<?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    <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>assume </b><a NAME="E1:30"/><i><font color="Green">E17</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<i><font color="Green">E18</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> = 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="necklace.html#D9">NECKLACE:def 9</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</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="ref" href="neckla_2.html#D3">NECKLA_2:def 3</a></i>
;<br/>
<i><font color="Green">E19</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="neckla_2.html#K1">union_of</a> <span class="p2">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> = 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="neckla_2.html#D2">NECKLA_2:def 2</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="necklace.html#D9">NECKLACE:def 9</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</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="ref" href="necklace.html#D9">NECKLACE:def 9</a></i>
;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> = the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="neckla_2.html#K1">union_of</a> <span class="p2">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <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>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <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>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = <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>1</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>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = <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>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<i><font color="Green">E20</font></i>: <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> = 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="neckla_2.html#D3">NECKLA_2:def 3</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="neckla_2.html#D3">NECKLA_2:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><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>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p3"><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>1</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></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><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>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="xboole_0.html#K2">\/</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="zfmisc_1.html#T121">ZFMISC_1:121</a></i>
;<br/>
<a NAME="E5:30"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> iff <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="neckla_2.html#K1">union_of</a> <span class="p2">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>set </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span>;<br/>
<b>thus </b><a NAME="E1:30_4"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="neckla_2.html#K1">union_of</a> <span class="p2">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_4_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:30_4_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <span class="p2">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="necklace.html#D9">NECKLACE:def 9</a></i>;<br/>
<a NAME="E3:30_4_1"/><b>then </b><i><font color="Green">E21</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E4:30_4_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="subset_1.html#D5">SUBSET_1:def 5</a></i>;<br/>
<a NAME="E5:30_4_1"/><b>then </b><i><font color="Green">E22</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E6:30_4_1"/><b>then </b><i><font color="Green">E23</font></i>: 
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><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>1</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></span>)</span> <a href="xboole_0.html#K2">\/</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>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="neckla_2.html#D3">NECKLA_2:def 3</a></i>;<br/>
<a NAME="E7:30_4_1"/><i><font color="Green">E24</font></i>: 
( not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> &amp; not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <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> &amp; not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</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> &amp; not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">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>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_4_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_4_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <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> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</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> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">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>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 ;<br/>

<a NAME="E2:30_4_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</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> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">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>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E3:30_4_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</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>1</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> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">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>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:30_4_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E6:30_4_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:30_4_1"/>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E3:30_4_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="neckla_2.html#D3">NECKLA_2:def 3</a></i>;<br/>
<a NAME="E9:30_4_1"/><b>then </b>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="sysrel.html#T32">SYSREL:32</a></i>;<br/>
<a NAME="E10:30_4_1"/><b>then </b><i><font color="Green">E25</font></i>: 
( not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</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="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E11:30_4_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2"><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>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><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>1</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></span>)</span> <a href="xboole_0.html#K2">\/</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>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">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="txt" href="neckla_3.html#E4:30"><i><font color="Green">E20</font></i></a>, <a class="txt" href="neckla_3.html#E5:30_4_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E12:30_4_1"/><b>then </b><i><font color="Green">E26</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</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>1</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> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">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>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">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_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_4_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_4_1_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</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> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">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>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">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> )
 </span><b>by </b><i><a class="txt" href="neckla_3.html#E12:30_4_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_4_1_2_1"><b>suppose </b></a><a NAME="E1:30_4_1_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><div class="add"><a NAME="E2:30_4_1_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E7:30_4_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:30_4_1_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="subset_1.html#D5">SUBSET_1:def 5</a></i>;<br/><a NAME="E4:30_4_1_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E10:30_4_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E5:30_4_1_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="necklace.html#D9">NECKLACE:def 9</a></i>;<br/><a NAME="E6:30_4_1_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E7:30_4_1_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="neckla_2.html#K1">union_of</a> <span class="p2">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="neckla_2.html#D2">NECKLA_2:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_4_1_2_2"><b>suppose </b></a><a NAME="E1:30_4_1_2_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</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>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:30_4_1_2_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="neckla_2.html#K1">union_of</a> <span class="p2">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E7:30_4_1"><i><font color="Green">E24</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_4_1_2_3"><b>suppose </b></a><a NAME="E1:30_4_1_2_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">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>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:30_4_1_2_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="neckla_2.html#K1">union_of</a> <span class="p2">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E7:30_4_1"><i><font color="Green">E24</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_4_1_2_4"><b>suppose </b></a><a NAME="E1:30_4_1_2_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">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>
 ;<br/><div class="add"><a NAME="E2:30_4_1_2_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">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> <a href="subset_1.html#K6">\</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="neckla_3.html#E7:30_4_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:30_4_1_2_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <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> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="subset_1.html#D5">SUBSET_1:def 5</a></i>;<br/><a NAME="E4:30_4_1_2_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E10:30_4_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E5:30_4_1_2_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="necklace.html#D9">NECKLACE:def 9</a></i>;<br/><a NAME="E6:30_4_1_2_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E7:30_4_1_2_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="neckla_2.html#K1">union_of</a> <span class="p2">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="neckla_2.html#D2">NECKLA_2:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>assume </b><a NAME="E2:30_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="neckla_2.html#K1">union_of</a> <span class="p2">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E3:30_4"/><b>then </b><i><font color="Green">E21</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="neckla_2.html#D2">NECKLA_2:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_4_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_4_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 </span><b>by </b><i><a class="txt" href="neckla_3.html#E3:30_4"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_4_2_1"><b>suppose </b></a><a NAME="E1:30_4_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:30_4_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="necklace.html#D9">NECKLACE:def 9</a></i>;<br/><a NAME="E3:30_4_2_1"/><b>then </b><i><font color="Green">E22</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a>  &amp; not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</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="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:30_4_2_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> &amp; not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</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="subset_1.html#D5">SUBSET_1:def 5</a></i>;<br/><a NAME="E5:30_4_2_1"/><b>then </b><i><font color="Green">E23</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> &amp; not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</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="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E6:30_4_2_1"/><i><font color="Green">E24</font></i>: 
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_4_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_4_2_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:30_4_2_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="neckla_2.html#D3">NECKLA_2:def 3</a></i>;<br/>
<a NAME="E3:30_4_2_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="sysrel.html#T32">SYSREL:32</a></i>;<br/>
<a NAME="E4:30_4_2_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</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="neckla_3.html#E3:30_4_2_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E5:30_4_2_1_1"/><b>then </b><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>11</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>12</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="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E6:30_4_2_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</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="txt" href="neckla_3.html#E5:30_4_2_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E7:30_4_2_1_1"/><b>then </b>
not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E5:30_4_2_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E8:30_4_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:30"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:30_4_2_1"/><i><font color="Green">E25</font></i>: 
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_4_2_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_4_2_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:30_4_2_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><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>1</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></span>)</span> <a href="xboole_0.html#K2">\/</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>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="neckla_2.html#D3">NECKLA_2:def 3</a></i>;<br/>
<a NAME="E3:30_4_2_1_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</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>1</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> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">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>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E4:30_4_2_1_2"/><b>then </b><i><font color="Green">E26</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</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> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">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>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_4_2_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_4_2_1_2_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <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> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</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> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">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>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 </span><b>by </b><i><a class="txt" href="neckla_3.html#E5:30_4_2_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="neckla_3.html#E4:30_4_2_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_4_2_1_2_1_1"><b>suppose </b></a><a NAME="E1:30_4_2_1_2_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <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/><div class="add"><a NAME="E2:30_4_2_1_2_1_1"/><b>then </b><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>11</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>12</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="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E3:30_4_2_1_2_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</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="txt" href="neckla_3.html#E5:30_4_2_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:30_4_2_1_2_1_1"/><b>then </b>
not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:30_4_2_1_2_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E5:30_4_2_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:30"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_4_2_1_2_1_2"><b>suppose </b></a><a NAME="E1:30_4_2_1_2_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</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>
 ;<br/><div class="add"><a NAME="E2:30_4_2_1_2_1_2"/><b>then </b><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>12</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="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E3:30_4_2_1_2_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</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="txt" href="neckla_3.html#E5:30_4_2_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:30_4_2_1_2_1_2"/><b>then </b>
not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:30_4_2_1_2_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E5:30_4_2_1_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:30"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_4_2_1_2_1_3"><b>suppose </b></a><a NAME="E1:30_4_2_1_2_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">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>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><div class="add"><a NAME="E2:30_4_2_1_2_1_3"/><b>then </b><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>11</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>12</sub></font> <a href="hidden.html#R2">in</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="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E3:30_4_2_1_2_1_3"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</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="txt" href="neckla_3.html#E5:30_4_2_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:30_4_2_1_2_1_3"/><b>then </b>
not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:30_4_2_1_2_1_3"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E5:30_4_2_1_2_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:30"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E8:30_4_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><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>1</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> <a href="xboole_0.html#K2">\/</a> <span class="p3"><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>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><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></span>)</span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E5:30_4_2_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E9:30_4_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E4:30"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xboole_1.html#T113">XBOOLE_1:113</a></i>;<br/><a NAME="E10:30_4_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E7:30_4_2_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E11:30_4_2_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E6:30_4_2_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="subset_1.html#D5">SUBSET_1:def 5</a></i>;<br/><a NAME="E12:30_4_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <span class="p2">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E13:30_4_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="necklace.html#D9">NECKLACE:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_4_2_2"><b>suppose </b></a><a NAME="E1:30_4_2_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:30_4_2_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="necklace.html#D9">NECKLACE:def 9</a></i>;<br/><a NAME="E3:30_4_2_2"/><b>then </b><i><font color="Green">E22</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <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> <a href="subset_1.html#K3">`</a>  &amp; not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</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="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:30_4_2_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">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> <a href="subset_1.html#K6">\</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> &amp; not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</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="ref" href="subset_1.html#D5">SUBSET_1:def 5</a></i>;<br/><a NAME="E5:30_4_2_2"/><b>then </b><i><font color="Green">E23</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">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> &amp; not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <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> &amp; not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</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="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E6:30_4_2_2"/><i><font color="Green">E24</font></i>: 
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_4_2_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_4_2_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:30_4_2_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="neckla_2.html#D3">NECKLA_2:def 3</a></i>;<br/>
<a NAME="E3:30_4_2_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="sysrel.html#T32">SYSREL:32</a></i>;<br/>
<a NAME="E4:30_4_2_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</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="txt" href="neckla_3.html#E3:30_4_2_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E5:30_4_2_2_1"/><b>then </b><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</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="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E6:30_4_2_2_1"/>
<font color="Maroon">c<sub>12</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="neckla_3.html#E5:30_4_2_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E7:30_4_2_2_1"/><b>then </b>
not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E5:30_4_2_2_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E8:30_4_2_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:30"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:30_4_2_2"/><i><font color="Green">E25</font></i>: 
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_4_2_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_4_2_2_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:30_4_2_2_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><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>1</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></span>)</span> <a href="xboole_0.html#K2">\/</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>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="neckla_2.html#D3">NECKLA_2:def 3</a></i>;<br/>
<a NAME="E3:30_4_2_2_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</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>1</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> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">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>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E4:30_4_2_2_2"/><b>then </b><i><font color="Green">E26</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</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> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">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>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_4_2_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_4_2_2_2_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</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> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">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>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 </span><b>by </b><i><a class="txt" href="neckla_3.html#E5:30_4_2_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="neckla_3.html#E4:30_4_2_2_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_4_2_2_2_1_1"><b>suppose </b></a><a NAME="E1:30_4_2_2_2_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:30_4_2_2_2_1_1"/><b>then </b><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</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="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E3:30_4_2_2_2_1_1"/>
<font color="Maroon">c<sub>11</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="neckla_3.html#E5:30_4_2_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:30_4_2_2_2_1_1"/><b>then </b>
not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:30_4_2_2_2_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E5:30_4_2_2_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:30"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_4_2_2_2_1_2"><b>suppose </b></a><a NAME="E1:30_4_2_2_2_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</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>
 ;<br/><div class="add"><a NAME="E2:30_4_2_2_2_1_2"/><b>then </b><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>12</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="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E3:30_4_2_2_2_1_2"/>
<font color="Maroon">c<sub>11</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="neckla_3.html#E5:30_4_2_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:30_4_2_2_2_1_2"/><b>then </b>
not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:30_4_2_2_2_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E5:30_4_2_2_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:30"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_4_2_2_2_1_3"><b>suppose </b></a><a NAME="E1:30_4_2_2_2_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">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>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><div class="add"><a NAME="E2:30_4_2_2_2_1_3"/><b>then </b><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>11</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>12</sub></font> <a href="hidden.html#R2">in</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="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E3:30_4_2_2_2_1_3"/>
<font color="Maroon">c<sub>12</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="neckla_3.html#E5:30_4_2_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:30_4_2_2_2_1_3"/><b>then </b>
not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:30_4_2_2_2_1_3"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E5:30_4_2_2_2_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:30"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E8:30_4_2_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E4:30"><i><font color="Green">E20</font></i></a>, <a class="txt" href="neckla_3.html#E5:30_4_2_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E9:30_4_2_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E7:30_4_2_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E10:30_4_2_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E6:30_4_2_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="subset_1.html#D5">SUBSET_1:def 5</a></i>;<br/><a NAME="E11:30_4_2_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <span class="p2">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E12:30_4_2_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="necklace.html#D9">NECKLACE:def 9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E6:30"/>
 <a href="necklace.html#K3">ComplRelStr</a> <span class="p1">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1">union_of</a> <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:30"><i><font color="Green">E18</font></i></a>, <a class="txt" href="neckla_3.html#E3:30"><i><font color="Green">E19</font></i></a>, <a class="ref" href="relat_1.html#D2">RELAT_1:def 2</a></i>;<br/>


</div>
