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<b>let </b><font color="Maroon" title="c1">IL</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">N</font> be  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>set </b><font color="Maroon" title="c3">T</font> =  <a href="ami_1.html#K1" title="AMI_1:func.1">Trivial-AMI</a> <font color="Maroon" title="c1">IL</font>,<font color="Maroon" title="c2">N</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">la</font> = <font color="Maroon" title="c1">IL</font> as   <a href="ami_1.html#NM1" title="AMI_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="ami_1.html#K1" title="AMI_1:func.1">Trivial-AMI</a> <font color="Maroon" title="c1">IL</font>,<font color="Maroon" title="c2">N</font></span>)</span> <b>by </b><i><a class="ref" href="ami_1.html#D2" target="_self" title="AMI_1:def.2">Def2</a></i>;<br/>
<a NAME="E2:67"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">IL</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c1">IL</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">IL</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E3:67"/><b>then </b><i><font color="Green" title="E33">A1</font></i>: 
<font color="Maroon" title="c1">IL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">IL</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c1">IL</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<i><font color="Green" title="E34">A2</font></i>:  <a href="ami_1.html#K3" title="AMI_1:func.3">ObjectKind</a> <font color="Maroon" title="c4">la</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">IL</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">IL</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c1">IL</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">IL</font>
<b>by </b><i><a class="ref" href="ami_1.html#D2" target="_self" title="AMI_1:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">IL</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c1">IL</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">IL</font>
<b>by </b><i><a class="txt" href="ami_1.html#E3:67"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon" title="c1">IL</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></i>
;<br/>
<b>consider </b><font color="Maroon" title="c5">l</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">IL</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">a</font> = <font color="Maroon" title="c5">l</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_1.html#K3" title="AMI_1:func.3">ObjectKind</a> <font color="Maroon" title="c4">la</font> <b>by </b><i><a class="txt" href="ami_1.html#E4:67"><i><font color="Green" title="E34">A2</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">la</font> <a href="ami_1.html#K15" title="AMI_1:func.15">.--&gt;</a> <font color="Maroon" title="c6">a</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="ami_1.html#D27" target="_self" title="AMI_1:def.27">AMI_1:def 27</a><br/>

<b>thus </b><a NAME="E7:67"/>
not <font color="Maroon" title="c4">la</font> <a href="ami_1.html#K15" title="AMI_1:func.15">.--&gt;</a> <font color="Maroon" title="c6">a</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c7">s1</font>, <font color="Maroon" title="c8">s2</font> be   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <span class="p1">(<span class="default"><a href="ami_1.html#K1" title="AMI_1:func.1">Trivial-AMI</a> <font color="Maroon" title="c1">IL</font>,<font color="Maroon" title="c2">N</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="ami_1.html#D25" target="_self" title="AMI_1:def.25">AMI_1:def 25</a><br/>

<b>assume </b><a NAME="E8:67"/><i><font color="Green" title="E35">A3</font></i>: 
( <font color="Maroon" title="c4">la</font> <a href="ami_1.html#K15" title="AMI_1:func.15">.--&gt;</a> <font color="Maroon" title="c6">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">s1</font> &amp; <font color="Maroon" title="c4">la</font> <a href="ami_1.html#K15" title="AMI_1:func.15">.--&gt;</a> <font color="Maroon" title="c6">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">s2</font> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c7">s1</font>,$1</span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">la</font> <a href="ami_1.html#K15" title="AMI_1:func.15">.--&gt;</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c8">s2</font>,$1</span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">la</font> <a href="ami_1.html#K15" title="AMI_1:func.15">.--&gt;</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span>;<br/>
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c7">s1</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">la</font> <a href="ami_1.html#K15" title="AMI_1:func.15">.--&gt;</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c7">s1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">la</font> <a href="ami_1.html#K15" title="AMI_1:func.15">.--&gt;</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#T13" target="_self" title="AMI_1:th.13">Tx0</a></i>
<br/>.= 
<font color="Maroon" title="c8">s2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">la</font> <a href="ami_1.html#K15" title="AMI_1:func.15">.--&gt;</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="ami_1.html#E8:67"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="grfunc_1.html#T94" title="GRFUNC_1:th.94">GRFUNC_1:94</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c8">s2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">la</font> <a href="ami_1.html#K15" title="AMI_1:func.15">.--&gt;</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#T13" target="_self" title="AMI_1:th.13">Tx0</a></i>
;<br/>
<a NAME="E10:67"/><b>then </b><i><font color="Green" title="E36">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<div><i><font color="Green" title="E37">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="67_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:67_3"/><i><font color="Green" title="E38">A6</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">i</font>]
 ;<br/><b>set </b><font color="Maroon" title="c10">s3</font> =  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c7">s1</font>,<font color="Maroon" title="c9">i</font>;<br/><b>set </b><font color="Maroon" title="c11">s4</font> =  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c8">s2</font>,<font color="Maroon" title="c9">i</font>;<br/><span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c7">s1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">la</font> <a href="ami_1.html#K15" title="AMI_1:func.15">.--&gt;</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Following</a> <span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c7">s1</font>,<font color="Maroon" title="c9">i</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">la</font> <a href="ami_1.html#K15" title="AMI_1:func.15">.--&gt;</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#T14" target="_self" title="AMI_1:th.14">Tx1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c7">s1</font>,<font color="Maroon" title="c9">i</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">la</font> <a href="ami_1.html#K15" title="AMI_1:func.15">.--&gt;</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#T9" target="_self" title="AMI_1:th.9">Th9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Following</a> <span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c8">s2</font>,<font color="Maroon" title="c9">i</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">la</font> <a href="ami_1.html#K15" title="AMI_1:func.15">.--&gt;</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="ami_1.html#E1:67_3"><i><font color="Green" title="E38">A6</font></i></a>, <a class="ref" href="ami_1.html#T9" target="_self" title="AMI_1:th.9">Th9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c8">s2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">la</font> <a href="ami_1.html#K15" title="AMI_1:func.15">.--&gt;</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#T14" target="_self" title="AMI_1:th.14">Tx1</a></i>
;<br/><b>hence </b><a NAME="E3:67_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E12:67"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="ami_1.html#E10:67"><i><font color="Green" title="E36">A4</font></i></a>, <a class="txt" href="ami_1.html#E11:67"><i><font color="Green" title="E37">A5</font></i></a>);</i><br/>


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