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<b>reconsider </b><font color="Maroon" title="c6">R</font> = <span class="p1">(<span class="default"><a href="scmnorm.html#K12" title="SCMNORM:func.12">Result</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">h</font></span>)</span> <a href="card_3.html#K11" title="CARD_3:func.11">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">p</font></span>)</span> as   <a href="ami_1.html#NM8" title="AMI_1:NM.8">FinPartState</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="ref" href="ami_1.html#T62" title="AMI_1:th.62">AMI_1:62</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">R</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">p'</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <font color="Maroon" title="c2">S</font>  st <font color="Maroon" title="c4">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">p'</font> holds <br/><font color="Maroon" title="c6">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmnorm.html#K12" title="SCMNORM:func.12">Result</a> <font color="Maroon" title="c3">f</font>,<font color="Olive" title="b1">p'</font></span>)</span> <a href="card_3.html#K11" title="CARD_3:func.11">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">p</font></span>)</span></span><br/>

<b>let </b><font color="Maroon" title="c7">p'</font> be   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</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"> ( <font color="Maroon" title="c4">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">p'</font> implies <font color="Maroon" title="c6">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmnorm.html#K12" title="SCMNORM:func.12">Result</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c7">p'</font></span>)</span> <a href="card_3.html#K11" title="CARD_3:func.11">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">p</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E2:35_1_1"/><i><font color="Green" title="E12">A4</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">p'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmnorm.html#K12" title="SCMNORM:func.12">Result</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c7">p'</font></span>)</span> <a href="card_3.html#K11" title="CARD_3:func.11">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">p</font></span>)</span></span><br/>

<a NAME="E3:35_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="scmnorm.html#R1" title="SCMNORM:pred.1">halts_on</a> <font color="Maroon" title="c5">h</font>
 
<b>by </b><i><a class="txt" href="#E1:35"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="#E3:35"><i><font color="Green" title="E11">A10</font></i></a>, <a class="ref" href="scmnorm.html#D9" target="_self" title="SCMNORM:def.9">Def9</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">k1</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> <b> such that </b><br/><a NAME="E5:35_1_1"/><i><font color="Green" title="E13">A5</font></i>: 
(  <a href="scmnorm.html#K12" title="SCMNORM:func.12">Result</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">h</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="c3">f</font>,<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c8">k1</font> &amp;  <a href="scmnorm.html#K9" title="SCMNORM:func.9">CurInstr</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><a href="scmnorm.html#K12" title="SCMNORM:func.12">Result</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">h</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="c2">S</font> )
 <b>by </b><i><a class="ref" href="scmnorm.html#D10" target="_self" title="SCMNORM:def.10">Def10</a></i>;<br/>
<a NAME="E6:35_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="scmnorm.html#R1" title="SCMNORM:pred.1">halts_on</a> <font color="Maroon" title="c7">p'</font>
 
<b>by </b><i><a class="txt" href="#E1:35"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="#E2:35_1_1"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="scmnorm.html#D9" target="_self" title="SCMNORM:def.9">Def9</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">k2</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> <b> such that </b><br/><a NAME="E8:35_1_1"/><i><font color="Green" title="E14">A6</font></i>: 
(  <a href="scmnorm.html#K12" title="SCMNORM:func.12">Result</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c7">p'</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="c3">f</font>,<font color="Maroon" title="c7">p'</font>,<font color="Maroon" title="c9">k2</font> &amp;  <a href="scmnorm.html#K9" title="SCMNORM:func.9">CurInstr</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><a href="scmnorm.html#K12" title="SCMNORM:func.12">Result</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c7">p'</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="c2">S</font> )
 <b>by </b><i><a class="ref" href="scmnorm.html#D10" target="_self" title="SCMNORM:def.10">Def10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_1_1_1_1"/>
( <font color="Maroon" title="c8">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k2</font> or <font color="Maroon" title="c8">k1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">k2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:35_1_1_1_1_1"/>
<font color="Maroon" title="c8">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmnorm.html#K12" title="SCMNORM:func.12">Result</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c7">p'</font></span>)</span> <a href="card_3.html#K11" title="CARD_3:func.11">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">p</font></span>)</span></span><br/><div class="add"><a NAME="E2:35_1_1_1_1_1"/><b>then </b>
 <a href="scmnorm.html#K12" title="SCMNORM:func.12">Result</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">h</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="c3">f</font>,<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c9">k2</font>
 <b>by </b><i><a class="txt" href="#E5:35_1_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="scmnorm.html#T7" target="_self" title="SCMNORM:th.7">Th7</a></i>;<br/><b>hence </b><a NAME="E3:35_1_1_1_1_1"/>
<font color="Maroon" title="c6">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmnorm.html#K12" title="SCMNORM:func.12">Result</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c7">p'</font></span>)</span> <a href="card_3.html#K11" title="CARD_3:func.11">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:35"><i><font color="Green" title="E11">A10</font></i></a>, <a class="txt" href="#E2:35_1_1"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="#E8:35_1_1"><i><font color="Green" title="E14">A6</font></i></a>, <a class="ref" href="scmnorm.html#D7" target="_self" title="SCMNORM:def.7">Def7</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:35_1_1_1_1_2"/>
<font color="Maroon" title="c8">k1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">k2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmnorm.html#K12" title="SCMNORM:func.12">Result</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c7">p'</font></span>)</span> <a href="card_3.html#K11" title="CARD_3:func.11">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">p</font></span>)</span></span><br/><div class="add"><a NAME="E2:35_1_1_1_1_2"/><b>then </b>
 <a href="scmnorm.html#K12" title="SCMNORM:func.12">Result</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c7">p'</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="c3">f</font>,<font color="Maroon" title="c7">p'</font>,<font color="Maroon" title="c8">k1</font>
 <b>by </b><i><a class="txt" href="#E8:35_1_1"><i><font color="Green" title="E14">A6</font></i></a>, <a class="ref" href="scmnorm.html#T7" target="_self" title="SCMNORM:th.7">Th7</a></i>;<br/><b>hence </b><a NAME="E3:35_1_1_1_1_2"/>
<font color="Maroon" title="c6">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmnorm.html#K12" title="SCMNORM:func.12">Result</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c7">p'</font></span>)</span> <a href="card_3.html#K11" title="CARD_3:func.11">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:35"><i><font color="Green" title="E11">A10</font></i></a>, <a class="txt" href="#E2:35_1_1"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="#E5:35_1_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="scmnorm.html#D7" target="_self" title="SCMNORM:def.7">Def7</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></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E10:35_1_1"/>
<font color="Maroon" title="c6">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmnorm.html#K12" title="SCMNORM:func.12">Result</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c7">p'</font></span>)</span> <a href="card_3.html#K11" title="CARD_3:func.11">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">p</font></span>)</span>
 ; <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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