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

<b>let </b><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">j</font> implies for <font color="Olive" title="b1">N</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <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/> for <font color="Olive" title="b2">S</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="ami_1.html#V4" title="AMI_1:attr.4">halting</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="Olive" title="b1">N</font><br/> for <font color="Olive" title="b3">f</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instructions</a> of <font color="Olive" title="b2">S</font><br/> for <font color="Olive" title="b4">s</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <font color="Olive" title="b2">S</font>  st  <a href="scmnorm.html#K9" title="SCMNORM:func.9">CurInstr</a> <font color="Olive" title="b3">f</font>,<span class="p1">(<span class="default"><a href="scmnorm.html#K11" title="SCMNORM:func.11">Comput</a> <font color="Olive" title="b3">f</font>,<font color="Olive" title="b4">s</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K5" title="AMI_1:func.5">halt</a> <font color="Olive" title="b2">S</font> holds <br/> <a href="scmnorm.html#K11" title="SCMNORM:func.11">Comput</a> <font color="Olive" title="b3">f</font>,<font color="Olive" title="b4">s</font>,<font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmnorm.html#K11" title="SCMNORM:func.11">Comput</a> <font color="Olive" title="b3">f</font>,<font color="Olive" title="b4">s</font>,<font color="Maroon" title="c1">i</font> )</span><br/>



<b>assume </b><a NAME="E1:33"/>
<font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">j</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">N</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <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/> for <font color="Olive" title="b2">S</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="ami_1.html#V4" title="AMI_1:attr.4">halting</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="Olive" title="b1">N</font><br/> for <font color="Olive" title="b3">f</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instructions</a> of <font color="Olive" title="b2">S</font><br/> for <font color="Olive" title="b4">s</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <font color="Olive" title="b2">S</font>  st  <a href="scmnorm.html#K9" title="SCMNORM:func.9">CurInstr</a> <font color="Olive" title="b3">f</font>,<span class="p1">(<span class="default"><a href="scmnorm.html#K11" title="SCMNORM:func.11">Comput</a> <font color="Olive" title="b3">f</font>,<font color="Olive" title="b4">s</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K5" title="AMI_1:func.5">halt</a> <font color="Olive" title="b2">S</font> holds <br/> <a href="scmnorm.html#K11" title="SCMNORM:func.11">Comput</a> <font color="Olive" title="b3">f</font>,<font color="Olive" title="b4">s</font>,<font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmnorm.html#K11" title="SCMNORM:func.11">Comput</a> <font color="Olive" title="b3">f</font>,<font color="Olive" title="b4">s</font>,<font color="Maroon" title="c1">i</font></span><br/>

<b>then consider </b><font color="Maroon" title="c3">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:33"/><i><font color="Green" title="E14">A1</font></i>: 
<font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">k</font> = <font color="Maroon" title="c3">k</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E5:33"/><i><font color="Green" title="E15">A2</font></i>: 
<font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">k</font>
 
<b>by </b><i><a class="txt" href="scmnorm.html#E3:33"><i><font color="Green" title="E14">A1</font></i></a></i>;<br/>
<b>let </b><font color="Maroon" title="c5">N</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <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#V4" title="AMI_1:attr.4">halting</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="c5">N</font><br/> for <font color="Olive" title="b2">f</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instructions</a> of <font color="Olive" title="b1">S</font><br/> for <font color="Olive" title="b3">s</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <font color="Olive" title="b1">S</font>  st  <a href="scmnorm.html#K9" title="SCMNORM:func.9">CurInstr</a> <font color="Olive" title="b2">f</font>,<span class="p1">(<span class="default"><a href="scmnorm.html#K11" title="SCMNORM:func.11">Comput</a> <font color="Olive" title="b2">f</font>,<font color="Olive" title="b3">s</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K5" title="AMI_1:func.5">halt</a> <font color="Olive" title="b1">S</font> holds <br/> <a href="scmnorm.html#K11" title="SCMNORM:func.11">Comput</a> <font color="Olive" title="b2">f</font>,<font color="Olive" title="b3">s</font>,<font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmnorm.html#K11" title="SCMNORM:func.11">Comput</a> <font color="Olive" title="b2">f</font>,<font color="Olive" title="b3">s</font>,<font color="Maroon" title="c1">i</font></span><br/><b>let </b><font color="Maroon" title="c6">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="ami_1.html#V4" title="AMI_1:attr.4">halting</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="c5">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">f</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instructions</a> of <font color="Maroon" title="c6">S</font><br/> for <font color="Olive" title="b2">s</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <font color="Maroon" title="c6">S</font>  st  <a href="scmnorm.html#K9" title="SCMNORM:func.9">CurInstr</a> <font color="Olive" title="b1">f</font>,<span class="p1">(<span class="default"><a href="scmnorm.html#K11" title="SCMNORM:func.11">Comput</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">s</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K5" title="AMI_1:func.5">halt</a> <font color="Maroon" title="c6">S</font> holds <br/> <a href="scmnorm.html#K11" title="SCMNORM:func.11">Comput</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">s</font>,<font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmnorm.html#K11" title="SCMNORM:func.11">Comput</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">s</font>,<font color="Maroon" title="c1">i</font></span><br/><b>let </b><font color="Maroon" title="c7">f</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instructions</a> of <font color="Maroon" title="c6">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">s</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <font color="Maroon" title="c6">S</font>  st  <a href="scmnorm.html#K9" title="SCMNORM:func.9">CurInstr</a> <font color="Maroon" title="c7">f</font>,<span class="p1">(<span class="default"><a href="scmnorm.html#K11" title="SCMNORM:func.11">Comput</a> <font color="Maroon" title="c7">f</font>,<font color="Olive" title="b1">s</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K5" title="AMI_1:func.5">halt</a> <font color="Maroon" title="c6">S</font> holds <br/> <a href="scmnorm.html#K11" title="SCMNORM:func.11">Comput</a> <font color="Maroon" title="c7">f</font>,<font color="Olive" title="b1">s</font>,<font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmnorm.html#K11" title="SCMNORM:func.11">Comput</a> <font color="Maroon" title="c7">f</font>,<font color="Olive" title="b1">s</font>,<font color="Maroon" title="c1">i</font></span><br/><b>let </b><font color="Maroon" title="c8">s</font> be   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <font color="Maroon" title="c6">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="scmnorm.html#K9" title="SCMNORM:func.9">CurInstr</a> <font color="Maroon" title="c7">f</font>,<span class="p1">(<span class="default"><a href="scmnorm.html#K11" title="SCMNORM:func.11">Comput</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">s</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K5" title="AMI_1:func.5">halt</a> <font color="Maroon" title="c6">S</font> implies  <a href="scmnorm.html#K11" title="SCMNORM:func.11">Comput</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">s</font>,<font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmnorm.html#K11" title="SCMNORM:func.11">Comput</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">s</font>,<font color="Maroon" title="c1">i</font> )</span><br/>







<b>assume </b><a NAME="E6:33"/><i><font color="Green" title="E16">A3</font></i>: 
 <a href="scmnorm.html#K9" title="SCMNORM:func.9">CurInstr</a> <font color="Maroon" title="c7">f</font>,<span class="p1">(<span class="default"><a href="scmnorm.html#K11" title="SCMNORM:func.11">Comput</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">s</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K5" title="AMI_1:func.5">halt</a> <font color="Maroon" title="c6">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="scmnorm.html#K11" title="SCMNORM:func.11">Comput</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">s</font>,<font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmnorm.html#K11" title="SCMNORM:func.11">Comput</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">s</font>,<font color="Maroon" title="c1">i</font></span><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> <a href="scmnorm.html#K11" title="SCMNORM:func.11">Comput</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> $1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmnorm.html#K11" title="SCMNORM:func.11">Comput</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">s</font>,<font color="Maroon" title="c1">i</font>;<br/>
<a NAME="E7:33"/><i><font color="Green" title="E17">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="E18">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">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"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/><b>assume </b><a NAME="E1:33_1"/><i><font color="Green" title="E19">A6</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/> <a href="scmnorm.html#K11" title="SCMNORM:func.11">Comput</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> = 
 <a href="scmnorm.html#K11" title="SCMNORM:func.11">Comput</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>

<br/>.= 
 <a href="scmnorm.html#K10" title="SCMNORM:func.10">Following</a> <font color="Maroon" title="c7">f</font>,<span class="p1">(<span class="default"><a href="scmnorm.html#K11" title="SCMNORM:func.11">Comput</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="scmnorm.html#T5" target="_self" title="SCMNORM:th.5">Tx1</a></i>
<br/>.= 
 <a href="scmnorm.html#K11" title="SCMNORM:func.11">Comput</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">s</font>,<font color="Maroon" title="c1">i</font>
<b>by </b><i><a class="txt" href="scmnorm.html#E6:33"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="scmnorm.html#E1:33_1"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="ami_1.html#D8" title="AMI_1:def.8">AMI_1:def 8</a></i>
;<br/><b>hence </b><a NAME="E3:33_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ; <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>
<a NAME="E9:33"/>
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  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</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="scmnorm.html#E7:33"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="scmnorm.html#E8:33"><i><font color="Green" title="E18">A5</font></i></a>);<br/></i>
<b>hence </b><a NAME="E10:33"/>
 <a href="scmnorm.html#K11" title="SCMNORM:func.11">Comput</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">s</font>,<font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmnorm.html#K11" title="SCMNORM:func.11">Comput</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">s</font>,<font color="Maroon" title="c1">i</font>
 <b>by </b><i><a class="txt" href="scmnorm.html#E5:33"><i><font color="Green" title="E15">A2</font></i></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>
