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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">A</font> being <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>  <a href="orders_4.html#M1" title="ORDERS_4:mode.1">Chain</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">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/><font color="Olive" title="b1">A</font>, <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> $1 <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a> ;<br/>
<a NAME="E1:17"/><i><font color="Green" title="E9">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">A</font> be  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>  <a href="orders_4.html#M1" title="ORDERS_4:mode.1">Chain</a> ;<br/>

<b>assume </b><a NAME="E1:17_1"/>
 <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">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<a NAME="E2:17_1"/><b>then </b><i><font color="Green" title="E10">A2</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c2">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>;<br/>
<b>take </b>
<font color="Maroon" title="c2">f</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D8" title="WAYBEL_1:def.8">WAYBEL_1:def 8</a><br/>

<a NAME="E4:17_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/>
<a NAME="E5:17_1"/><b>then </b>
( <font color="Maroon" title="c1">A</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> &amp;  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> )
 
<b>by </b><i><a class="txt" href="orders_4.html#E2:17_1"><i><font color="Green" title="E10">A2</font></i></a>, <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="E6:17_1"/>
<font color="Maroon" title="c2">f</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a>
 <b>by </b><i><a class="ref" href="waybel_0.html#D38" title="WAYBEL_0:def.38">WAYBEL_0:def 38</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:17"/><i><font color="Green" title="E10">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="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><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="E1:17_2"/><i><font color="Green" title="E11">A4</font></i>: 
for <font color="Olive" title="b1">A</font> being <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>  <a href="orders_4.html#M1" title="ORDERS_4:mode.1">Chain</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">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Olive" title="b1">A</font>, <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c1">n</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a> 
 ;<br/>

<b>let </b><font color="Maroon" title="c2">A</font> be  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>  <a href="orders_4.html#M1" title="ORDERS_4:mode.1">Chain</a> ;<br/>

<b>assume </b><a NAME="E2:17_2"/><i><font color="Green" title="E12">A5</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="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/>

<a NAME="E3:17_2"/>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/>
<a NAME="E4:17_2"/><b>then </b>
<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E5:17_2"/><b>then </b><i><font color="Green" title="E13">A6</font></i>: 
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 or <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></i>;<br/>
<a NAME="E6:17_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="orders_4.html#E2:17_2"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="card_4.html#T17" title="CARD_4:th.17">CARD_4:17</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">A</font> = <font color="Maroon" title="c2">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="orders_4.html#M1" title="ORDERS_4:mode.1">Chain</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>set </b><font color="Maroon" title="c4">b</font> =  <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c3">A</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_2_1"/>
( <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><b>by </b><i><a class="txt" href="orders_4.html#E5:17_2"><i><font color="Green" title="E13">A6</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_1_1"><b>suppose </b></a><a NAME="E1:17_2_1_1"/><i><font color="Green" title="E14">A7</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:17_2_1_1"/><i><font color="Green" title="E15">A8</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">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="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="orders_4.html#E2:17_2"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="card_2.html#T60" title="CARD_2:th.60">CARD_2:60</a></i>;<br/><a NAME="E4:17_2_1_1"/>
<font color="Maroon" title="c3">A</font>, <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> 1 <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c6">g</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">A</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>
<a NAME="E1:17_2_1_1_1"/><i><font color="Green" title="E16">A9</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">A</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T14" title="FUNCOP_1:th.14">FUNCOP_1:14</a></i>;<br/>
<a NAME="E2:17_2_1_1_1"/><i><font color="Green" title="E17">A10</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</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 <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a>, <a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">g</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">A</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">A</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> 1</span>)</span> ;<br/>
<a NAME="E4:17_2_1_1_1"/>
<font color="Maroon" title="c7">g</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:17_2_1_1_1_1"/><i><font color="Green" title="E18">A11</font></i>: 
<font color="Maroon" title="c7">g</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x1</font>, <font color="Maroon" title="c9">x2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">A</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D1" title="WAYBEL_1:def.1">WAYBEL_1:def 1</a><br/>

<b>assume </b><a NAME="E1:17_2_1_1_1_1_1"/>
<font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x2</font>
 ;<br/>

<a NAME="E2:17_2_1_1_1_1_1"/>
( <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Maroon" title="c9">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> )
 
<b>by </b><i><a class="txt" href="orders_4.html#E3:17_2_1_1"><i><font color="Green" title="E15">A8</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:17_2_1_1_1_1_1"/>
<font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x2</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:17_2_1_1_1_1"/>
for <font color="Olive" title="b1">e</font>, <font color="Olive" title="b2">f</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">A</font> holds <br/> ( <font color="Olive" title="b1">e</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">f</font> iff <font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">e</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">f</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">e</font>, <font color="Maroon" title="c9">f</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">A</font>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:17_2_1_1_1_1_2_1"/>
<font color="Maroon" title="c8">e</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c9">f</font>
 ;<br/><a NAME="E2:17_2_1_1_1_1_2_1"/>
( <font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><b>hence </b><a NAME="E3:17_2_1_1_1_1_2_1"/>
<font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">e</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">f</font>
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:17_2_1_1_1_1_2"/>
<font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">e</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">f</font>
 ;<br/>

<a NAME="E3:17_2_1_1_1_1_2"/>
<font color="Maroon" title="c8">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font>
 
<b>by </b><i><a class="txt" href="orders_4.html#E3:17_2_1_1"><i><font color="Green" title="E15">A8</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:17_2_1_1_1_1_2"/>
<font color="Maroon" title="c8">e</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c9">f</font>
 <b>by </b><i><a class="txt" href="orders_4.html#E3:17_2_1_1"><i><font color="Green" title="E15">A8</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:17_2_1_1_1_1"/>
<font color="Maroon" title="c7">g</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a>
 <b>by </b><i><a class="txt" href="orders_4.html#E1:17_2_1_1_1"><i><font color="Green" title="E16">A9</font></i></a>, <a class="txt" href="orders_4.html#E2:17_2_1_1_1"><i><font color="Green" title="E17">A10</font></i></a>, <a class="txt" href="orders_4.html#E1:17_2_1_1_1_1"><i><font color="Green" title="E18">A11</font></i></a>, <a class="ref" href="waybel_0.html#T66" title="WAYBEL_0:th.66">WAYBEL_0:66</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:17_2_1_1_1"/>
<font color="Maroon" title="c3">A</font>, <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> 1 <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a> 
 <b>by </b><i><a class="ref" href="waybel_1.html#D8" title="WAYBEL_1:def.8">WAYBEL_1:def 8</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:17_2_1_1"/>
<font color="Maroon" title="c2">A</font>, <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a> 
 <b>by </b><i><a class="txt" href="orders_4.html#E1:17_2_1_1"><i><font color="Green" title="E14">A7</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_1_2"><b>suppose </b></a><a NAME="E1:17_2_1_2"/>
<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:17_2_1_2"/><b>then </b><i><font color="Green" title="E14">A12</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><i><font color="Green" title="E15">A13</font></i>:  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c3">A</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> = 
<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="c3">A</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"><span class="p3">(<span class="default"><a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c3">A</font></span>)</span></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/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
<b>by </b><i><a class="txt" href="orders_4.html#E2:17_2"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="card_1.html#T79" title="CARD_1:th.79">CARD_1:79</a></i>
<br/>.= 
<font color="Maroon" title="c1">n</font>

;<br/><b>then reconsider </b><font color="Maroon" title="c5">Ab</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">A</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"><span class="p2">(<span class="default"><a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c3">A</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">A</font> <b>by </b><i><a class="txt" href="orders_4.html#E2:17_2_1_2"><i><font color="Green" title="E14">A12</font></i></a>, <a class="ref" href="card_4.html#T17" title="CARD_4:th.17">CARD_4:17</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">B</font> =  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c5">Ab</font> as  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>  <a href="orders_4.html#M1" title="ORDERS_4:mode.1">Chain</a>  <b>by </b><i><a class="ref" href="orders_4.html#D1" target="_self" title="ORDERS_4:def.1">Def1</a></i>;<br/><a NAME="E6:17_2_1_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="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="orders_4.html#E3:17_2_1_2"><i><font color="Green" title="E15">A13</font></i></a>, <a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>;<br/><a NAME="E7:17_2_1_2"/><b>then </b>
<font color="Maroon" title="c6">B</font>, <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c1">n</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a> 
 <b>by </b><i><a class="txt" href="orders_4.html#E1:17_2"><i><font color="Green" title="E11">A4</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c6">B</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c1">n</font></span>)</span><b> such that </b><br/><a NAME="E9:17_2_1_2"/><i><font color="Green" title="E16">A14</font></i>: 
<font color="Maroon" title="c7">f</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a>
 <b>by </b><i><a class="ref" href="waybel_1.html#D8" title="WAYBEL_1:def.8">WAYBEL_1:def 8</a></i>;<br/><a NAME="E10:17_2_1_2"/><i><font color="Green" title="E17">A15</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">B</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="c3">A</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"><span class="p2">(<span class="default"><a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c3">A</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <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="E11:17_2_1_2"/>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E12:17_2_1_2"/><b>then </b>
<font color="Maroon" title="c1">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T40" title="NAT_1:th.40">NAT_1:40</a></i>;<br/><a NAME="E13:17_2_1_2"/><b>then </b><i><font color="Green" title="E18">A16</font></i>: 
<font color="Maroon" title="c1">n</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 <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/><a NAME="E14:17_2_1_2"/><i><font color="Green" title="E19">A17</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">B</font>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c3">A</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="orders_4.html#R1" title="ORDERS_4:pred.1">form_upper_lower_partition_of</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="orders_4.html#E10:17_2_1_2"><i><font color="Green" title="E17">A15</font></i></a>, <a class="ref" href="orders_4.html#T3" target="_self" title="ORDERS_4:th.3">Th3</a></i>;<br/><a NAME="E15:17_2_1_2"/><i><font color="Green" title="E20">A18</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/><i><font color="Green" title="E21">A19</font></i>: <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 = 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="nat_1.html#T39" title="NAT_1:th.39">NAT_1:39</a></i>
<br/>.= 
<font color="Maroon" title="c1">n</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">n</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="ordinal1.html#D1" title="ORDINAL1:def.1">ORDINAL1:def 1</a></i>
;<br/><a NAME="E17:17_2_1_2"/><b>then </b><i><font color="Green" title="E22">A20</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">n</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</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 <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="orders_4.html#E15:17_2_1_2"><i><font color="Green" title="E20">A18</font></i></a>, <a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/><a NAME="E18:17_2_1_2"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>  st <font color="Olive" title="b1">a</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 <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">n</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/><font color="Olive" title="b1">a</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b2">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c9">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>;<br/>

<b>assume </b><a NAME="E1:17_2_1_2_3"/>
( <font color="Maroon" title="c8">a</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 <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp; <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">n</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ;<br/>

<a NAME="E2:17_2_1_2_3"/><b>then </b><i><font color="Green" title="E23">A21</font></i>: 
( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/>
<a NAME="E3:17_2_1_2_3"/><b>then </b>
( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <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>  : <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> </span>}</span>  &amp; <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="ref" href="axioms.html#T30" title="AXIOMS:th.30">AXIOMS:30</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">h</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="E5:17_2_1_2_3"/><i><font color="Green" title="E24">A22</font></i>: 
( <font color="Maroon" title="c10">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> &amp; <font color="Maroon" title="c10">h</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> )
 ;<br/>
<a NAME="E6:17_2_1_2_3"/>
<font color="Maroon" title="c8">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_2_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_2_1_2_3_1"/>
not <font color="Maroon" title="c8">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">b</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:17_2_1_2_3_1"/><i><font color="Green" title="E25">A23</font></i>: 
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">a</font> &amp; not <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">b</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E4:17_2_1_2_3_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <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>  : <font color="Olive" title="b1">w</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">h</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="orders_4.html#E5:17_2_1_2_3"><i><font color="Green" title="E24">A22</font></i></a>, <a class="txt" href="orders_4.html#E3:17_2_1_2_3_1"><i><font color="Green" title="E25">A23</font></i></a>, <a class="ref" href="axioms.html#T30" title="AXIOMS:th.30">AXIOMS:30</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">w'</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="E6:17_2_1_2_3_1"/><i><font color="Green" title="E26">A24</font></i>: 
( <font color="Maroon" title="c12">w'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font> &amp; <font color="Maroon" title="c12">w'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">h</font> )
 ;<br/>
<a NAME="E7:17_2_1_2_3_1"/>
<font color="Maroon" title="c12">w'</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="orders_4.html#E5:17_2_1_2_3"><i><font color="Green" title="E24">A22</font></i></a>, <a class="txt" href="orders_4.html#E6:17_2_1_2_3_1"><i><font color="Green" title="E26">A24</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:17_2_1_2_3_1"/><b>then </b>
<font color="Maroon" title="c12">w'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">t</font> where <font color="Olive" title="b1">t</font> is    <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>  : <font color="Olive" title="b1">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E9:17_2_1_2_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="orders_4.html#E2:17_2_1_2_3"><i><font color="Green" title="E23">A21</font></i></a>, <a class="txt" href="orders_4.html#E3:17_2_1_2_3_1"><i><font color="Green" title="E25">A23</font></i></a>, <a class="txt" href="orders_4.html#E6:17_2_1_2_3_1"><i><font color="Green" title="E26">A24</font></i></a>, <a class="ref" href="axioms.html#T30" title="AXIOMS:th.30">AXIOMS:30</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:17_2_1_2_3"/><b>then </b><i><font color="Green" title="E25">A25</font></i>: 
<font color="Maroon" title="c8">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c9">b</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/>
<a NAME="E8:17_2_1_2_3"/>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b</font>
 
<b>by </b><i><a class="txt" href="orders_4.html#E2:17_2_1_2_3"><i><font color="Green" title="E23">A21</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:17_2_1_2_3"/>
<font color="Maroon" title="c8">a</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c9">b</font>
 <b>by </b><i><a class="txt" href="orders_4.html#E7:17_2_1_2_3"><i><font color="Green" title="E25">A25</font></i></a>, <a class="ref" href="orders_2.html#D10" title="ORDERS_2:def.10">ORDERS_2:def 10</a></i>;<br/>


</div><b>end;</b></div><a NAME="E19:17_2_1_2"/><b>then </b><i><font color="Green" title="E23">A26</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">n</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="orders_4.html#R1" title="ORDERS_4:pred.1">form_upper_lower_partition_of</a>  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="orders_4.html#E17:17_2_1_2"><i><font color="Green" title="E22">A20</font></i></a>, <a class="ref" href="orders_4.html#D3" target="_self" title="ORDERS_4:def.3">Def3</a></i>;<br/><b>set </b><font color="Maroon" title="c8">X</font> =  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c3">A</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/><a NAME="E20:17_2_1_2"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c3">A</font></span>)</span></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> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c3">A</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c9">b'</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c3">A</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c3">A</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/><a NAME="E22:17_2_1_2"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">n</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="orders_4.html#E16:17_2_1_2"><i><font color="Green" title="E21">A19</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c10">n'</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">n</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/><b>set </b><font color="Maroon" title="c11">X'</font> =  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c9">b'</font>;<br/><b>set </b><font color="Maroon" title="c12">Y'</font> =  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">n'</font>;<br/><a NAME="E24:17_2_1_2"/><i><font color="Green" title="E24">A27</font></i>: 
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="c9">b'</font></span>)</span> <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"><span class="p2">(<span class="default"><a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c3">A</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <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>set </b><font color="Maroon" title="c13">g</font> = 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="c9">b'</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c1">n</font>;<br/><a NAME="E25:17_2_1_2"/><i><font color="Green" title="E25">A28</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c9">b'</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c1">n</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 <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c9">b'</font></span>)</span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c14">g</font> = 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="c9">b'</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c1">n</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of 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="c9">b'</font></span>)</span> <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E27:17_2_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">n</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/><a NAME="E28:17_2_1_2"/><b>then </b><i><font color="Green" title="E26">A29</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">g</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 <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">n'</font></span>)</span>
 <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="E29:17_2_1_2"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">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 <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c9">b'</font></span>)</span> holds <br/><font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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 <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">n'</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:17_2_1_2_4"/>
<font color="Maroon" title="c15">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 <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c9">b'</font></span>)</span>
 ;<br/>

<a NAME="E2:17_2_1_2_4"/><b>then </b>
<font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">g</font>
 
<b>by </b><i><a class="txt" href="orders_4.html#E25:17_2_1_2"><i><font color="Green" title="E25">A28</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E3:17_2_1_2_4"/>
<font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">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 <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">n'</font></span>)</span>
 <b>by </b><i><a class="txt" href="orders_4.html#E28:17_2_1_2"><i><font color="Green" title="E26">A29</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E30:17_2_1_2"/><b>then </b><i><font color="Green" title="E27">A30</font></i>: 
( 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="c10">n'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n'</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="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c3">A</font></span>)</span></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> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c3">A</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c14">g</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of 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="c9">b'</font></span>)</span>,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="c10">n'</font></span>)</span> )
 <b>by </b><i><a class="txt" href="orders_4.html#E25:17_2_1_2"><i><font color="Green" title="E25">A28</font></i></a>, <a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a>, <a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c15">g</font> = <font color="Maroon" title="c14">g</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c3">A</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">n'</font></span>)</span> <b>by </b><i><a class="txt" href="orders_4.html#E24:17_2_1_2"><i><font color="Green" title="E24">A27</font></i></a></i>;<br/><a NAME="E32:17_2_1_2"/><i><font color="Green" title="E28">A31</font></i>: 
<font color="Maroon" title="c15">g</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="orders_4.html#E30:17_2_1_2"><i><font color="Green" title="E27">A30</font></i></a>, <a class="ref" href="partfun1.html#T70" title="PARTFUN1:th.70">PARTFUN1:70</a></i>;<br/><a NAME="E33:17_2_1_2"/>
<font color="Maroon" title="c15">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">n'</font>
 <b>by </b><i><a class="txt" href="orders_4.html#E30:17_2_1_2"><i><font color="Green" title="E27">A30</font></i></a>, <a class="ref" href="funct_2.html#T61" title="FUNCT_2:th.61">FUNCT_2:61</a></i>;<br/><a NAME="E34:17_2_1_2"/><b>then </b>
<font color="Maroon" title="c15">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E35:17_2_1_2"/><b>then </b><i><font color="Green" title="E29">A32</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">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 <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">n'</font></span>)</span>
 <b>by </b><i><a class="txt" href="orders_4.html#E30:17_2_1_2"><i><font color="Green" title="E27">A30</font></i></a>, <a class="ref" href="funct_2.html#T62" title="FUNCT_2:th.62">FUNCT_2:62</a></i>;<br/><a NAME="E36:17_2_1_2"/><i><font color="Green" title="E30">A33</font></i>: 
<font color="Maroon" title="c15">g</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_2_5"><b>proof </b></a><div class="add">

<a NAME="E1:17_2_1_2_5"/>
for <font color="Olive" title="b1">e</font>, <font color="Olive" title="b2">f</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c3">A</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> holds <br/> ( <font color="Olive" title="b1">e</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">f</font> iff <font color="Maroon" title="c15">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">e</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">f</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_2_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">e</font>, <font color="Maroon" title="c17">f</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c3">A</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>;<br/>

<a NAME="E1:17_2_1_2_5_1"/>
<font color="Maroon" title="c16">e</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 <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c3">A</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 
;<br/>
<a NAME="E2:17_2_1_2_5_1"/><b>then </b><i><font color="Green" title="E31">A34</font></i>: 
<font color="Maroon" title="c16">e</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"><span class="p2">(<span class="default"><a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c3">A</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<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>reconsider </b><font color="Maroon" title="c18">e1</font> = <font color="Maroon" title="c16">e</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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="c9">b'</font></span>)</span> <b>by </b><i><a class="txt" href="orders_4.html#E24:17_2_1_2"><i><font color="Green" title="E24">A27</font></i></a>, <a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>;<br/>
<a NAME="E4:17_2_1_2_5_1"/>
<font color="Maroon" title="c17">f</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 <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c3">A</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 
;<br/>
<a NAME="E5:17_2_1_2_5_1"/><b>then </b><i><font color="Green" title="E32">A35</font></i>: 
<font color="Maroon" title="c17">f</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"><span class="p2">(<span class="default"><a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c3">A</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<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>reconsider </b><font color="Maroon" title="c19">f1</font> = <font color="Maroon" title="c17">f</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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="c9">b'</font></span>)</span> <b>by </b><i><a class="txt" href="orders_4.html#E24:17_2_1_2"><i><font color="Green" title="E24">A27</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>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:17_2_1_2_5_1_1"/>
<font color="Maroon" title="c16">e</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c17">f</font>
 ;<br/><a NAME="E2:17_2_1_2_5_1_1"/><i><font color="Green" title="E33">A36</font></i>: 
<font color="Maroon" title="c15">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><a NAME="E3:17_2_1_2_5_1_1"/>
<font color="Maroon" title="c15">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><b>hence </b><a NAME="E4:17_2_1_2_5_1_1"/>
<font color="Maroon" title="c15">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">e</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">f</font>
 <b>by </b><i><a class="txt" href="orders_4.html#E2:17_2_1_2_5_1_1"><i><font color="Green" title="E33">A36</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E8:17_2_1_2_5_1"/>
<font color="Maroon" title="c15">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">e</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">f</font>
 ;<br/>

<a NAME="E9:17_2_1_2_5_1"/>
<font color="Maroon" title="c16">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="txt" href="orders_4.html#E2:17_2_1_2_5_1"><i><font color="Green" title="E31">A34</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E10:17_2_1_2_5_1"/>
<font color="Maroon" title="c16">e</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c17">f</font>
 <b>by </b><i><a class="txt" href="orders_4.html#E5:17_2_1_2_5_1"><i><font color="Green" title="E32">A35</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:17_2_1_2_5"/>
<font color="Maroon" title="c15">g</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a>
 <b>by </b><i><a class="txt" href="orders_4.html#E32:17_2_1_2"><i><font color="Green" title="E28">A31</font></i></a>, <a class="txt" href="orders_4.html#E35:17_2_1_2"><i><font color="Green" title="E29">A32</font></i></a>, <a class="ref" href="waybel_0.html#T66" title="WAYBEL_0:th.66">WAYBEL_0:66</a></i>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon" title="c16">A2</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c1">n</font></span>)</span>, <font color="Maroon" title="c17">B2</font> = <font color="Maroon" title="c10">n'</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <b>by </b><i><a class="txt" href="orders_4.html#E13:17_2_1_2"><i><font color="Green" title="E18">A16</font></i></a>, <a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/><a NAME="E38:17_2_1_2"/>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E39:17_2_1_2"/><b>then </b>
( <font color="Maroon" title="c16">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c1">n</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  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="orders_4.html#T1" target="_self" title="ORDERS_4:th.1">Th1</a>, <a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/><a NAME="E40:17_2_1_2"/><b>then </b>
(  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c16">A2</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Ab</font> &amp; <font color="Maroon" title="c15">g</font> is   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c3">A</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c17">B2</font></span>)</span> )
 <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="c18">h</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">A</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span><b> such that </b><br/><a NAME="E42:17_2_1_2"/><i><font color="Green" title="E31">A37</font></i>: 
( <font color="Maroon" title="c18">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c15">g</font> &amp; <font color="Maroon" title="c18">h</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a> )
 <b>by </b><i><a class="txt" href="orders_4.html#E9:17_2_1_2"><i><font color="Green" title="E16">A14</font></i></a>, <a class="txt" href="orders_4.html#E14:17_2_1_2"><i><font color="Green" title="E19">A17</font></i></a>, <a class="txt" href="orders_4.html#E19:17_2_1_2"><i><font color="Green" title="E23">A26</font></i></a>, <a class="txt" href="orders_4.html#E36:17_2_1_2"><i><font color="Green" title="E30">A33</font></i></a>, <a class="ref" href="orders_4.html#T5" target="_self" title="ORDERS_4:th.5">Th5</a></i>;<br/><b>thus </b><a NAME="E43:17_2_1_2"/>
<font color="Maroon" title="c2">A</font>, <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a> 
 <b>by </b><i><a class="txt" href="orders_4.html#E42:17_2_1_2"><i><font color="Green" title="E31">A37</font></i></a>, <a class="ref" href="waybel_1.html#D8" title="WAYBEL_1:def.8">WAYBEL_1:def 8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:17"/>
for <font color="Olive" title="b1">n</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">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="orders_4.html#E1:17"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="orders_4.html#E2:17"><i><font color="Green" title="E10">A3</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:17"/>
for <font color="Olive" title="b1">A</font> being <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>  <a href="orders_4.html#M1" title="ORDERS_4:mode.1">Chain</a> <br/> for <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">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="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">n</font> holds <br/><font color="Olive" title="b1">A</font>, <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Olive" title="b2">n</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a> 
 ;<br/>


</div>
