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<div class="add">

<b>let </b><font color="Maroon" title="c1">G</font> be  <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="E1:54"/><i><font color="Green" title="E31">A1</font></i>: 
<font color="Maroon" title="c1">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/>

<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 <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  <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; <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>  holds <br/> <a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <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="E2:54"/><i><font color="Green" title="E32">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 ( for <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">n</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:54_1"/><i><font color="Green" title="E33">A3</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 <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 ;<br/>

<b>let </b><font color="Maroon" title="c3">G</font> be  <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><b>that </b><br/><a NAME="E2:54_1"/><i><font color="Green" title="E34">A4</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="c3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font>
 <b>and </b><br/><a NAME="E3:54_1"/><i><font color="Green" title="E35">A5</font></i>: 
<font color="Maroon" title="c3">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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_1_1"/>
( <font color="Maroon" title="c3">G</font> is  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#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  or  ex <font color="Olive" title="b1">G1</font>, <font color="Olive" title="b2">G2</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">G1</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">G2</font> &amp; <font color="Olive" title="b1">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>  &amp; <font color="Olive" title="b2">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>  &amp; ( <font color="Maroon" title="c3">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="Olive" title="b1">G1</font>,<font color="Olive" title="b2">G2</font> or <font color="Maroon" title="c3">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="Olive" title="b1">G1</font>,<font color="Olive" title="b2">G2</font> ) ) )
 </span><b>by </b><i><a class="txt" href="neckla_3.html#E3:54_1"><i><font color="Green" title="E35">A5</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="54_1_1_1"><b>suppose </b></a><a NAME="E1:54_1_1_1"/>
<font color="Maroon" title="c3">G</font> is  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#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_1_1_1"/>
 <a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c3">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#E3:54_1"><i><font color="Green" title="E35">A5</font></i></a>, <a class="ref" href="neckla_3.html#T15" target="_self" title="NECKLA_3:th.15">Th15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_2"><b>suppose </b></a><a NAME="E1:54_1_1_2"/>
 ex <font color="Olive" title="b1">G1</font>, <font color="Olive" title="b2">G2</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">G1</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">G2</font> &amp; <font color="Olive" title="b1">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>  &amp; <font color="Olive" title="b2">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>  &amp; ( <font color="Maroon" title="c3">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="Olive" title="b1">G1</font>,<font color="Olive" title="b2">G2</font> or <font color="Maroon" title="c3">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="Olive" title="b1">G1</font>,<font color="Olive" title="b2">G2</font> ) )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c4">G1</font>, <font color="Maroon" title="c5">G2</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="E3:54_1_1_2"/><i><font color="Green" title="E36">A6</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="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="c5">G2</font> &amp; <font color="Maroon" title="c4">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>  &amp; <font color="Maroon" title="c5">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>and </b><br/><a NAME="E4:54_1_1_2"/><i><font color="Green" title="E37">A7</font></i>: 
( <font color="Maroon" title="c3">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> or <font color="Maroon" title="c3">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> )
 ;<br/><a NAME="E5:54_1_1_2"/><i><font color="Green" title="E38">A8</font></i>: 
( not <font color="Maroon" title="c4">G1</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> &amp; <font color="Maroon" title="c4">G1</font> is <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:54_1_1_2"><i><font color="Green" title="E36">A6</font></i></a>, <a class="ref" href="neckla_2.html#T4" title="NECKLA_2:th.4">NECKLA_2:4</a></i>;<br/><a NAME="E6:54_1_1_2"/><b>then </b>
( not 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="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; 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="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a>, <a class="ref" href="struct_0.html#D11" title="STRUCT_0:def.11">STRUCT_0:def 11</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">n1</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> <b> such that </b><br/><a NAME="E8:54_1_1_2"/><i><font color="Green" title="E39">A9</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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">n1</font>
 <b>by </b><i><a class="ref" href="card_4.html#T4" title="CARD_4:th.4">CARD_4:4</a></i>;<br/><a NAME="E9:54_1_1_2"/><i><font color="Green" title="E40">A10</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n1</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E8:54_1_1_2"><i><font color="Green" title="E39">A9</font></i></a>, <a class="ref" href="card_1.html#T66" title="CARD_1:th.66">CARD_1:66</a></i>;<br/><a NAME="E10:54_1_1_2"/><i><font color="Green" title="E41">A11</font></i>: 
( not <font color="Maroon" title="c5">G2</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> &amp; <font color="Maroon" title="c5">G2</font> is <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:54_1_1_2"><i><font color="Green" title="E36">A6</font></i></a>, <a class="ref" href="neckla_2.html#T4" title="NECKLA_2:th.4">NECKLA_2:4</a></i>;<br/><a NAME="E11:54_1_1_2"/><b>then </b>
( not 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="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; 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="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a>, <a class="ref" href="struct_0.html#D11" title="STRUCT_0:def.11">STRUCT_0:def 11</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">n2</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> <b> such that </b><br/><a NAME="E13:54_1_1_2"/><i><font color="Green" title="E42">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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">n2</font>
 <b>by </b><i><a class="ref" href="card_4.html#T4" title="CARD_4:th.4">CARD_4:4</a></i>;<br/><a NAME="E14:54_1_1_2"/><i><font color="Green" title="E43">A13</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n2</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E13:54_1_1_2"><i><font color="Green" title="E42">A12</font></i></a>, <a class="ref" href="card_1.html#T66" title="CARD_1:th.66">CARD_1:66</a></i>;<br/><b>thus </b><a NAME="E15:54_1_1_2"/>
 <a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c3">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="54_1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_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="E1:54_1_1_2_1_1"/>
( <font color="Maroon" title="c3">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> or <font color="Maroon" title="c3">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> )
 </span><b>by </b><i><a class="txt" href="neckla_3.html#E4:54_1_1_2"><i><font color="Green" title="E37">A7</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:54_1_1_2_1_1_1"/><i><font color="Green" title="E44">A14</font></i>: 
<font color="Maroon" title="c3">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>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c8">G1</font> = <font color="Maroon" title="c4">G1</font> as  <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="ref" href="neckla_3.html#T9" target="_self" title="NECKLA_3:th.9">Th9</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">G2</font> = <font color="Maroon" title="c5">G2</font> as  <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#E1:54_1_1_2_1_1_1"><i><font color="Green" title="E44">A14</font></i></a>, <a class="ref" href="neckla_3.html#T9" target="_self" title="NECKLA_3:th.9">Th9</a></i>;<br/><a NAME="E4:54_1_1_2_1_1_1"/><i><font color="Green" title="E45">A15</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">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="c8">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="c9">G2</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:54_1_1_2_1_1_1"><i><font color="Green" title="E44">A14</font></i></a>, <a class="ref" href="neckla_2.html#D2" title="NECKLA_2:def.2">NECKLA_2:def 2</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="c8">G1</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <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#E5:54_1_1_2"><i><font color="Green" title="E38">A8</font></i></a>, <a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</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="c11">cG2</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">G2</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <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#E10:54_1_1_2"><i><font color="Green" title="E41">A11</font></i></a>, <a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a>, <a class="ref" href="struct_0.html#D11" title="STRUCT_0:def.11">STRUCT_0:def 11</a></i>;<br/><a NAME="E7:54_1_1_2_1_1_1"/><i><font color="Green" title="E46">A16</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="c3">G</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> <font color="Maroon" title="c10">cG1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">cG2</font></span>)</span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:54_1_1_2"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="neckla_3.html#E4:54_1_1_2_1_1_1"><i><font color="Green" title="E45">A15</font></i></a>, <a class="ref" href="card_2.html#T53" title="CARD_2:th.53">CARD_2:53</a></i>;<br/><a NAME="E8:54_1_1_2_1_1_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">cG1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n1</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E8:54_1_1_2"><i><font color="Green" title="E39">A9</font></i></a>, <a class="ref" href="card_1.html#T66" title="CARD_1:th.66">CARD_1:66</a></i>;<br/><a NAME="E9:54_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">n1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E10:54_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E47">A17</font></i>: 
<font color="Maroon" title="c6">n1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E11:54_1_1_2_1_1_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">cG2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n2</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E13:54_1_1_2"><i><font color="Green" title="E42">A12</font></i></a>, <a class="ref" href="card_1.html#T66" title="CARD_1:th.66">CARD_1:66</a></i>;<br/><a NAME="E12:54_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c7">n2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E13:54_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E48">A18</font></i>: 
<font color="Maroon" title="c7">n2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E14:54_1_1_2_1_1_1"/>
<font color="Maroon" title="c6">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:54_1_1_2_1_1_1_1"/>
not <font color="Maroon" title="c6">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font>
 ;<br/>

<a NAME="E2:54_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">n1</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E13:54_1_1_2_1_1_1"><i><font color="Green" title="E48">A18</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E3:54_1_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:54_1"><i><font color="Green" title="E34">A4</font></i></a>, <a class="txt" href="neckla_3.html#E8:54_1_1_2"><i><font color="Green" title="E39">A9</font></i></a>, <a class="txt" href="neckla_3.html#E14:54_1_1_2"><i><font color="Green" title="E43">A13</font></i></a>, <a class="txt" href="neckla_3.html#E7:54_1_1_2_1_1_1"><i><font color="Green" title="E46">A16</font></i></a>, <a class="ref" href="card_1.html#T66" title="CARD_1:th.66">CARD_1:66</a></i>;<br/>


</div><b>end;</b></div><a NAME="E15:54_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E49">A19</font></i>: 
 <a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c8">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#E1:54_1"><i><font color="Green" title="E33">A3</font></i></a>, <a class="txt" href="neckla_3.html#E3:54_1_1_2"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="neckla_3.html#E9:54_1_1_2"><i><font color="Green" title="E40">A10</font></i></a></i>;<br/><a NAME="E16:54_1_1_2_1_1_1"/>
<font color="Maroon" title="c7">n2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_2_1_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:54_1_1_2_1_1_1_2"/>
not <font color="Maroon" title="c7">n2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font>
 ;<br/>

<a NAME="E2:54_1_1_2_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n2</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E10:54_1_1_2_1_1_1"><i><font color="Green" title="E47">A17</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E3:54_1_1_2_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:54_1"><i><font color="Green" title="E34">A4</font></i></a>, <a class="txt" href="neckla_3.html#E8:54_1_1_2"><i><font color="Green" title="E39">A9</font></i></a>, <a class="txt" href="neckla_3.html#E14:54_1_1_2"><i><font color="Green" title="E43">A13</font></i></a>, <a class="txt" href="neckla_3.html#E7:54_1_1_2_1_1_1"><i><font color="Green" title="E46">A16</font></i></a>, <a class="ref" href="card_1.html#T66" title="CARD_1:th.66">CARD_1:66</a></i>;<br/>


</div><b>end;</b></div><a NAME="E17:54_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E50">A20</font></i>: 
 <a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c9">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#E1:54_1"><i><font color="Green" title="E33">A3</font></i></a>, <a class="txt" href="neckla_3.html#E3:54_1_1_2"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="neckla_3.html#E14:54_1_1_2"><i><font color="Green" title="E43">A13</font></i></a></i>;<br/><a NAME="E18:54_1_1_2_1_1_1"/><i><font color="Green" title="E51">A21</font></i>: 
 <a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c8">G1</font></span>)</span>,<span class="p1">(<span class="default"><a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c9">G2</font></span>)</span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:54_1_1_2"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="neckla_3.html#E1:54_1_1_2_1_1_1"><i><font color="Green" title="E44">A14</font></i></a>, <a class="ref" href="neckla_3.html#T17" target="_self" title="NECKLA_3:th.17">Th17</a></i>;<br/><a NAME="E19:54_1_1_2_1_1_1"/>
( 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="c8">G1</font></span>)</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="c8">G1</font> &amp; 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="c9">G2</font></span>)</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="c9">G2</font> )
 <b>by </b><i><a class="ref" href="necklace.html#D9" title="NECKLACE:def.9">NECKLACE:def 9</a></i>;<br/><b>hence </b><a NAME="E20:54_1_1_2_1_1_1"/>
 <a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c3">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#E3:54_1_1_2"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="neckla_3.html#E15:54_1_1_2_1_1_1"><i><font color="Green" title="E49">A19</font></i></a>, <a class="txt" href="neckla_3.html#E17:54_1_1_2_1_1_1"><i><font color="Green" title="E50">A20</font></i></a>, <a class="txt" href="neckla_3.html#E18:54_1_1_2_1_1_1"><i><font color="Green" title="E51">A21</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="54_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:54_1_1_2_1_1_2"/><i><font color="Green" title="E44">A22</font></i>: 
<font color="Maroon" title="c3">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>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c8">G1</font> = <font color="Maroon" title="c4">G1</font> as  <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="ref" href="neckla_3.html#T9" target="_self" title="NECKLA_3:th.9">Th9</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">G2</font> = <font color="Maroon" title="c5">G2</font> as  <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#E1:54_1_1_2_1_1_2"><i><font color="Green" title="E44">A22</font></i></a>, <a class="ref" href="neckla_3.html#T9" target="_self" title="NECKLA_3:th.9">Th9</a></i>;<br/><a NAME="E4:54_1_1_2_1_1_2"/><i><font color="Green" title="E45">A23</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">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="c8">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="c9">G2</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:54_1_1_2_1_1_2"><i><font color="Green" title="E44">A22</font></i></a>, <a class="ref" href="neckla_2.html#D3" title="NECKLA_2:def.3">NECKLA_2:def 3</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="c8">G1</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <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#E5:54_1_1_2"><i><font color="Green" title="E38">A8</font></i></a>, <a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</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="c11">cG2</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">G2</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <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#E10:54_1_1_2"><i><font color="Green" title="E41">A11</font></i></a>, <a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a>, <a class="ref" href="struct_0.html#D11" title="STRUCT_0:def.11">STRUCT_0:def 11</a></i>;<br/><a NAME="E7:54_1_1_2_1_1_2"/><i><font color="Green" title="E46">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="c3">G</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> <font color="Maroon" title="c10">cG1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">cG2</font></span>)</span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:54_1_1_2"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="neckla_3.html#E4:54_1_1_2_1_1_2"><i><font color="Green" title="E45">A23</font></i></a>, <a class="ref" href="card_2.html#T53" title="CARD_2:th.53">CARD_2:53</a></i>;<br/><a NAME="E8:54_1_1_2_1_1_2"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">cG1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n1</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E8:54_1_1_2"><i><font color="Green" title="E39">A9</font></i></a>, <a class="ref" href="card_1.html#T66" title="CARD_1:th.66">CARD_1:66</a></i>;<br/><a NAME="E9:54_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">n1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E10:54_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E47">A25</font></i>: 
<font color="Maroon" title="c6">n1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E11:54_1_1_2_1_1_2"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">cG2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n2</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E13:54_1_1_2"><i><font color="Green" title="E42">A12</font></i></a>, <a class="ref" href="card_1.html#T66" title="CARD_1:th.66">CARD_1:66</a></i>;<br/><a NAME="E12:54_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c7">n2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E13:54_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E48">A26</font></i>: 
<font color="Maroon" title="c7">n2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E14:54_1_1_2_1_1_2"/>
<font color="Maroon" title="c6">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:54_1_1_2_1_1_2_1"/>
not <font color="Maroon" title="c6">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font>
 ;<br/>

<a NAME="E2:54_1_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">n1</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E13:54_1_1_2_1_1_2"><i><font color="Green" title="E48">A26</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E3:54_1_1_2_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:54_1"><i><font color="Green" title="E34">A4</font></i></a>, <a class="txt" href="neckla_3.html#E8:54_1_1_2"><i><font color="Green" title="E39">A9</font></i></a>, <a class="txt" href="neckla_3.html#E14:54_1_1_2"><i><font color="Green" title="E43">A13</font></i></a>, <a class="txt" href="neckla_3.html#E7:54_1_1_2_1_1_2"><i><font color="Green" title="E46">A24</font></i></a>, <a class="ref" href="card_1.html#T66" title="CARD_1:th.66">CARD_1:66</a></i>;<br/>


</div><b>end;</b></div><a NAME="E15:54_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E49">A27</font></i>: 
 <a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c8">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#E1:54_1"><i><font color="Green" title="E33">A3</font></i></a>, <a class="txt" href="neckla_3.html#E3:54_1_1_2"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="neckla_3.html#E9:54_1_1_2"><i><font color="Green" title="E40">A10</font></i></a></i>;<br/><a NAME="E16:54_1_1_2_1_1_2"/>
<font color="Maroon" title="c7">n2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_2_1_1_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:54_1_1_2_1_1_2_2"/>
not <font color="Maroon" title="c7">n2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font>
 ;<br/>

<a NAME="E2:54_1_1_2_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n2</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E10:54_1_1_2_1_1_2"><i><font color="Green" title="E47">A25</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E3:54_1_1_2_1_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:54_1"><i><font color="Green" title="E34">A4</font></i></a>, <a class="txt" href="neckla_3.html#E8:54_1_1_2"><i><font color="Green" title="E39">A9</font></i></a>, <a class="txt" href="neckla_3.html#E14:54_1_1_2"><i><font color="Green" title="E43">A13</font></i></a>, <a class="txt" href="neckla_3.html#E7:54_1_1_2_1_1_2"><i><font color="Green" title="E46">A24</font></i></a>, <a class="ref" href="card_1.html#T66" title="CARD_1:th.66">CARD_1:66</a></i>;<br/>


</div><b>end;</b></div><a NAME="E17:54_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E50">A28</font></i>: 
 <a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c9">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#E1:54_1"><i><font color="Green" title="E33">A3</font></i></a>, <a class="txt" href="neckla_3.html#E3:54_1_1_2"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="neckla_3.html#E14:54_1_1_2"><i><font color="Green" title="E43">A13</font></i></a></i>;<br/><a NAME="E18:54_1_1_2_1_1_2"/><i><font color="Green" title="E51">A29</font></i>: 
 <a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c8">G1</font></span>)</span>,<span class="p1">(<span class="default"><a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c9">G2</font></span>)</span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:54_1_1_2"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="neckla_3.html#E1:54_1_1_2_1_1_2"><i><font color="Green" title="E44">A22</font></i></a>, <a class="ref" href="neckla_3.html#T18" target="_self" title="NECKLA_3:th.18">Th18</a></i>;<br/><a NAME="E19:54_1_1_2_1_1_2"/>
( 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="c8">G1</font></span>)</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="c8">G1</font> &amp; 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="c9">G2</font></span>)</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="c9">G2</font> )
 <b>by </b><i><a class="ref" href="necklace.html#D9" title="NECKLACE:def.9">NECKLACE:def 9</a></i>;<br/><b>hence </b><a NAME="E20:54_1_1_2_1_1_2"/>
 <a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c3">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#E3:54_1_1_2"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="neckla_3.html#E15:54_1_1_2_1_1_2"><i><font color="Green" title="E49">A27</font></i></a>, <a class="txt" href="neckla_3.html#E17:54_1_1_2_1_1_2"><i><font color="Green" title="E50">A28</font></i></a>, <a class="txt" href="neckla_3.html#E18:54_1_1_2_1_1_2"><i><font color="Green" title="E51">A29</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><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="E3:54"/><i><font color="Green" title="E33">A30</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#E2:54"><i><font color="Green" title="E32">A2</font></i></a>);<br/></i>
<a NAME="E4:54"/>
<font color="Maroon" title="c1">G</font> is <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E1:54"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="neckla_2.html#T4" title="NECKLA_2:th.4">NECKLA_2:4</a></i>;<br/>
<a NAME="E5:54"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="ref" href="struct_0.html#D11" title="STRUCT_0:def.11">STRUCT_0:def 11</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c2">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> <b> such that </b><br/><a NAME="E7:54"/><i><font color="Green" title="E34">A31</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="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">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>thus </b><a NAME="E8:54"/>
 <a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c1">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#E1:54"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="neckla_3.html#E3:54"><i><font color="Green" title="E33">A30</font></i></a>, <a class="txt" href="neckla_3.html#E7:54"><i><font color="Green" title="E34">A31</font></i></a></i>;<br/>


</div>
