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<b>consider </b><font color="Maroon" title="c4">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E2:80_1_1"/><i><font color="Green" title="E39">A2</font></i>: 
<font color="Maroon" title="c4">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> the <a href="ami_1.html#U4" title="AMI_1:sel.4">Object-Kind</a> of <font color="Maroon" title="c2">S</font>
 <b>and </b><br/><a NAME="E3:80_1_1"/><i><font color="Green" title="E40">A3</font></i>: 
<font color="Maroon" title="c3">s</font> <a href="funct_4.html#NR1" title="FUNCT_4:NR.1">&lt;=</a> <font color="Maroon" title="c4">h</font>
 <b>by </b><i><a class="ref" href="card_3.html#T70" title="CARD_3:th.70">CARD_3:70</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">h</font> = <font color="Maroon" title="c4">h</font> as   <a href="ami_1.html#NM4" title="AMI_1:NM.4">State</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="ami_1.html#E2:80_1_1"><i><font color="Green" title="E39">A2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">R</font> = <span class="p1">(<span class="default"><a href="ami_1.html#K11" title="AMI_1:func.11">Result</a> <font color="Maroon" title="c5">h</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#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">s</font></span>)</span> as   <a href="ami_1.html#NM5" title="AMI_1:NM.5">FinPartState</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="ref" href="ami_1.html#T62" target="_self" title="AMI_1:th.62">Th62</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">R</font>
;<br/>

<b>let </b><font color="Maroon" title="c7">s'</font> be   <a href="ami_1.html#NM4" title="AMI_1:NM.4">State</a> of <font color="Maroon" title="c2">S</font>;<br/>

<b>assume </b><a NAME="E6:80_1_1"/><i><font color="Green" title="E41">A4</font></i>: 
<font color="Maroon" title="c3">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">s'</font>
 ;<br/>

<a NAME="E7:80_1_1"/>
<font color="Maroon" title="c5">h</font> is <a href="ami_1.html#V8" title="AMI_1:attr.8">halting</a>
 
<b>by </b><i><a class="txt" href="ami_1.html#E1:80"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="ami_1.html#E3:80_1_1"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="ami_1.html#D26" target="_self" title="AMI_1:def.26">Def26</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="E9:80_1_1"/><i><font color="Green" title="E42">A5</font></i>: 
(  <a href="ami_1.html#K11" title="AMI_1:func.11">Result</a> <font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <font color="Maroon" title="c5">h</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c8">k1</font> &amp;  <a href="ami_1.html#K7" title="AMI_1:func.7">CurInstr</a> <span class="p1">(<span class="default"><a href="ami_1.html#K11" title="AMI_1:func.11">Result</a> <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="ami_1.html#D22" target="_self" title="AMI_1:def.22">Def22</a></i>;<br/>
<a NAME="E10:80_1_1"/>
<font color="Maroon" title="c7">s'</font> is <a href="ami_1.html#V8" title="AMI_1:attr.8">halting</a>
 
<b>by </b><i><a class="txt" href="ami_1.html#E1:80"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="ami_1.html#E6:80_1_1"><i><font color="Green" title="E41">A4</font></i></a>, <a class="ref" href="ami_1.html#D26" target="_self" title="AMI_1:def.26">Def26</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="E12:80_1_1"/><i><font color="Green" title="E43">A6</font></i>: 
(  <a href="ami_1.html#K11" title="AMI_1:func.11">Result</a> <font color="Maroon" title="c7">s'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <font color="Maroon" title="c7">s'</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c9">k2</font> &amp;  <a href="ami_1.html#K7" title="AMI_1:func.7">CurInstr</a> <span class="p1">(<span class="default"><a href="ami_1.html#K11" title="AMI_1:func.11">Result</a> <font color="Maroon" title="c7">s'</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="ami_1.html#D22" target="_self" title="AMI_1:def.22">Def22</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_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="E1:80_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="80_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:80_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>
 ;<br/><div class="add"><a NAME="E2:80_1_1_1_1_1"/><b>then </b>
 <a href="ami_1.html#K11" title="AMI_1:func.11">Result</a> <font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <font color="Maroon" title="c5">h</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c9">k2</font>
 <b>by </b><i><a class="txt" href="ami_1.html#E9:80_1_1"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="ami_1.html#T52" target="_self" title="AMI_1:th.52">Th52</a></i>;<br/><b>hence </b><a NAME="E3:80_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="ami_1.html#K11" title="AMI_1:func.11">Result</a> <font color="Maroon" title="c7">s'</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#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">s</font></span>)</span>
 <b>by </b><i><a class="txt" href="ami_1.html#E1:80"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="ami_1.html#E3:80_1_1"><i><font color="Green" title="E40">A3</font></i></a>, <a class="txt" href="ami_1.html#E6:80_1_1"><i><font color="Green" title="E41">A4</font></i></a>, <a class="txt" href="ami_1.html#E12:80_1_1"><i><font color="Green" title="E43">A6</font></i></a>, <a class="ref" href="ami_1.html#D25" target="_self" title="AMI_1:def.25">Def25</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:80_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>
 ;<br/><div class="add"><a NAME="E2:80_1_1_1_1_2"/><b>then </b>
 <a href="ami_1.html#K11" title="AMI_1:func.11">Result</a> <font color="Maroon" title="c7">s'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <font color="Maroon" title="c7">s'</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c8">k1</font>
 <b>by </b><i><a class="txt" href="ami_1.html#E12:80_1_1"><i><font color="Green" title="E43">A6</font></i></a>, <a class="ref" href="ami_1.html#T52" target="_self" title="AMI_1:th.52">Th52</a></i>;<br/><b>hence </b><a NAME="E3:80_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="ami_1.html#K11" title="AMI_1:func.11">Result</a> <font color="Maroon" title="c7">s'</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#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">s</font></span>)</span>
 <b>by </b><i><a class="txt" href="ami_1.html#E1:80"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="ami_1.html#E3:80_1_1"><i><font color="Green" title="E40">A3</font></i></a>, <a class="txt" href="ami_1.html#E6:80_1_1"><i><font color="Green" title="E41">A4</font></i></a>, <a class="txt" href="ami_1.html#E9:80_1_1"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="ami_1.html#D25" target="_self" title="AMI_1:def.25">Def25</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E14:80_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="ami_1.html#K11" title="AMI_1:func.11">Result</a> <font color="Maroon" title="c7">s'</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#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">s</font></span>)</span>
 ;<br/>


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