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

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

<b>assume </b><a NAME="E1:54"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a> 
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>for <font color="Olive">G</font> being <a href="necklace.html#V3">irreflexive</a>  <a href="orders_2.html#L1">RelStr</a>   st  <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">G</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a>  holds <br/> <a href="necklace.html#K3">ComplRelStr</a> <font color="Olive">G</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a> ;<br/>
<a NAME="E2:54"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">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">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:54_1"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 ;<br/>

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

<b>assume </b><b>that </b><br/><a NAME="E2:54_1"/><i><font color="Green">E36</font></i>: 
 <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>and </b><br/><a NAME="E3:54_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">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">G</font> is  non <a href="struct_0.html#V3">empty</a> <a href="realset2.html#V3">trivial</a> <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  or  ex <font color="Olive">G1</font>, <font color="Olive">G2</font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">G1</font> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">G2</font> &amp; <font color="Olive">G1</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a>  &amp; <font color="Olive">G2</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a>  &amp; ( <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1">union_of</a> <font color="Olive">G1</font>,<font color="Olive">G2</font> or <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K2">sum_of</a> <font color="Olive">G1</font>,<font color="Olive">G2</font> ) ) )
 </span><b>by </b><i><a class="txt" href="neckla_3.html#E2:54_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="neckla_2.html#T6">NECKLA_2:6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_1"><b>suppose </b></a><a NAME="E1:54_1_1_1"/>
<font color="Maroon">G</font> is  non <a href="struct_0.html#V3">empty</a> <a href="realset2.html#V3">trivial</a> <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_1_1_1"/>
 <a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">G</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a> 
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:54_1"><i><font color="Green">E36</font></i></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">G1</font>, <font color="Olive">G2</font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">G1</font> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">G2</font> &amp; <font color="Olive">G1</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a>  &amp; <font color="Olive">G2</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a>  &amp; ( <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1">union_of</a> <font color="Olive">G1</font>,<font color="Olive">G2</font> or <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K2">sum_of</a> <font color="Olive">G1</font>,<font color="Olive">G2</font> ) )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">G1</font> being  <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">G2</font> being  <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a> <b> such that </b><br/><a NAME="E3:54_1_1_2"/><i><font color="Green">E41</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G1</font> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G2</font> &amp; <font color="Maroon">G1</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a>  &amp; <font color="Maroon">G2</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a>  )
 <b>and </b><br/><a NAME="E4:54_1_1_2"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1">union_of</a> <font color="Maroon">G1</font>,<font color="Maroon">G2</font> or <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">G1</font>,<font color="Maroon">G2</font> )
 ;<br/><a NAME="E5:54_1_1_2"/><i><font color="Green">E43</font></i>: 
( not <font color="Maroon">G1</font> is <a href="struct_0.html#V3">empty</a> &amp; <font color="Maroon">G1</font> is <a href="group_1.html#V6">finite</a> )
 <b>by </b><i>, <a class="ref" href="neckla_2.html#T4">NECKLA_2:4</a></i>;<br/><a NAME="E6:54_1_1_2"/><b>then </b>
( not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G1</font> is <a href="xboole_0.html#V1">empty</a> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G1</font> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i><a class="ref" href="group_1.html#D14">GROUP_1:def 14</a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">n1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:54_1_1_2"/><i><font color="Green">E60</font></i>: 
 <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G1</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="card_4.html#T4">CARD_4:4</a></i>;<br/><a NAME="E9:54_1_1_2"/><i><font color="Green">E61</font></i>: 
 <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n1</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T78">FINSEQ_1:78</a></i>;<br/><a NAME="E10:54_1_1_2"/><i><font color="Green">E62</font></i>: 
( not <font color="Maroon">G2</font> is <a href="struct_0.html#V3">empty</a> &amp; <font color="Maroon">G2</font> is <a href="group_1.html#V6">finite</a> )
 <b>by </b><i>, <a class="ref" href="neckla_2.html#T4">NECKLA_2:4</a></i>;<br/><a NAME="E11:54_1_1_2"/><b>then </b>
( not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G2</font> is <a href="xboole_0.html#V1">empty</a> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G2</font> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i><a class="ref" href="group_1.html#D14">GROUP_1:def 14</a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">n2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E13:54_1_1_2"/><i><font color="Green">E63</font></i>: 
 <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G2</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">n2</font>
 <b>by </b><i><a class="ref" href="card_4.html#T4">CARD_4:4</a></i>;<br/><a NAME="E14:54_1_1_2"/><i><font color="Green">E64</font></i>: 
 <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n2</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E5:54_1_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#T78">FINSEQ_1:78</a></i>;<br/><b>thus </b><a NAME="E15:54_1_1_2"/>
 <a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">G</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="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">G</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1">union_of</a> <font color="Maroon">G1</font>,<font color="Maroon">G2</font> or <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">G1</font>,<font color="Maroon">G2</font> )
 </span><b>by </b><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">E70</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1">union_of</a> <font color="Maroon">G1</font>,<font color="Maroon">G2</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">G1</font> = <font color="Maroon">G1</font> as  <a href="necklace.html#V3">irreflexive</a>  <a href="orders_2.html#L1">RelStr</a>  <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">G2</font> = <font color="Maroon">G2</font> as  <a href="necklace.html#V3">irreflexive</a>  <a href="orders_2.html#L1">RelStr</a>  <b>by </b><i>, </i>;<br/><a NAME="E4:54_1_1_2_1_1_1"/><i><font color="Green">E92</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G1</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G2</font>
 <b>by </b><i>, <a class="ref" href="neckla_2.html#D2">NECKLA_2:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">cG1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G1</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="neckla_3.html#E3:54_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="group_1.html#D14">GROUP_1:def 14</a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">cG2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G2</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="neckla_3.html#E4:54_1_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="group_1.html#D14">GROUP_1:def 14</a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><a NAME="E7:54_1_1_2_1_1_1"/><i><font color="Green">E93</font></i>: 
 <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">cG1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">cG2</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="card_2.html#T53">CARD_2:53</a></i>;<br/><a NAME="E8:54_1_1_2_1_1_1"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">cG1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n1</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T78">FINSEQ_1:78</a></i>;<br/><a NAME="E9:54_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">n1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="card_4.html#T17">CARD_4:17</a></i>;<br/><a NAME="E10:54_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E94</font></i>: 
<font color="Maroon">n1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E11:54_1_1_2_1_1_1"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">cG2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n2</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E5:54_1_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#T78">FINSEQ_1:78</a></i>;<br/><a NAME="E12:54_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">n2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="card_4.html#T17">CARD_4:17</a></i>;<br/><a NAME="E13:54_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E104</font></i>: 
<font color="Maroon">n2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E14:54_1_1_2_1_1_1"/>
<font color="Maroon">n1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">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">n1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E2:54_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T10">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#E1:54_1"><i><font color="Green">E35</font></i></a>, , <a class="ref" href="neckla_3.html#T3" target="_self">Th3</a>, , <a class="ref" href="finseq_1.html#T78">FINSEQ_1:78</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">E105</font></i>: 
 <a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">G1</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a> 
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:54"><i><font color="Green">E34</font></i></a>, , <a class="txt" href="neckla_3.html#E3:54_1_1_2"><i><font color="Green">E41</font></i></a></i>;<br/><a NAME="E16:54_1_1_2_1_1_1"/>
<font color="Maroon">n2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">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">n2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E2:54_1_1_2_1_1_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T10">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#E1:54_1"><i><font color="Green">E35</font></i></a>, , <a class="ref" href="neckla_3.html#T3" target="_self">Th3</a>, , <a class="ref" href="finseq_1.html#T78">FINSEQ_1:78</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">E106</font></i>: 
 <a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">G2</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a> 
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:54"><i><font color="Green">E34</font></i></a>, , <a class="ref" href="neckla_3.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E18:54_1_1_2_1_1_1"/><i><font color="Green">E107</font></i>: 
 <a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K2">sum_of</a> <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">G1</font></span>)</span>,<span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">G2</font></span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E19:54_1_1_2_1_1_1"/>
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">G1</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G1</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">G2</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G2</font> )
 <b>by </b><i><a class="ref" href="necklace.html#D9">NECKLACE:def 9</a></i>;<br/><b>hence </b><a NAME="E20:54_1_1_2_1_1_1"/>
 <a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">G</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a> 
 <b>by </b><i>, , , , <a class="ref" href="neckla_2.html#D5">NECKLA_2:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="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">E108</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">G1</font>,<font color="Maroon">G2</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">G1</font> = <font color="Maroon">G1</font> as  <a href="necklace.html#V3">irreflexive</a>  <a href="orders_2.html#L1">RelStr</a>  <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">G2</font> = <font color="Maroon">G2</font> as  <a href="necklace.html#V3">irreflexive</a>  <a href="orders_2.html#L1">RelStr</a>  <b>by </b><i>, </i>;<br/><a NAME="E4:54_1_1_2_1_1_2"/><i><font color="Green">E109</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G1</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G2</font>
 <b>by </b><i>, <a class="ref" href="neckla_2.html#D3">NECKLA_2:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">cG1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G1</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="neckla_3.html#E3:54_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="group_1.html#D14">GROUP_1:def 14</a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">cG2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G2</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="neckla_3.html#E4:54_1_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="group_1.html#D14">GROUP_1:def 14</a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><a NAME="E7:54_1_1_2_1_1_2"/><i><font color="Green">E110</font></i>: 
 <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">cG1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">cG2</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="card_2.html#T53">CARD_2:53</a></i>;<br/><a NAME="E8:54_1_1_2_1_1_2"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">cG1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n1</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T78">FINSEQ_1:78</a></i>;<br/><a NAME="E9:54_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">n1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="card_4.html#T17">CARD_4:17</a></i>;<br/><a NAME="E10:54_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E111</font></i>: 
<font color="Maroon">n1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E11:54_1_1_2_1_1_2"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">cG2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n2</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E5:54_1_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#T78">FINSEQ_1:78</a></i>;<br/><a NAME="E12:54_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">n2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="card_4.html#T17">CARD_4:17</a></i>;<br/><a NAME="E13:54_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E112</font></i>: 
<font color="Maroon">n2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E14:54_1_1_2_1_1_2"/>
<font color="Maroon">n1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">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">n1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E2:54_1_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T10">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#E1:54_1"><i><font color="Green">E35</font></i></a>, , <a class="ref" href="neckla_3.html#T3" target="_self">Th3</a>, , <a class="ref" href="finseq_1.html#T78">FINSEQ_1:78</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">E113</font></i>: 
 <a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">G1</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a> 
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:54"><i><font color="Green">E34</font></i></a>, , <a class="txt" href="neckla_3.html#E3:54_1_1_2"><i><font color="Green">E41</font></i></a></i>;<br/><a NAME="E16:54_1_1_2_1_1_2"/>
<font color="Maroon">n2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">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">n2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E2:54_1_1_2_1_1_2_2"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T10">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#E1:54_1"><i><font color="Green">E35</font></i></a>, , <a class="ref" href="neckla_3.html#T3" target="_self">Th3</a>, , <a class="ref" href="finseq_1.html#T78">FINSEQ_1:78</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">E114</font></i>: 
 <a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">G2</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a> 
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:54"><i><font color="Green">E34</font></i></a>, , <a class="ref" href="neckla_3.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E18:54_1_1_2_1_1_2"/><i><font color="Green">E115</font></i>: 
 <a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1">union_of</a> <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">G1</font></span>)</span>,<span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">G2</font></span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E19:54_1_1_2_1_1_2"/>
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">G1</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G1</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">G2</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G2</font> )
 <b>by </b><i><a class="ref" href="necklace.html#D9">NECKLACE:def 9</a></i>;<br/><b>hence </b><a NAME="E20:54_1_1_2_1_1_2"/>
 <a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">G</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a> 
 <b>by </b><i>, , , , <a class="ref" href="neckla_2.html#D5">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">E116</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S4">NAT_1:sch 4</a>();<br/></i>
<a NAME="E4:54"/>
<font color="Maroon">G</font> is <a href="group_1.html#V6">finite</a>
 
<b>by </b><i>, <a class="ref" href="neckla_2.html#T4">NECKLA_2:4</a></i>;<br/>
<a NAME="E5:54"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="group_1.html#D14">GROUP_1:def 14</a></i>;<br/>
<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:54"/><i><font color="Green">E117</font></i>: 
 <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="card_4.html#T4">CARD_4:4</a></i>;<br/>
<b>thus </b><a NAME="E8:54"/>
 <a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">G</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a> 
 <b>by </b><i>, <a class="txt" href="neckla_3.html#E2:54"><i><font color="Green">E34</font></i></a>, </i>;<br/>


</div>
