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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  <a href="ami_3.html#V1">programmed</a>  <a href="ami_1.html#M1">FinPartState</a> of  <a href="scmpds_2.html#K1">SCMPDS</a> ;<br/>



<a NAME="E1:47"/><i><font color="Green">E21</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="scmfsa_4.html#K1">+*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/>
<a NAME="E2:47"/><i><font color="Green">E22</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K6" target="_self">Shift</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmpds_3.html#K2" target="_self">inspos</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> :  <a href="scmpds_3.html#K2" target="_self">inspos</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="scmpds_3.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E3:47"/><i><font color="Green">E23</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K6" target="_self">Shift</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmpds_3.html#K2" target="_self">inspos</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> :  <a href="scmpds_3.html#K2" target="_self">inspos</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="scmpds_3.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E4:47"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="scmpds_3.html#K6" target="_self">Shift</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="scmpds_3.html#K6" target="_self">Shift</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmpds_3.html#K2" target="_self">inspos</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> :  <a href="scmpds_3.html#K2" target="_self">inspos</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:47_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="scmpds_3.html#K6" target="_self">Shift</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="scmpds_3.html#K6" target="_self">Shift</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:47_1_1"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K6" target="_self">Shift</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K6" target="_self">Shift</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:47_1_1"/><i><font color="Green">E24</font></i>: 
( ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="scmpds_3.html#K2" target="_self">inspos</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="scmpds_3.html#K2" target="_self">inspos</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="scmpds_3.html#K2" target="_self">inspos</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="scmpds_3.html#K2" target="_self">inspos</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> ) )
 <b>by </b><i><a class="txt" href="scmpds_3.html#E2:47"><i><font color="Green">E22</font></i></a>, <a class="txt" href="scmpds_3.html#E3:47"><i><font color="Green">E23</font></i></a></i>;<br/><a NAME="E5:47_1_1"/>
 <a href="scmpds_3.html#K2" target="_self">inspos</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="scmpds_3.html#E4:47_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E6:47_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmpds_3.html#K2" target="_self">inspos</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> :  <a href="scmpds_3.html#K2" target="_self">inspos</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> </span>}</span> 
 <b>by </b><i><a class="txt" href="scmpds_3.html#E4:47_1_1"><i><font color="Green">E24</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:47_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmpds_3.html#K2" target="_self">inspos</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> :  <a href="scmpds_3.html#K2" target="_self">inspos</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:47_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="scmpds_3.html#K2" target="_self">inspos</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E5:47_1"/><i><font color="Green">E25</font></i>: 
 <a href="scmpds_3.html#K2" target="_self">inspos</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/>
<a NAME="E6:47_1"/>
(  <a href="scmpds_3.html#K2" target="_self">inspos</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> or  <a href="scmpds_3.html#K2" target="_self">inspos</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="scmpds_3.html#E5:47_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E7:47_1"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K6" target="_self">Shift</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K6" target="_self">Shift</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="scmpds_3.html#E2:47"><i><font color="Green">E22</font></i></a>, <a class="txt" href="scmpds_3.html#E3:47"><i><font color="Green">E23</font></i></a>, <a class="txt" href="scmpds_3.html#E4:47_1"><i><font color="Green">E24</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:47_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="scmpds_3.html#K6" target="_self">Shift</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="scmpds_3.html#K6" target="_self">Shift</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:47"/><b>then </b><i><font color="Green">E24</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_3.html#K6" target="_self">Shift</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="scmfsa_4.html#K1">+*</a> <span class="p2">(<span class="default"><a href="scmpds_3.html#K6" target="_self">Shift</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmpds_3.html#K2" target="_self">inspos</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> :  <a href="scmpds_3.html#K2" target="_self">inspos</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="scmfsa_4.html#K1">+*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> </span>}</span> 
 
<b>by </b><i><a class="txt" href="scmpds_3.html#E1:47"><i><font color="Green">E21</font></i></a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:47_2"/><i><font color="Green">E25</font></i>: 
 <a href="scmpds_3.html#K2" target="_self">inspos</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="scmfsa_4.html#K1">+*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_2_1"/>
(  <a href="scmpds_3.html#K2" target="_self">inspos</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> or  not  <a href="scmpds_3.html#K2" target="_self">inspos</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_1_1"><b>suppose </b></a><a NAME="E1:47_2_1_1"/><i><font color="Green">E26</font></i>: 
 <a href="scmpds_3.html#K2" target="_self">inspos</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:47_2_1_1"/><b>then </b>
 <a href="scmpds_3.html#K2" target="_self">inspos</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K6" target="_self">Shift</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="scmpds_3.html#E2:47"><i><font color="Green">E22</font></i></a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_3.html#K6" target="_self">Shift</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="scmfsa_4.html#K1">+*</a> <span class="p2">(<span class="default"><a href="scmpds_3.html#K6" target="_self">Shift</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K2" target="_self">inspos</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="scmpds_3.html#K6" target="_self">Shift</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K2" target="_self">inspos</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K2" target="_self">inspos</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="scmpds_3.html#E1:47_2_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="scmpds_3.html#D7" target="_self">Def7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="scmfsa_4.html#K1">+*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K2" target="_self">inspos</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="scmpds_3.html#E1:47_2_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_1_2"><b>suppose </b></a><a NAME="E1:47_2_1_2"/><i><font color="Green">E26</font></i>: 
not  <a href="scmpds_3.html#K2" target="_self">inspos</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2_1_2_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><a NAME="E1:47_2_1_2_1"/><i><font color="Green">E27</font></i>: 
 <a href="scmpds_3.html#K2" target="_self">inspos</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="scmpds_3.html#K2" target="_self">inspos</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E2:47_2_1_2_1"/><i><font color="Green">E28</font></i>: 
 <a href="scmpds_3.html#K2" target="_self">inspos</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E3:47_2_1_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="scmpds_3.html#E1:47_2_1_2_1"><i><font color="Green">E27</font></i></a></i>;<br/><b>hence </b><a NAME="E4:47_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="scmpds_3.html#E1:47_2_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="scmpds_3.html#E2:47_2_1_2_1"><i><font color="Green">E28</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E3:47_2_1_2"/><b>then </b><i><font color="Green">E27</font></i>: 
not  <a href="scmpds_3.html#K2" target="_self">inspos</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K6" target="_self">Shift</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="scmpds_3.html#E2:47"><i><font color="Green">E22</font></i></a></i>;<br/><a NAME="E4:47_2_1_2"/>
 <a href="scmpds_3.html#K2" target="_self">inspos</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="scmpds_3.html#E1:47_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/><a NAME="E5:47_2_1_2"/><b>then </b><i><font color="Green">E28</font></i>: 
 <a href="scmpds_3.html#K2" target="_self">inspos</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="scmpds_3.html#E1:47_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/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_3.html#K6" target="_self">Shift</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="scmfsa_4.html#K1">+*</a> <span class="p2">(<span class="default"><a href="scmpds_3.html#K6" target="_self">Shift</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K2" target="_self">inspos</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="scmpds_3.html#K6" target="_self">Shift</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K2" target="_self">inspos</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scmpds_3.html#E3:47_2_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K2" target="_self">inspos</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="scmpds_3.html#E5:47_2_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="scmpds_3.html#D7" target="_self">Def7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="scmfsa_4.html#K1">+*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K2" target="_self">inspos</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="scmpds_3.html#E1:47_2_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:47"/>
 <a href="scmpds_3.html#K6" target="_self">Shift</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="scmfsa_4.html#K1">+*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K6" target="_self">Shift</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="scmfsa_4.html#K1">+*</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K6" target="_self">Shift</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="scmpds_3.html#E5:47"><i><font color="Green">E24</font></i></a>, <a class="ref" href="scmpds_3.html#D7" target="_self">Def7</a></i>;<br/>


</div>
