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

<b>let </b><font color="Maroon" title="c1">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">S</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="ami_1.html#V5" title="AMI_1:attr.5">IC-Ins-separated</a>  <a href="ami_1.html#L1" title="AMI_1:struct.1">AMI-Struct</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c1">N</font><br/> for <font color="Olive" title="b2">l1</font>, <font color="Olive" title="b3">l2</font> being   <a href="ami_1.html#M2" title="AMI_1:mode.2">Instruction-Location</a> of <font color="Olive" title="b1">S</font><br/> for <font color="Olive" title="b4">k</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 <br/> (  <a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">l1</font> <a href="scmnorm.html#K2" title="SCMNORM:func.2">+</a> <font color="Olive" title="b4">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">l2</font> <a href="scmnorm.html#K2" title="SCMNORM:func.2">+</a> <font color="Olive" title="b4">k</font></span>)</span> iff  <a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <font color="Olive" title="b2">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <font color="Olive" title="b3">l2</font> )</span><br/><b>let </b><font color="Maroon" title="c2">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="ami_1.html#V5" title="AMI_1:attr.5">IC-Ins-separated</a>  <a href="ami_1.html#L1" title="AMI_1:struct.1">AMI-Struct</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c1">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">l1</font>, <font color="Olive" title="b2">l2</font> being   <a href="ami_1.html#M2" title="AMI_1:mode.2">Instruction-Location</a> of <font color="Maroon" title="c2">S</font><br/> for <font color="Olive" title="b3">k</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 <br/> (  <a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">l1</font> <a href="scmnorm.html#K2" title="SCMNORM:func.2">+</a> <font color="Olive" title="b3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">l2</font> <a href="scmnorm.html#K2" title="SCMNORM:func.2">+</a> <font color="Olive" title="b3">k</font></span>)</span> iff  <a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <font color="Olive" title="b1">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <font color="Olive" title="b2">l2</font> )</span><br/><b>let </b><font color="Maroon" title="c3">l1</font>, <font color="Maroon" title="c4">l2</font> be    <a href="ami_1.html#M2" title="AMI_1:mode.2">Instruction-Location</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">k</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 <br/> (  <a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">l1</font> <a href="scmnorm.html#K2" title="SCMNORM:func.2">+</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">l2</font> <a href="scmnorm.html#K2" title="SCMNORM:func.2">+</a> <font color="Olive" title="b1">k</font></span>)</span> iff  <a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <font color="Maroon" title="c3">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <font color="Maroon" title="c4">l2</font> )</span><br/><b>let </b><font color="Maroon" title="c5">k</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">l1</font> <a href="scmnorm.html#K2" title="SCMNORM:func.2">+</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">l2</font> <a href="scmnorm.html#K2" title="SCMNORM:func.2">+</a> <font color="Maroon" title="c5">k</font></span>)</span> iff  <a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <font color="Maroon" title="c3">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <font color="Maroon" title="c4">l2</font> )</span><br/>







<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <font color="Maroon" title="c3">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <font color="Maroon" title="c4">l2</font> implies  <a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">l1</font> <a href="scmnorm.html#K2" title="SCMNORM:func.2">+</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">l2</font> <a href="scmnorm.html#K2" title="SCMNORM:func.2">+</a> <font color="Maroon" title="c5">k</font></span>)</span> )</span>
<div class="add"><b>assume </b><a NAME="E1:38_1"/><i><font color="Green" title="E17">A1</font></i>: 
 <a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">l1</font> <a href="scmnorm.html#K2" title="SCMNORM:func.2">+</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">l2</font> <a href="scmnorm.html#K2" title="SCMNORM:func.2">+</a> <font color="Maroon" title="c5">k</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <font color="Maroon" title="c3">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <font color="Maroon" title="c4">l2</font></span><br/><a NAME="E2:38_1"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <font color="Maroon" title="c2">S</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">l1</font> <a href="scmnorm.html#K2" title="SCMNORM:func.2">+</a> <font color="Maroon" title="c5">k</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">l2</font> <a href="scmnorm.html#K2" title="SCMNORM:func.2">+</a> <font color="Maroon" title="c5">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="scmnorm.html#E1:38_1"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="funct_4.html#T87" title="FUNCT_4:th.87">FUNCT_4:87</a></i>;<br/><a NAME="E3:38_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <font color="Maroon" title="c2">S</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">l1</font> <a href="scmnorm.html#K2" title="SCMNORM:func.2">+</a> <font color="Maroon" title="c5">k</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <font color="Maroon" title="c2">S</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">l2</font> <a href="scmnorm.html#K2" title="SCMNORM:func.2">+</a> <font color="Maroon" title="c5">k</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="funct_4.html#T87" title="FUNCT_4:th.87">FUNCT_4:87</a></i>;<br/><a NAME="E4:38_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <font color="Maroon" title="c2">S</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">l1</font> <a href="scmnorm.html#K2" title="SCMNORM:func.2">+</a> <font color="Maroon" title="c5">k</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <font color="Maroon" title="c2">S</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">l2</font> <a href="scmnorm.html#K2" title="SCMNORM:func.2">+</a> <font color="Maroon" title="c5">k</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T6" title="ZFMISC_1:th.6">ZFMISC_1:6</a></i>;<br/><a NAME="E5:38_1"/><b>then </b>
<font color="Maroon" title="c3">l1</font> <a href="scmnorm.html#K2" title="SCMNORM:func.2">+</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">l2</font> <a href="scmnorm.html#K2" title="SCMNORM:func.2">+</a> <font color="Maroon" title="c5">k</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E6:38_1"/>
 <a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <font color="Maroon" title="c3">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <font color="Maroon" title="c4">l2</font>
 ; <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>assume </b><a NAME="E2:38"/>
 <a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <font color="Maroon" title="c3">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <font color="Maroon" title="c4">l2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">l1</font> <a href="scmnorm.html#K2" title="SCMNORM:func.2">+</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">l2</font> <a href="scmnorm.html#K2" title="SCMNORM:func.2">+</a> <font color="Maroon" title="c5">k</font></span>)</span></span><br/>

<a NAME="E3:38"/><b>then </b>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <font color="Maroon" title="c2">S</font></span>)</span>,<font color="Maroon" title="c3">l1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <font color="Maroon" title="c4">l2</font>
 
<b>by </b><i><a class="ref" href="funct_4.html#T87" title="FUNCT_4:th.87">FUNCT_4:87</a></i>;<br/>
<a NAME="E4:38"/><b>then </b>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <font color="Maroon" title="c2">S</font></span>)</span>,<font color="Maroon" title="c3">l1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <font color="Maroon" title="c2">S</font></span>)</span>,<font color="Maroon" title="c4">l2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T87" title="FUNCT_4:th.87">FUNCT_4:87</a></i>;<br/>
<a NAME="E5:38"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <font color="Maroon" title="c2">S</font></span>)</span>,<font color="Maroon" title="c3">l1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <font color="Maroon" title="c2">S</font></span>)</span>,<font color="Maroon" title="c4">l2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T6" title="ZFMISC_1:th.6">ZFMISC_1:6</a></i>;<br/>
<b>hence </b><a NAME="E6:38"/>
 <a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">l1</font> <a href="scmnorm.html#K2" title="SCMNORM:func.2">+</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">l2</font> <a href="scmnorm.html#K2" title="SCMNORM:func.2">+</a> <font color="Maroon" title="c5">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</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>
