:62_1_1_4_1"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c7">A</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/>
<a NAME="E3:62_1_1_4_1"/><b>then </b>
<font color="Maroon" title="c7">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>;<br/>
<a NAME="E4:62_1_1_4_1"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T66" title="ZFMISC_1:th.66">ZFMISC_1:66</a></i>;<br/>
<b>hence </b><a NAME="E5:62_1_1_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:62_1_1_4"><i><font color="Green" title="E39">A28</font></i></a>, <a class="ref" href="struct_0.html#D9" title="STRUCT_0:def.9">STRUCT_0:def 9</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c9">R</font> =  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c7">A</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a> <a href="necklace.html#V3" title="NECKLACE:attr.3">irreflexive</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ;<br/><a NAME="E9:62_1_1_4"/><i><font color="Green" title="E42">A31</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">R</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</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#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="card_2.html#T63" title="CARD_2:th.63">CARD_2:63</a></i>;<br/>
<a NAME="E2:62_1_1_4_2"/><b>then </b><i><font color="Green" title="E43">A32</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="txt" href="neckla_3.html#E2:62_1"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="card_2.html#T60" title="CARD_2:th.60">CARD_2:60</a></i>;<br/>
<a NAME="E3:62_1_1_4_2"/>
<font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<b>hence </b><a NAME="E4:62_1_1_4_2"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">R</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:62_1_1_4_2"><i><font color="Green" title="E43">A32</font></i></a>, <a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:62_1_1_4"/><i><font color="Green" title="E43">A33</font></i>: 
<font color="Maroon" title="c9">R</font> is <a href="neckla_2.html#V1" title="NECKLA_2:attr.1">N-free</a>
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:62_1"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="neckla_3.html#T23" target="_self" title="NECKLA_3:th.23">Th23</a></i>;<br/><a NAME="E11:62_1_1_4"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes2.html#K13" title="CLASSES2:func.13">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" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E5:62_1_1_4"><i><font color="Green" title="E41">A30</font></i></a>, <a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">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" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes2.html#K13" title="CLASSES2:func.13">FinSETS</a> 
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:62_1"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="classes1.html#T6" title="CLASSES1:th.6">CLASSES1:6</a>, <a class="ref" href="classes2.html#D2" title="CLASSES2:def.2">CLASSES2:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E12:62_1_1_4"/><b>then </b><i><font color="Green" title="E44">A34</font></i>: 
<font color="Maroon" title="c9">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a> 
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:62_1"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="neckla_3.html#E9:62_1_1_4"><i><font color="Green" title="E42">A31</font></i></a>, <a class="txt" href="neckla_3.html#E10:62_1_1_4"><i><font color="Green" title="E43">A33</font></i></a></i>;<br/><b>thus </b><a NAME="E13:62_1_1_4"/>
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">G</font> #) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">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" title="c9">R</font> is  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  or  ex <font color="Olive" title="b1">R1</font>, <font color="Olive" title="b2">R2</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">R1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">R2</font> &amp; <font color="Olive" title="b1">R1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  &amp; <font color="Olive" title="b2">R2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  &amp; ( <font color="Maroon" title="c9">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Olive" title="b1">R1</font>,<font color="Olive" title="b2">R2</font> or <font color="Maroon" title="c9">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Olive" title="b1">R1</font>,<font color="Olive" title="b2">R2</font> ) ) )
 </span><b>by </b><i><a class="txt" href="neckla_3.html#E12:62_1_1_4"><i><font color="Green" title="E44">A34</font></i></a>, <a class="ref" href="neckla_2.html#T6" title="NECKLA_2:th.6">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" title="c9">R</font> is  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">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" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">R</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>
 <b>by </b><i><a class="ref" href="struct_0.html#D9" title="STRUCT_0:def.9">STRUCT_0:def 9</a></i>;<br/><a NAME="E3:62_1_1_4_4_1_1"/><b>then </b><i><font color="Green" title="E45">A35</font></i>: 
<font color="Maroon" title="c7">A</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>
 <b>by </b><i><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">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" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><a NAME="E5:62_1_1_4_4_1_1"/><b>then </b>
not <font color="Maroon" title="c7">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:62_1_1_4_4_1_1"/><i><font color="Green" title="E46">A36</font></i>: 
<font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</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" title="E45">A35</font></i></a>, <a class="ref" href="realset1.html#D4" title="REALSET1:def.4">REALSET1:def 4</a></i>;<br/><a NAME="E8:62_1_1_4_4_1_1"/><i><font color="Green" title="E47">A37</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</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" title="E46">A36</font></i></a>, <a class="ref" href="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a></i>;<br/><a NAME="E9:62_1_1_4_4_1_1"/><i><font color="Green" title="E48">A38</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</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" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">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" title="c11">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:62_1_1_4_4_1_1_1_1"/>
<font color="Maroon" title="c11">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/>

<a NAME="E2:62_1_1_4_4_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c11">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> or <font color="Maroon" title="c11">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">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" title="c11">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>
 ;<br/>


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

<b>let </b><font color="Maroon" title="c11">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:62_1_1_4_4_1_1_1"/>
<font color="Maroon" title="c11">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>
 ;<br/>

<b>hence </b><a NAME="E3:62_1_1_4_4_1_1_1"/>
<font color="Maroon" title="c11">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">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" title="E49">A39</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</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" title="E47">A37</font></i></a>, <a class="ref" href="enumset1.html#T41" title="ENUMSET1:th.41">ENUMSET1:41</a></i>;<br/><a NAME="E11:62_1_1_4_4_1_1"/>
<font color="Maroon" title="c10">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:62_1_1_4"><i><font color="Green" title="E39">A28</font></i></a>, <a class="txt" href="neckla_3.html#E8:62_1_1_4_4_1_1"><i><font color="Green" title="E47">A37</font></i></a>, <a class="txt" href="neckla_3.html#E9:62_1_1_4_4_1_1"><i><font color="Green" title="E48">A38</font></i></a>, <a class="ref" href="struct_0.html#D9" title="STRUCT_0:def.9">STRUCT_0:def 9</a></i>;<br/><a NAME="E12:62_1_1_4_4_1_1"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="E49">A39</font></i></a>, <a class="ref" href="card_2.html#T76" title="CARD_2:th.76">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" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">G</font> #) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a> 
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:62_1"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="neckla_3.html#T34" target="_self" title="NECKLA_3:th.34">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" title="b1">R1</font>, <font color="Olive" title="b2">R2</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">R1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">R2</font> &amp; <font color="Olive" title="b1">R1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  &amp; <font color="Olive" title="b2">R2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  &amp; ( <font color="Maroon" title="c9">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Olive" title="b1">R1</font>,<font color="Olive" title="b2">R2</font> or <font color="Maroon" title="c9">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Olive" title="b1">R1</font>,<font color="Olive" title="b2">R2</font> ) )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c10">R1</font>, <font color="Maroon" title="c11">R2</font> being  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> <b> such that </b><br/><a NAME="E3:62_1_1_4_4_1_2"/><i><font color="Green" title="E45">A40</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">R1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">R2</font>
 <b>and </b><br/><a NAME="E4:62_1_1_4_4_1_2"/><i><font color="Green" title="E46">A41</font></i>: 
( <font color="Maroon" title="c10">R1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  &amp; <font color="Maroon" title="c11">R2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  &amp; ( <font color="Maroon" title="c9">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Maroon" title="c10">R1</font>,<font color="Maroon" title="c11">R2</font> or <font color="Maroon" title="c9">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Maroon" title="c10">R1</font>,<font color="Maroon" title="c11">R2</font> ) )
 ;<br/><b>thus </b><a NAME="E5:62_1_1_4_4_1_2"/>
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">G</font> #) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">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" title="c9">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Maroon" title="c10">R1</font>,<font color="Maroon" title="c11">R2</font> or <font color="Maroon" title="c9">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Maroon" title="c10">R1</font>,<font color="Maroon" title="c11">R2</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" title="E46">A41</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" title="E47">A42</font></i>: 
<font color="Maroon" title="c9">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Maroon" title="c10">R1</font>,<font color="Maroon" title="c11">R2</font>
 ;<br/><div class="add"><a NAME="E2:62_1_1_4_4_1_2_1_1_1"/>
<font color="Maroon" title="c10">R1</font> is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c9">R</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:62_1_1_4_4_1_2"><i><font color="Green" title="E45">A40</font></i></a>, <a class="txt" href="neckla_3.html#E4:62_1_1_4_4_1_2"><i><font color="Green" title="E46">A41</font></i></a>, <a class="ref" href="neckla_3.html#T10" target="_self" title="NECKLA_3:th.10">Th10</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c12">R1</font> = <font color="Maroon" title="c10">R1</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="txt" href="neckla_3.html#E4:62_1_1_4_4_1_2"><i><font color="Green" title="E46">A41</font></i></a>, <a class="ref" href="neckla_2.html#T4" title="NECKLA_2:th.4">NECKLA_2:4</a>, <a class="ref" href="yellow_6.html#T16" title="YELLOW_6:th.16">YELLOW_6:16</a></i>;<br/><a NAME="E4:62_1_1_4_4_1_2_1_1_1"/>
<font color="Maroon" title="c11">R2</font> is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c9">R</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:62_1_1_4_4_1_2"><i><font color="Green" title="E45">A40</font></i></a>, <a class="txt" href="neckla_3.html#E4:62_1_1_4_4_1_2"><i><font color="Green" title="E46">A41</font></i></a>, <a class="ref" href="neckla_3.html#T10" target="_self" title="NECKLA_3:th.10">Th10</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c13">R2</font> = <font color="Maroon" title="c11">R2</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="txt" href="neckla_3.html#E4:62_1_1_4_4_1_2"><i><font color="Green" title="E46">A41</font></i></a>, <a class="ref" href="neckla_2.html#T4" title="NECKLA_2:th.4">NECKLA_2:4</a>, <a class="ref" href="yellow_6.html#T16" title="YELLOW_6:th.16">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" title="YELLOW_0:func.5">subrelstr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Maroon" title="c12">R1</font>,<font color="Maroon" title="c13">R2</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" title="E47">A42</font></i></a></i>;<br/><a NAME="E7:62_1_1_4_4_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">G</font> <a href="necklace.html#R2" title="NECKLACE:pred.2">embeds</a>  <a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:62_1_1_4"><i><font color="Green" title="E39">A28</font></i></a>, <a class="txt" href="neckla_3.html#E3:62_1_1_4_4_1_2"><i><font color="Green" title="E45">A40</font></i></a>, <a class="ref" href="neckla_3.html#T40" target="_self" title="NECKLA_3:th.40">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" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">G</font> #) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a> 
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:62_1"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="neckla_2.html#D1" title="NECKLA_2:def.1">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" title="c9">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Maroon" title="c10">R1</font>,<font color="Maroon" title="c11">R2</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" title="E47">A43</font></i>: 
 <a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c9">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <span class="p1">(<span class="default"><a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c10">R1</font></span>)</span>,<span class="p1">(<span class="default"><a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c11">R2</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" title="E45">A40</font></i></a>, <a class="ref" href="neckla_3.html#T18" target="_self" title="NECKLA_3:th.18">Th18</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">G'</font> =  <a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c2">G</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a> <a href="necklace.html#V3" title="NECKLACE:attr.3">irreflexive</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ;<br/><a NAME="E4:62_1_1_4_4_1_2_1_1_2"/><i><font color="Green" title="E48">A44</font></i>: 
not  <a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c10">R1</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">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" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c10">R1</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">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" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c10">R1</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">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" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">R1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="ref" href="necklace.html#D9" title="NECKLACE:def.9">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" title="c10">R1</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>
 
<b>by </b><i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">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" title="E46">A41</font></i></a>, <a class="ref" href="neckla_2.html#T4" title="NECKLA_2:th.4">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" title="E49">A45</font></i>: 
not  <a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c11">R2</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">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" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c11">R2</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">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" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c11">R2</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">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" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">R2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="ref" href="necklace.html#D9" title="NECKLACE:def.9">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" title="c11">R2</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>
 
<b>by </b><i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">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" title="E46">A41</font></i></a>, <a class="ref" href="neckla_2.html#T4" title="NECKLA_2:th.4">NECKLA_2:4</a></i>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon" title="c13">R11</font> =  <a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c10">R1</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">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" title="E48">A44</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">R22</font> =  <a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c11">R2</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">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" title="E49">A45</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">x'</font> = <font color="Maroon" title="c5">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c12">G'</font> <b>by </b><i><a class="ref" href="necklace.html#D9" title="NECKLACE:def.9">NECKLACE:def 9</a></i>;<br/><a NAME="E9:62_1_1_4_4_1_2_1_1_2"/><i><font color="Green" title="E50">A46</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c13">R11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">R1</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c14">R22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">R2</font> )
 <b>by </b><i><a class="ref" href="necklace.html#D9" title="NECKLACE:def.9">NECKLACE:def 9</a></i>;<br/><a NAME="E10:62_1_1_4_4_1_2_1_1_2"/><i><font color="Green" title="E51">A47</font></i>: 
( <font color="Maroon" title="c12">G'</font> is <a href="neckla_3.html#V1" title="NECKLA_3:attr.1">path-connected</a> &amp;  <a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c12">G'</font> is <a href="neckla_3.html#V1" title="NECKLA_3:attr.1">path-connected</a> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:62_1_1_4"><i><font color="Green" title="E39">A28</font></i></a>, <a class="ref" href="neckla_3.html#T16" target="_self" title="NECKLA_3:th.16">Th16</a></i>;<br/><a NAME="E11:62_1_1_4_4_1_2_1_1_2"/><i><font color="Green" title="E52">A48</font></i>: 
not <font color="Maroon" title="c12">G'</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">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" title="c12">G'</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c16">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">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" title="E53">A49</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">G'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">x1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="group_6.html#D2" title="GROUP_6:def.2">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" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">x1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</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" title="E53">A49</font></i></a>, <a class="ref" href="necklace.html#D9" title="NECKLACE:def.9">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" title="E39">A28</font></i></a>, <a class="ref" href="group_6.html#D2" title="GROUP_6:def.2">GROUP_6:def 2</a></i>;<br/>


</div><b>end;</b></div> <a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c9">R</font> = 
 <a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>

<br/>.= 
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c12">G'</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c15">x'</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="neckla_3.html#T20" target="_self" title="NECKLA_3:th.20">Th20</a></i>
;<br/><a NAME="E13:62_1_1_4_4_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E53">A50</font></i>: 
<font color="Maroon" title="c12">G'</font> <a href="necklace.html#R2" title="NECKLACE:pred.2">embeds</a>  <a href="necklace.html#K4" title="NECKLACE:func.4">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" title="E45">A40</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" title="E47">A43</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" title="E50">A46</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" title="E51">A47</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" title="E52">A48</font></i></a>, <a class="ref" href="neckla_3.html#T40" target="_self" title="NECKLA_3:th.40">Th40</a></i>;<br/><a NAME="E14:62_1_1_4_4_1_2_1_1_2"/>
<font color="Maroon" title="c12">G'</font> is <a href="neckla_2.html#V1" title="NECKLA_2:attr.1">N-free</a>
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:62_1"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="neckla_3.html#T25" target="_self" title="NECKLA_3:th.25">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" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">G</font> #) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">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" title="E53">A50</font></i></a>, <a class="ref" href="neckla_2.html#D1" title="NECKLA_2:def.1">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" title="E37">A51</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S4" title="NAT_1:sch.4">NAT_1:sch 4</a>(<a class="txt" href="neckla_3.html#E1:62"><i><font color="Green" title="E36">A1</font></i></a>);<br/></i>
<b>let </b><font color="Maroon" title="c1">R</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a> <a href="necklace.html#V3" title="NECKLACE:attr.3">irreflexive</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/>

<b>assume </b><a NAME="E3:62"/><i><font color="Green" title="E38">A52</font></i>: 
<font color="Maroon" title="c1">R</font> is <a href="neckla_2.html#V1" title="NECKLA_2:attr.1">N-free</a>
 ;<br/>

<a NAME="E4:62"/>
 ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">n</font>
 
<b>by </b><i><a class="ref" href="card_4.html#T4" title="CARD_4:th.4">CARD_4:4</a></i>;<br/>
<b>hence </b><a NAME="E5:62"/>
(  not the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes2.html#K13" title="CLASSES2:func.13">FinSETS</a>  or  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> #) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  )
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:62"><i><font color="Green" title="E37">A51</font></i></a>, <a class="txt" href="neckla_3.html#E3:62"><i><font color="Green" title="E38">A52</font></i></a></i>;<br/>


</div>
