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

<b>let </b><font color="Maroon">R</font> be  <a href="scmring1.html#V1">good</a> <a href="funcsdom.html#NM1">Ring</a>;<br/>

<b>assume </b><a NAME="E1:47"/><i><font color="Green">E45</font></i>: 
not <font color="Maroon">R</font> is <a href="realset2.html#V3">trivial</a>
 ;<br/>

<b>let </b><font color="Maroon">p</font> be  <a href="ami_1.html#V11">autonomic</a>  <a href="ami_1.html#M1">FinPartState</a> of  <a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font>;<br/>

<b>assume </b><a NAME="E2:47"/>
 <a href="ami_5.html#K7">DataPart</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E3:47"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="ami_5.html#K7">DataPart</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>;<br/>
<b>assume </b><a NAME="E4:47"/><i><font color="Green">E61</font></i>: 
not  <a href="ami_1.html#K2">IC</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E5:47"/>
not <font color="Maroon">p</font> is <a href="ami_1.html#V11">autonomic</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">d1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="ami_5.html#K7">DataPart</a> <font color="Maroon">p</font></span>)</span>;<br/>
<a NAME="E2:47_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">d1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="ami_5.html#K7">DataPart</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:47_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="ami_5.html#K7">DataPart</a> <font color="Maroon">p</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="ami_3.html#T37">AMI_3:37</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">d1</font> = <font color="Maroon">d1</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E5:47_1"/>
 <a href="ami_5.html#K7">DataPart</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="relat_1.html#K7">|</a> <a href="ami_2.html#K2">SCM-Data-Loc</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E6:47_1"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="ami_5.html#K7">DataPart</a> <font color="Maroon">p</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="ami_2.html#K2">SCM-Data-Loc</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T87">RELAT_1:87</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">d1</font> = <font color="Maroon">d1</font> as    <a href="scmring2.html#M1">Data-Location</a> of <font color="Maroon">R</font> <b>by </b><i>, <a class="ref" href="scmring2.html#T1">SCMRING2:1</a></i>;<br/>
<b>consider </b><font color="Maroon">d2</font> being    <a href="subset_1.html#M1">Element</a> of <a href="ami_2.html#K2">SCM-Data-Loc</a>  <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font></span>)</span>;<br/>
<a NAME="E9:47_1"/>
not  <a href="ami_2.html#K2">SCM-Data-Loc</a>  <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<a NAME="E10:47_1"/><b>then </b><i><font color="Green">E70</font></i>: 
<a href="ami_2.html#K2">SCM-Data-Loc</a>  <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/>
<a NAME="E11:47_1"/><b>then </b>
<font color="Maroon">d2</font> <a href="hidden.html#R2">in</a>  <a href="ami_2.html#K2">SCM-Data-Loc</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">d2</font> = <font color="Maroon">d2</font> as    <a href="scmring2.html#M1">Data-Location</a> of <font color="Maroon">R</font> <b>by </b><i><a class="ref" href="scmring2.html#T1">SCMRING2:1</a></i>;<br/>
<b>consider </b><font color="Maroon">il</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="ami_1.html#U2">Instruction-Locations</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font></span>)</span>;<br/>
<a NAME="E14:47_1"/>
not the <a href="ami_1.html#U2">Instruction-Locations</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<a NAME="E15:47_1"/><b>then </b><i><font color="Green">E74</font></i>: 
the <a href="ami_1.html#U2">Instruction-Locations</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">il</font> = <font color="Maroon">il</font> as   <a href="ami_1.html#NM2">Instruction-Location</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span> <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">e</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E18:47_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="anproj_1.html#D8">ANPROJ_1:def 8</a></i>;<br/>
<b>set </b><font color="Maroon">p1</font> = <font color="Maroon">p</font> <a href="ami_1.html#K17">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <span class="p4">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">p2</font> = <font color="Maroon">p</font> <a href="ami_1.html#K17">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon">s1</font> being   <a href="ami_1.html#NM4">State</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span><b> such that </b><br/><a NAME="E20:47_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">p</font> <a href="ami_1.html#K17">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <span class="p4">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">s1</font>
 <b>by </b><i><a class="ref" href="ami_3.html#T39">AMI_3:39</a></i>;<br/>
<b>consider </b><font color="Maroon">s2</font> being   <a href="ami_1.html#NM4">State</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span><b> such that </b><br/><a NAME="E22:47_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">p</font> <a href="ami_1.html#K17">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">s2</font>
 <b>by </b><i><a class="ref" href="ami_3.html#T39">AMI_3:39</a></i>;<br/>
<b>take </b>
<font color="Maroon">s1</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="ami_1.html#D25">AMI_1:def 25</a><br/>

<b>take </b>
<font color="Maroon">s2</font>
;<br/>

<a NAME="E23:47_1"/><i><font color="Green">E79</font></i>: 
not <font color="Maroon">d2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E24:47_1"/><i><font color="Green">E80</font></i>: 
not <font color="Maroon">il</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="txt" href="scmring4.html#E1"><i><font color="Green">Lemma38</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E25:47_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <span class="p4">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1"><b>proof </b></a><div class="add">

 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <span class="p4">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span></span>)</span> = 
<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"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <span class="p4">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></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"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></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"><span class="p3">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <span class="p4">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2">IC</a> <span class="p3">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i><a class="ref" href="ami_3.html#T34">AMI_3:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <span class="p4">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2">IC</a> <span class="p3">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></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"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">il</font></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <span class="p4">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2">IC</a> <span class="p3">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">il</font></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d2</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2">IC</a> <span class="p3">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>
;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K3">/\</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"><span class="p4">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p5">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p4">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <span class="p5">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">il</font></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d2</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K2">IC</a> <span class="p4">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">il</font></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d2</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">il</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d2</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">il</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i>, </i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i>, </i>
;<br/> <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/>


</div><b>end;</b></div>
<a NAME="E26:47_1"/><b>then </b>
<font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">p</font> <a href="ami_1.html#K17">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <span class="p4">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T33">FUNCT_4:33</a></i>;<br/>
<b>hence </b><a NAME="E27:47_1"/>
<font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">s1</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>

<a NAME="E28:47_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2"><b>proof </b></a><div class="add">

 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span></span>)</span> = 
<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"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <font color="Maroon">e</font></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"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></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"><span class="p3">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <font color="Maroon">e</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2">IC</a> <span class="p3">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i><a class="ref" href="ami_3.html#T34">AMI_3:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <font color="Maroon">e</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2">IC</a> <span class="p3">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></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"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">il</font></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <font color="Maroon">e</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2">IC</a> <span class="p3">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">il</font></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d2</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2">IC</a> <span class="p3">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>
;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K3">/\</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"><span class="p4">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p5">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p4">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">il</font></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d2</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K2">IC</a> <span class="p4">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">il</font></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d2</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">il</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d2</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">il</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i>, </i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i>, </i>
;<br/> <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/>


</div><b>end;</b></div>
<a NAME="E29:47_1"/><b>then </b>
<font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">p</font> <a href="ami_1.html#K17">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T33">FUNCT_4:33</a></i>;<br/>
<b>hence </b><a NAME="E30:47_1"/>
<font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">s2</font>
 <b>by </b><i><a class="txt" href="scmring4.html#E2"><i><font color="Green">Lemma41</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>

<b>take </b>
1
;<br/>

<a NAME="E31:47_1"/>
 <a href="ami_5.html#K7">DataPart</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="ami_5.html#T62">AMI_5:62</a></i>;<br/>
<a NAME="E32:47_1"/><b>then </b><i><font color="Green">E81</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="ami_5.html#K7">DataPart</a> <font color="Maroon">p</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T25">RELAT_1:25</a></i>;<br/>
<a NAME="E33:47_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="ami_3.html#T36">AMI_3:36</a></i>;<br/>
<a NAME="E34:47_1"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="ami_3.html#T37">AMI_3:37</a></i>;<br/>
<a NAME="E35:47_1"/><b>then </b><i><font color="Green">E82</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> 1</span>)</span> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E36:47_1"/><i><font color="Green">E83</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2">IC</a> <span class="p3">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="ami_3.html#T34">AMI_3:34</a></i>;<br/>
<a NAME="E37:47_1"/><b>then </b><i><font color="Green">E84</font></i>: 
 <a href="ami_1.html#K2">IC</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</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="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E38:47_1"/><i><font color="Green">E85</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <span class="p4">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</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"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <span class="p4">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></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"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/>
<a NAME="E39:47_1"/><b>then </b><i><font color="Green">E113</font></i>: 
 <a href="ami_1.html#K2">IC</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span> <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"><span class="p3">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <span class="p4">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="scmring4.html#T2" target="_self">Th2</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E40:47_1"/><i><font color="Green">E114</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p5">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p4">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <span class="p5">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</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">p</font></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"><span class="p4">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p5">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p4">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <span class="p5">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></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/>
<a NAME="E41:47_1"/><b>then </b><i><font color="Green">E115</font></i>: 
 <a href="ami_1.html#K2">IC</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p5">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p4">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <span class="p5">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="scmring4.html#T5" target="_self">Th5</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<i><font color="Green">E116</font></i>:  <a href="ami_1.html#K6">IC</a> <font color="Maroon">s1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p5">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p4">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <span class="p5">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <span class="p2">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <span class="p4">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <span class="p2">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="scmring4.html#T5" target="_self">Th5</a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <span class="p2">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="scmring4.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">il</font>
<b>by </b><i><a class="ref" href="ami_3.html#T50">AMI_3:50</a></i>
;<br/>
<a NAME="E43:47_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">il</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<a NAME="E44:47_1"/><b>then </b><i><font color="Green">E117</font></i>: 
<font color="Maroon">il</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">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E45:47_1"/><i><font color="Green">E118</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d2</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<a NAME="E46:47_1"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">il</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E47:47_1"/><b>then </b><i><font color="Green">E120</font></i>: 
not <font color="Maroon">il</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">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E48:47_1"/><i><font color="Green">E121</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</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> <span class="p2">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></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"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></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/>
<a NAME="E49:47_1"/><b>then </b><i><font color="Green">E122</font></i>: 
<font color="Maroon">il</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"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="scmring4.html#E11"><i><font color="Green">Lemma53</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E50:47_1"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">il</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ami_1.html#K2">IC</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="ami_1.html#T48">AMI_1:48</a></i>;<br/>
<a NAME="E51:47_1"/><b>then </b><i><font color="Green">E124</font></i>: 
not <font color="Maroon">il</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E52:47_1"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">il</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"><span class="p3">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <span class="p4">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="scmring4.html#E12"><i><font color="Green">Lemma58</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E53:47_1"/><b>then </b>
<font color="Maroon">il</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">p</font> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p5">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p4">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <span class="p5">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="scmring4.html#T7" target="_self">Th7</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>then </b><i><font color="Green">E126</font></i>: <font color="Maroon">s1</font> <a href="ami_1.html#K13">.</a> <font color="Maroon">il</font> = 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p5">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p4">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <span class="p5">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">il</font>
<b>by </b><i>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <span class="p4">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">il</font>
<b>by </b><i><a class="txt" href="scmring4.html#E4:47"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">il</font>
<b>by </b><i><a class="ref" href="scmring4.html#T8" target="_self">Th8</a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">il</font>
<b>by </b><i>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/>
<a NAME="E55:47_1"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">d2</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">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E56:47_1"/><b>then </b><i><font color="Green">E129</font></i>: 
<font color="Maroon">d2</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"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="scmring4.html#E3:47"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E57:47_1"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">d2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ami_1.html#K2">IC</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="scmring3.html#T3">SCMRING3:3</a></i>;<br/>
<a NAME="E58:47_1"/><b>then </b><i><font color="Green">E131</font></i>: 
not <font color="Maroon">d2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E59:47_1"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">d2</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"><span class="p3">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <span class="p4">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E60:47_1"/><b>then </b>
<font color="Maroon">d2</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">p</font> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p5">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p4">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <span class="p5">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="scmring4.html#T7" target="_self">Th7</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>then </b><i><font color="Green">E133</font></i>: <font color="Maroon">s1</font> <a href="scmring2.html#K3">.</a> <font color="Maroon">d2</font> = 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p5">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p4">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <span class="p5">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">d2</font>
<b>by </b><i>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <span class="p4">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">d2</font>
<b>by </b><i><a class="ref" href="scmring4.html#T11" target="_self">Th11</a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">d2</font>
<b>by </b><i><a class="ref" href="scmring4.html#T10" target="_self">Th10</a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">d2</font>
<b>by </b><i><a class="ref" href="scmring4.html#T9" target="_self">Th9</a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="scmring2.html#K3">.</a> <font color="Maroon">d1</font> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K9">Following</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> 0</span>)</span></span>)</span> <a href="scmring2.html#K3">.</a> <font color="Maroon">d1</font>
<b>by </b><i><a class="ref" href="ami_1.html#D19">AMI_1:def 19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K9">Following</a> <font color="Maroon">s1</font></span>)</span> <a href="scmring2.html#K3">.</a> <font color="Maroon">d1</font>
<b>by </b><i><a class="ref" href="ami_1.html#D19">AMI_1:def 19</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
<b>by </b><i>, <a class="txt" href="scmring4.html#E2:47_1"><i><font color="Green">E62</font></i></a>, , <a class="ref" href="scmring2.html#T13">SCMRING2:13</a></i>
;<br/>
<a NAME="E63:47_1"/><b>then </b><i><font color="Green">E134</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> 1</span>)</span> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">d1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="scmring4.html#T1" target="_self">Th1</a>, <a class="txt" href="scmring4.html#E1:47"><i><font color="Green">E45</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E64:47_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="ami_3.html#T36">AMI_3:36</a></i>;<br/>
<a NAME="E65:47_1"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="ami_3.html#T37">AMI_3:37</a></i>;<br/>
<a NAME="E66:47_1"/><b>then </b><i><font color="Green">E135</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> 1</span>)</span> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E67:47_1"/><i><font color="Green">E136</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2">IC</a> <span class="p3">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="ami_3.html#T34">AMI_3:34</a></i>;<br/>
<a NAME="E68:47_1"/><b>then </b><i><font color="Green">E137</font></i>: 
 <a href="ami_1.html#K2">IC</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</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="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E69:47_1"/><i><font color="Green">E138</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</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"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <font color="Maroon">e</font></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"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/>
<a NAME="E70:47_1"/><b>then </b><i><font color="Green">E139</font></i>: 
 <a href="ami_1.html#K2">IC</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span> <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"><span class="p3">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E71:47_1"/><i><font color="Green">E140</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p5">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p4">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</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">p</font></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"><span class="p4">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p5">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p4">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></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/>
<a NAME="E72:47_1"/><b>then </b><i><font color="Green">E141</font></i>: 
 <a href="ami_1.html#K2">IC</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p5">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p4">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="scmring4.html#E15:47_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<i><font color="Green">E142</font></i>:  <a href="ami_1.html#K6">IC</a> <font color="Maroon">s2</font> = 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p5">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p4">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <span class="p2">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scmring4.html#E2"><i><font color="Green">Lemma41</font></i></a>, <a class="txt" href="scmring4.html#E18:47_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <span class="p2">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scmring4.html#E15:47_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <span class="p2">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</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">il</font>
<b>by </b><i><a class="ref" href="ami_3.html#T50">AMI_3:50</a></i>
;<br/>
<a NAME="E74:47_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">il</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<a NAME="E75:47_1"/><b>then </b><i><font color="Green">E143</font></i>: 
<font color="Maroon">il</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">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E76:47_1"/><i><font color="Green">E144</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <font color="Maroon">e</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d2</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<a NAME="E77:47_1"/><b>then </b><i><font color="Green">E145</font></i>: 
not <font color="Maroon">il</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">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <font color="Maroon">e</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E78:47_1"/><i><font color="Green">E146</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></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"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <font color="Maroon">e</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/>
<a NAME="E79:47_1"/><b>then </b><i><font color="Green">E147</font></i>: 
<font color="Maroon">il</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"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <font color="Maroon">e</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="scmring4.html#E22:47_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E80:47_1"/><i><font color="Green">E148</font></i>: 
not <font color="Maroon">il</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span>
 
<b>by </b><i><a class="txt" href="scmring4.html#E13"><i><font color="Green">Lemma59</font></i></a>, <a class="ref" href="scmring4.html#T13" target="_self">Th13</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E81:47_1"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">il</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"><span class="p3">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="scmring4.html#E35:47_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E82:47_1"/><b>then </b>
<font color="Maroon">il</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">p</font> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p5">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p4">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</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/>
<b>then </b><i><font color="Green">E150</font></i>: <font color="Maroon">s2</font> <a href="ami_1.html#K13">.</a> <font color="Maroon">il</font> = 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p5">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p4">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">il</font>
<b>by </b><i><a class="txt" href="scmring4.html#E2"><i><font color="Green">Lemma41</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">il</font>
<b>by </b><i><a class="txt" href="scmring4.html#E37:47_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">il</font>
<b>by </b><i><a class="txt" href="scmring4.html#E36:47_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">il</font>
<b>by </b><i><a class="txt" href="scmring4.html#E24:47_1"><i><font color="Green">E80</font></i></a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/>
<a NAME="E84:47_1"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">d2</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">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <font color="Maroon">e</font></span>)</span>
 
<b>by </b><i><a class="txt" href="scmring4.html#E23:47_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E85:47_1"/><b>then </b><i><font color="Green">E159</font></i>: 
<font color="Maroon">d2</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"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <font color="Maroon">e</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="scmring4.html#E32:47_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E86:47_1"/><i><font color="Green">E160</font></i>: 
not <font color="Maroon">d2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="scmring4.html#T13" target="_self">Th13</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E87:47_1"/><i><font color="Green">E161</font></i>: 
<font color="Maroon">d2</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"><span class="p3">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E88:47_1"/><b>then </b>
<font color="Maroon">d2</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">p</font> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p5">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p4">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</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/>
<b>then </b><i><font color="Green">E162</font></i>: <font color="Maroon">s2</font> <a href="scmring2.html#K3">.</a> <font color="Maroon">d2</font> = 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p5">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p4">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">d2</font>
<b>by </b><i><a class="txt" href="scmring4.html#E2"><i><font color="Green">Lemma41</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <font color="Maroon">il</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">d2</font>
<b>by </b><i>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">il</font> <a href="ami_3.html#K14">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">d1</font> <a href="scmring2.html#K4">:=</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">d2</font>
<b>by </b><i>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">d2</font> <a href="scmring4.html#K2" target="_self">.--&gt;</a> <font color="Maroon">e</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">d2</font>
<b>by </b><i>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="scmring2.html#K3">.</a> <font color="Maroon">d1</font> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K9">Following</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> 0</span>)</span></span>)</span> <a href="scmring2.html#K3">.</a> <font color="Maroon">d1</font>
<b>by </b><i><a class="ref" href="ami_1.html#D19">AMI_1:def 19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K9">Following</a> <font color="Maroon">s2</font></span>)</span> <a href="scmring2.html#K3">.</a> <font color="Maroon">d1</font>
<b>by </b><i><a class="ref" href="ami_1.html#D19">AMI_1:def 19</a></i>
<br/>.= 
<font color="Maroon">e</font>
<b>by </b><i><a class="txt" href="scmring4.html#E20:47_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="scmring4.html#E38:47_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="scmring4.html#T14" target="_self">Th14</a>, <a class="ref" href="scmring2.html#T13">SCMRING2:13</a></i>
;<br/>
<b>hence </b><a NAME="E91:47_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> 1</span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> 1</span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="scmring4.html#T1" target="_self">Th1</a>, , <a class="txt" href="scmring4.html#E10:47_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:47"/>
contradiction
 ;<br/>


</div>
