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<span class="kw">let </span><font color="Maroon" title="c5">s1</font>, <font color="Maroon" title="c6">s2</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c5">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">s</font>,<font color="Olive" title="b1">k</font> &amp;  <a href="ami_1.html#K7" title="AMI_1:func.7">CurInstr</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">s1</font> <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> ) &amp;  ex <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c6">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">s</font>,<font color="Olive" title="b1">k</font> &amp;  <a href="ami_1.html#K7" title="AMI_1:func.7">CurInstr</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c6">s2</font> <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> ) implies <font color="Maroon" title="c5">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s2</font> )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c7">k1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><a NAME="E2:18_1_1"/><span class="lab"><font color="Green" title="E2">A2</font></span>: 
<font color="Maroon" title="c5">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c7">k1</font>
 <span class="kw">and </span><br/><a NAME="E3:18_1_1"/><span class="lab"><font color="Green" title="E3">A3</font></span>: 
 <a href="ami_1.html#K7" title="AMI_1:func.7">CurInstr</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">s1</font> <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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <br/>(  not <font color="Maroon" title="c6">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">s</font>,<font color="Olive" title="b1">k</font> or  not  <a href="ami_1.html#K7" title="AMI_1:func.7">CurInstr</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c6">s2</font> <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> ) or <font color="Maroon" title="c5">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s2</font> )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c8">k2</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><a NAME="E5:18_1_1"/><span class="lab"><font color="Green" title="E4">A4</font></span>: 
<font color="Maroon" title="c6">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c8">k2</font>
 <span class="kw">and </span><br/><a NAME="E6:18_1_1"/><span class="lab"><font color="Green" title="E5">A5</font></span>: 
 <a href="ami_1.html#K7" title="AMI_1:func.7">CurInstr</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c6">s2</font> <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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s2</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c9">k1</font> = <font color="Maroon" title="c7">k1</font>, <font color="Maroon" title="c10">k2</font> = <font color="Maroon" title="c8">k2</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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<a NAME="E8:18_1_1"/>
( <font color="Maroon" title="c9">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k2</font> or <font color="Maroon" title="c10">k2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k1</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E9:18_1_1"/>
<font color="Maroon" title="c5">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_1_1"><span class="lab"><font color="Green" title="E2">A2</font></span></a>, <a class="txt" href="#E3:18_1_1"><span class="lab"><font color="Green" title="E3">A3</font></span></a>, <a class="txt" href="#E5:18_1_1"><span class="lab"><font color="Green" title="E4">A4</font></span></a>, <a class="txt" href="#E6:18_1_1"><span class="lab"><font color="Green" title="E5">A5</font></span></a>, <a class="ref" href="ami_1.html#T52" title="AMI_1:th.52">AMI_1:52</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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