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

<b>set </b><font color="Maroon">P</font> = <font color="Maroon">I</font> <a href="scmfsa_4.html#K1">+*</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K6">Shift</a> <font color="Maroon">J</font>,<span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span></span>)</span>;<br/>
<a NAME="E1:38_1_1"/>
<font color="Maroon">I</font> <a href="scmfsa_4.html#K1">+*</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K6">Shift</a> <font color="Maroon">J</font>,<span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span></span>)</span> is <a href="scmpds_3.html#V1">initial</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="scmpds_3.html#D5">SCMPDS_3:def 5</a><br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:38_1_1_1"/><i><font color="Green">E30</font></i>: 
 <a href="scmpds_3.html#K2">inspos</a> <font color="Maroon">n</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">I</font> <a href="scmfsa_4.html#K1">+*</a> <span class="p2">(<span class="default"><a href="scmpds_3.html#K6">Shift</a> <font color="Maroon">J</font>,<span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E2:38_1_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 ;<br/>

<b>set </b><font color="Maroon">D</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmpds_3.html#K2">inspos</a> <span class="p3">(<span class="default"><font color="Olive">l</font> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="scmpds_3.html#K2">inspos</a> <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">J</font> </span>}</span> ;<br/>
<a NAME="E3:38_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K6">Shift</a> <font color="Maroon">J</font>,<span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</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">inspos</a> <span class="p3">(<span class="default"><font color="Olive">l</font> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="scmpds_3.html#K2">inspos</a> <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">J</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="scmpds_3.html#D7">SCMPDS_3:def 7</a></i>;<br/>
<a NAME="E4:38_1_1_1"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="scmfsa_4.html#K1">+*</a> <span class="p2">(<span class="default"><a href="scmpds_3.html#K6">Shift</a> <font color="Maroon">J</font>,<span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span></span>)</span></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">I</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmpds_3.html#K2">inspos</a> <span class="p3">(<span class="default"><font color="Olive">l</font> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="scmpds_3.html#K2">inspos</a> <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">J</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_1_1_1_1"/>
(  <a href="scmpds_3.html#K2">inspos</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> or  <a href="scmpds_3.html#K2">inspos</a> <font color="Maroon">n</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">inspos</a> <span class="p3">(<span class="default"><font color="Olive">l</font> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="scmpds_3.html#K2">inspos</a> <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">J</font> </span>}</span>  )
 </span><b>by </b><i>, , <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="38_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:38_1_1_1_1_1"/>
 <a href="scmpds_3.html#K2">inspos</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font>
 ;<br/><div class="add"><a NAME="E2:38_1_1_1_1_1"/><b>then </b>
 <a href="scmpds_3.html#K2">inspos</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font>
 <b>by </b><i>, <a class="ref" href="scmpds_3.html#D5">SCMPDS_3:def 5</a></i>;<br/><b>hence </b><a NAME="E3:38_1_1_1_1_1"/>
 <a href="scmpds_3.html#K2">inspos</a> <font color="Maroon">m</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">I</font> <a href="scmfsa_4.html#K1">+*</a> <span class="p2">(<span class="default"><a href="scmpds_3.html#K6">Shift</a> <font color="Maroon">J</font>,<span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:38_1_1_1_1_2"/>
 <a href="scmpds_3.html#K2">inspos</a> <font color="Maroon">n</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">inspos</a> <span class="p3">(<span class="default"><font color="Olive">l</font> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="scmpds_3.html#K2">inspos</a> <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">J</font> </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">l</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:38_1_1_1_1_2"/><i><font color="Green">E38</font></i>: 
 <a href="scmpds_3.html#K2">inspos</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_3.html#K2">inspos</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:38_1_1_1_1_2"/><i><font color="Green">E39</font></i>: 
 <a href="scmpds_3.html#K2">inspos</a> <font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">J</font>
 ;<br/><a NAME="E5:38_1_1_1_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_1_1_1_1_2_1_1"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">I</font> or <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">I</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:38_1_1_1_1_2_1_1_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">I</font>
 ;<br/><div class="add"><a NAME="E2:38_1_1_1_1_2_1_1_1"/><b>then </b>
 <a href="scmpds_3.html#K2">inspos</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:38_1_1_1_1_2_1_1_1"/>
 <a href="scmpds_3.html#K2">inspos</a> <font color="Maroon">m</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">I</font> <a href="scmfsa_4.html#K1">+*</a> <span class="p2">(<span class="default"><a href="scmpds_3.html#K6">Shift</a> <font color="Maroon">J</font>,<span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:38_1_1_1_1_2_1_1_2"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">I</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">l1</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:38_1_1_1_1_2_1_1_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">l1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><b>reconsider </b><font color="Maroon">l1</font> = <font color="Maroon">l1</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E5:38_1_1_1_1_2_1_1_2"/>
<font color="Maroon">l1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">l</font>
 <b>by </b><i>, , , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E6:38_1_1_1_1_2_1_1_2"/><b>then </b>
 <a href="scmpds_3.html#K2">inspos</a> <font color="Maroon">l1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">J</font>
 <b>by </b><i>, <a class="ref" href="scmpds_3.html#D5">SCMPDS_3:def 5</a></i>;<br/><b>hence </b><a NAME="E7:38_1_1_1_1_2_1_1_2"/>
 <a href="scmpds_3.html#K2">inspos</a> <font color="Maroon">m</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">inspos</a> <span class="p3">(<span class="default"><font color="Olive">l</font> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="scmpds_3.html#K2">inspos</a> <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">J</font> </span>}</span> 
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:38_1_1_1_1_2"/>
 <a href="scmpds_3.html#K2">inspos</a> <font color="Maroon">m</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">I</font> <a href="scmfsa_4.html#K1">+*</a> <span class="p2">(<span class="default"><a href="scmpds_3.html#K6">Shift</a> <font color="Maroon">J</font>,<span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span></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></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E2:38_1_1"/>
<font color="Maroon">I</font> <a href="scmfsa_4.html#K1">+*</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K6">Shift</a> <font color="Maroon">J</font>,<span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span></span>)</span> is   <a href="scmpds_4.html#NM1">Program-block</a>
 ;<br/>


</div>
