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

<b>set </b><font color="Maroon">i</font> = <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span></span>)</span> <a href="ami_3.html#K14">.--&gt;</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K13">goto</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">C</font> =  <a href="scmfsa8b.html#K1">if=0</a> <font color="Maroon">a</font>,<a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> ,<span class="p1">(<span class="default"><a href="scmfsa8b.html#K2">if&gt;0</a> <font color="Maroon">a</font>,<a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> ,<span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="scmfsa6a.html#K4">';'</a> <span class="p3">(<span class="default"><a href="scmfsa8a.html#K2">Goto</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">P</font> = <span class="p1">(<span class="default"><a href="scmfsa8b.html#K1">if=0</a> <font color="Maroon">a</font>,<a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> ,<span class="p2">(<span class="default"><a href="scmfsa8b.html#K2">if&gt;0</a> <font color="Maroon">a</font>,<a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> ,<span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="scmfsa6a.html#K4">';'</a> <span class="p4">(<span class="default"><a href="scmfsa8a.html#K2">Goto</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="scmfsa_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span></span>)</span> <a href="ami_3.html#K14">.--&gt;</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K13">goto</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E1:5_1_1"/><i><font color="Green">E32</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="scmfsa8b.html#K1">if=0</a> <font color="Maroon">a</font>,<a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> ,<span class="p2">(<span class="default"><a href="scmfsa8b.html#K2">if&gt;0</a> <font color="Maroon">a</font>,<a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> ,<span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="scmfsa6a.html#K4">';'</a> <span class="p4">(<span class="default"><a href="scmfsa8a.html#K2">Goto</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <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="nat_1.html#K1">+</a> 11
 
<b>by </b><i/>;<br/>
<a NAME="E2:5_1_1"/>
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span> <a href="nat_1.html#K1">+</a> 4 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span> <a href="nat_1.html#K1">+</a> 11
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E3:5_1_1"/><b>then </b>
 <a href="scmfsa_2.html#K5">insloc</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</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="scmfsa8b.html#K1">if=0</a> <font color="Maroon">a</font>,<a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> ,<span class="p2">(<span class="default"><a href="scmfsa8b.html#K2">if&gt;0</a> <font color="Maroon">a</font>,<a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> ,<span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="scmfsa6a.html#K4">';'</a> <span class="p4">(<span class="default"><a href="scmfsa8a.html#K2">Goto</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="scmfsa6a.html#T15">SCMFSA6A:15</a></i>;<br/>
<a NAME="E4:5_1_1"/><b>then </b><i><font color="Green">E33</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa8b.html#K1">if=0</a> <font color="Maroon">a</font>,<a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> ,<span class="p2">(<span class="default"><a href="scmfsa8b.html#K2">if&gt;0</a> <font color="Maroon">a</font>,<a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> ,<span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="scmfsa6a.html#K4">';'</a> <span class="p4">(<span class="default"><a href="scmfsa8a.html#K2">Goto</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<i><font color="Green">E34</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa8b.html#K1">if=0</a> <font color="Maroon">a</font>,<a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> ,<span class="p3">(<span class="default"><a href="scmfsa8b.html#K2">if&gt;0</a> <font color="Maroon">a</font>,<a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> ,<span class="p4">(<span class="default"><font color="Maroon">I</font> <a href="scmfsa6a.html#K4">';'</a> <span class="p5">(<span class="default"><a href="scmfsa8a.html#K2">Goto</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="scmfsa_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span></span>)</span> <a href="ami_3.html#K14">.--&gt;</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K13">goto</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="scmfsa8b.html#K1">if=0</a> <font color="Maroon">a</font>,<a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> ,<span class="p3">(<span class="default"><a href="scmfsa8b.html#K2">if&gt;0</a> <font color="Maroon">a</font>,<a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> ,<span class="p4">(<span class="default"><font color="Maroon">I</font> <a href="scmfsa6a.html#K4">';'</a> <span class="p5">(<span class="default"><a href="scmfsa8a.html#K2">Goto</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></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"><span class="p3">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span></span>)</span> <a href="ami_3.html#K14">.--&gt;</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K13">goto</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="scmfsa8b.html#K1">if=0</a> <font color="Maroon">a</font>,<a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> ,<span class="p3">(<span class="default"><a href="scmfsa8b.html#K2">if&gt;0</a> <font color="Maroon">a</font>,<a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> ,<span class="p4">(<span class="default"><font color="Maroon">I</font> <a href="scmfsa6a.html#K4">';'</a> <span class="p5">(<span class="default"><a href="scmfsa8a.html#K2">Goto</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa8b.html#K1">if=0</a> <font color="Maroon">a</font>,<a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> ,<span class="p2">(<span class="default"><a href="scmfsa8b.html#K2">if&gt;0</a> <font color="Maroon">a</font>,<a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> ,<span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="scmfsa6a.html#K4">';'</a> <span class="p4">(<span class="default"><a href="scmfsa8a.html#K2">Goto</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
;<br/>
<a NAME="E6:5_1_1"/>
<span class="p1">(<span class="default"><a href="scmfsa8b.html#K1">if=0</a> <font color="Maroon">a</font>,<a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> ,<span class="p2">(<span class="default"><a href="scmfsa8b.html#K2">if&gt;0</a> <font color="Maroon">a</font>,<a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> ,<span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="scmfsa6a.html#K4">';'</a> <span class="p4">(<span class="default"><a href="scmfsa8a.html#K2">Goto</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="scmfsa_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span></span>)</span> <a href="ami_3.html#K14">.--&gt;</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K13">goto</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span> is <a href="scmfsa_4.html#V1">initial</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="scmfsa_4.html#D4">SCMFSA_4:def 4</a><br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>thus </b><a NAME="E1:5_1_1_2"/>
(  not  <a href="scmfsa_2.html#K5">insloc</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"><span class="p2">(<span class="default"><a href="scmfsa8b.html#K1">if=0</a> <font color="Maroon">a</font>,<a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> ,<span class="p3">(<span class="default"><a href="scmfsa8b.html#K2">if&gt;0</a> <font color="Maroon">a</font>,<a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> ,<span class="p4">(<span class="default"><font color="Maroon">I</font> <a href="scmfsa6a.html#K4">';'</a> <span class="p5">(<span class="default"><a href="scmfsa8a.html#K2">Goto</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="scmfsa_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span></span>)</span> <a href="ami_3.html#K14">.--&gt;</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K13">goto</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> or <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> or  <a href="scmfsa_2.html#K5">insloc</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"><span class="p2">(<span class="default"><a href="scmfsa8b.html#K1">if=0</a> <font color="Maroon">a</font>,<a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> ,<span class="p3">(<span class="default"><a href="scmfsa8b.html#K2">if&gt;0</a> <font color="Maroon">a</font>,<a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> ,<span class="p4">(<span class="default"><font color="Maroon">I</font> <a href="scmfsa6a.html#K4">';'</a> <span class="p5">(<span class="default"><a href="scmfsa8a.html#K2">Goto</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="scmfsa_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span></span>)</span> <a href="ami_3.html#K14">.--&gt;</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K13">goto</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="scmfsa_4.html#D4">SCMFSA_4:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:5_1_1"/>
<span class="p1">(<span class="default"><a href="scmfsa8b.html#K1">if=0</a> <font color="Maroon">a</font>,<a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> ,<span class="p2">(<span class="default"><a href="scmfsa8b.html#K2">if&gt;0</a> <font color="Maroon">a</font>,<a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> ,<span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="scmfsa6a.html#K4">';'</a> <span class="p4">(<span class="default"><a href="scmfsa8a.html#K2">Goto</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="scmfsa_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span></span>)</span> <a href="ami_3.html#K14">.--&gt;</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K13">goto</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span> is   <a href="scmfsa6a.html#NM1">Macro-Instruction</a>
 ;<br/>


</div>
