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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>set </b><font color="Maroon">c<sub>3</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> as    <a href="hidden.html#M1">set</a>  ;<br/>
<b>assume </b><a NAME="E2:34"/>
<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="waybel_1.html#R5">are_isomorphic</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E4:34"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>8</sub></font> is <a href="waybel_0.html#V23">isomorphic</a>
 <b>by </b><i><a class="ref" href="waybel_1.html#D8">WAYBEL_1:def 8</a></i>;<br/>
<a NAME="E5:34"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>8</sub></font> is <a href="orders_3.html#V5">monotone</a>
 
<b>by </b><i><a class="txt" href="necklace.html#E4:34"><i><font color="Green">E19</font></i></a>, <a class="ref" href="waybel_0.html#D38">WAYBEL_0:def 38</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as  <a href="funct_1.html#V2">one-to-one</a> <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="txt" href="necklace.html#E4:34"><i><font color="Green">E19</font></i></a>, <a class="ref" href="waybel_0.html#D38">WAYBEL_0:def 38</a></i>;<br/>
<a NAME="E7:34"/><i><font color="Green">E21</font></i>: 
<span class="p1"><a href="funct_3.html#K15">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="funct_3.html#K15">:]</a></span> is <a href="funct_1.html#V2">one-to-one</a>
 
;<br/>
<a NAME="E8:34"/><i><font color="Green">E22</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:34"/><i><font color="Green">E23</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="necklace.html#E4:34"><i><font color="Green">E19</font></i></a>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/>
<a NAME="E10:34"/>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R2">are_equipotent</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="funct_3.html#K16">:]</a></span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>7</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> =  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="funct_3.html#K16">:]</a></span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="funct_3.html#K16">:]</a></span>;<br/>
<i><font color="Green">E24</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="funct_3.html#K16">:]</a></span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="funct_3.html#K16">:]</a></span></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="necklace.html#E8:34"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<a NAME="E2:34_1"/><i><font color="Green">E25</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="funct_3.html#K16">:]</a></span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:34_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="funct_3.html#K16">:]</a></span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:34_1_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="funct_3.html#K16">:]</a></span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:34_1_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="funct_3.html#K16">:]</a></span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:34_1_2"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="necklace.html#E3:34_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E7:34_1_2"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="txt" href="necklace.html#E8:34"><i><font color="Green">E22</font></i></a>, <a class="txt" href="necklace.html#E3:34_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="necklace.html#E6:34_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E8:34_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="funct_3.html#K16">:]</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="necklace.html#E3:34_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="necklace.html#E4:34_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E9:34_1_2"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="txt" href="necklace.html#E6:34_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="necklace.html#E7:34_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>16</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="necklace.html#E3:34_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="necklace.html#E6:34_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E11:34_1_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="txt" href="necklace.html#E1:34_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="necklace.html#E3:34_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="necklace.html#E6:34_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E12:34_1_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="txt" href="necklace.html#E5:34"><i><font color="Green">E20</font></i></a>, <a class="ref" href="orders_3.html#D5">ORDERS_3:def 5</a></i>;<br/>
<b>hence </b><a NAME="E13:34_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="necklace.html#E9:34_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:34_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="funct_3.html#K16">:]</a></span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="funct_3.html#K16">:]</a></span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>7</sub></font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="funct_3.html#K16">:]</a></span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <b>by </b><i><a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="funct_3.html#K16">:]</a></span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="necklace.html#E2:34_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_2.html#T8">FUNCT_2:8</a></i>;<br/>
<a NAME="E6:34_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>14</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="necklace.html#E7:34"><i><font color="Green">E21</font></i></a>, <a class="ref" href="funct_1.html#T84">FUNCT_1:84</a></i>;<br/>
<a NAME="E7:34_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:34_1_3"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="necklace.html#E2:34_1"><i><font color="Green">E25</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:34_1_3"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:34_1_3"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E5:34_1_3"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="necklace.html#E2:34_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="necklace.html#E4:34_1_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>17</sub></font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="necklace.html#E2:34_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="necklace.html#E4:34_1_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E7:34_1_3"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="necklace.html#E2:34_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="necklace.html#E4:34_1_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>20</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:34_1_3"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E10:34_1_3"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="necklace.html#E9:34"><i><font color="Green">E23</font></i></a>, <a class="txt" href="necklace.html#E5:34_1_3"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>21</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:34_1_3"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E13:34_1_3"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="necklace.html#E9:34"><i><font color="Green">E23</font></i></a>, <a class="txt" href="necklace.html#E5:34_1_3"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>20</sub></font>, <font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>21</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="necklace.html#E9:34_1_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="necklace.html#E12:34_1_3"><i><font color="Green">E33</font></i></a></i>;<br/>
<a NAME="E15:34_1_3"/>
<font color="Maroon">c<sub>22</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>23</sub></font>
 
<b>by </b><i><a class="txt" href="necklace.html#E4:34"><i><font color="Green">E19</font></i></a>, <a class="txt" href="necklace.html#E7:34_1_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="necklace.html#E10:34_1_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="necklace.html#E13:34_1_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/>
<a NAME="E16:34_1_3"/><b>then </b><i><font color="Green">E35</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>24</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="tarski.html#K4">]</a></span>;<br/>
<a NAME="E17:34_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="necklace.html#E9:34_1_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="necklace.html#E12:34_1_3"><i><font color="Green">E33</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then </b><font color="Maroon">c<sub>15</sub></font> = 
<span class="p1"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="funct_3.html#K16">:]</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="necklace.html#E4:34_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="necklace.html#E10:34_1_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="necklace.html#E13:34_1_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_3.html#T86">FUNCT_3:86</a></i>
<br/>.= 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="necklace.html#E1:34_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="necklace.html#E16:34_1_3"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
;<br/>
<b>hence </b><a NAME="E19:34_1_3"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="necklace.html#E1:34_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="necklace.html#E16:34_1_3"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:34_1"/>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R2">are_equipotent</a> 
 <b>by </b><i><a class="txt" href="necklace.html#E1:34_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="necklace.html#E6:34_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:34"/>
 <a href="card_1.html#K1">Card</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/>


</div>
