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<b>set </b><font color="Maroon" title="c2">I</font> = <span class="p1">(<span class="default"><a href="scmpds_3.html#K2" title="SCMPDS_3:func.2">inspos</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="ami_1.html#K30" title="AMI_1:func.30">.--&gt;</a> <font color="Maroon" title="c1">i</font>;<br/>
<a NAME="E1:1_1_1"/><i><font color="Green" title="E1">A1</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_3.html#K2" title="SCMPDS_3:func.2">inspos</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="ami_1.html#K30" title="AMI_1:func.30">.--&gt;</a> <font color="Maroon" title="c1">i</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"><span class="p2">(<span class="default"><a href="scmpds_3.html#K2" title="SCMPDS_3:func.2">inspos</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></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/>
<b>reconsider </b><font color="Maroon" title="c3">I</font> = <span class="p1">(<span class="default"><a href="scmpds_3.html#K2" title="SCMPDS_3:func.2">inspos</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="ami_1.html#K30" title="AMI_1:func.30">.--&gt;</a> <font color="Maroon" title="c1">i</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c4">I</font> = <font color="Maroon" title="c3">I</font> as   <a href="ami_1.html#NM8" title="AMI_1:NM.8">FinPartState</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>  ;<br/>
<a NAME="E4:1_1_1"/>
( <font color="Maroon" title="c4">I</font> is <a href="scmnorm.html#V1" title="SCMNORM:attr.1">initial</a> &amp; <font color="Maroon" title="c4">I</font> is <a href="ami_1.html#V15" title="AMI_1:attr.15">programmed</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:1_1_1_1"/>
<font color="Maroon" title="c4">I</font> is <a href="scmnorm.html#V1" title="SCMNORM:attr.1">initial</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">I</font> is <a href="ami_1.html#V15" title="AMI_1:attr.15">programmed</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">m</font>, <font color="Maroon" title="c6">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="scmnorm.html#D1" title="SCMNORM:def.1">SCMNORM:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c6">n</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> <font color="Maroon" title="c4">I</font> or <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">m</font> or <font color="Maroon" title="c5">m</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> <font color="Maroon" title="c4">I</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_1_1"/><i><font color="Green" title="E2">A2</font></i>: 
<font color="Maroon" title="c6">n</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> <font color="Maroon" title="c4">I</font>
 <b>and </b><br/><a NAME="E2:1_1_1_1_1"/><i><font color="Green" title="E3">A3</font></i>: 
<font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">m</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> <font color="Maroon" title="c4">I</font></span><br/>

<b>thus </b><a NAME="E3:1_1_1_1_1"/>
<font color="Maroon" title="c5">m</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> <font color="Maroon" title="c4">I</font>
 <b>by </b><i><a class="txt" href="scmpds_4.html#E1:1_1_1"><i><font color="Green" title="E1">A1</font></i></a>, <a class="txt" href="scmpds_4.html#E1:1_1_1_1_1"><i><font color="Green" title="E2">A2</font></i></a>, <a class="txt" href="scmpds_4.html#E2:1_1_1_1_1"><i><font color="Green" title="E3">A3</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:1_1_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="scmpds_4.html#E1:1_1_1"><i><font color="Green" title="E1">A1</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="ami_1.html#D40" title="AMI_1:def.40">AMI_1:def 40</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:1_1_1"/>
<span class="p1">(<span class="default"><a href="scmpds_3.html#K2" title="SCMPDS_3:func.2">inspos</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="ami_1.html#K30" title="AMI_1:func.30">.--&gt;</a> <font color="Maroon" title="c1">i</font> is   <a href="scmnorm.html#NM2" title="SCMNORM:NM.2">Program</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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