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<b>let </b><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">j</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"> for <font color="Olive" title="b1">IL</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">N</font> being <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>   st <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> holds <br/>for <font color="Olive" title="b3">S</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="ami_1.html#V2" title="AMI_1:attr.2">stored-program</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#V6" title="AMI_1:attr.6">steady-programmed</a> <a href="ami_1.html#V7" title="AMI_1:attr.7">definite</a>  <a href="ami_1.html#L1" title="AMI_1:struct.1">AMI-Struct</a> of <font color="Olive" title="b1">IL</font>,<font color="Olive" title="b2">N</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="b3">S</font>  st <font color="Olive" title="b4">s</font> <a href="ami_1.html#R3" title="AMI_1:pred.3">halts_at</a>  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Olive" title="b4">s</font>,<font color="Maroon" title="c1">i</font></span>)</span> holds <br/><font color="Olive" title="b4">s</font> <a href="ami_1.html#R3" title="AMI_1:pred.3">halts_at</a>  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Olive" title="b4">s</font>,<font color="Maroon" title="c2">j</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">IL</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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">N</font> being <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>   st <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> holds <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#V2" title="AMI_1:attr.2">stored-program</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#V6" title="AMI_1:attr.6">steady-programmed</a> <a href="ami_1.html#V7" title="AMI_1:attr.7">definite</a>  <a href="ami_1.html#L1" title="AMI_1:struct.1">AMI-Struct</a> of <font color="Maroon" title="c3">IL</font>,<font color="Olive" title="b1">N</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="b2">S</font>  st <font color="Olive" title="b3">s</font> <a href="ami_1.html#R3" title="AMI_1:pred.3">halts_at</a>  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Olive" title="b3">s</font>,<font color="Maroon" title="c1">i</font></span>)</span> holds <br/><font color="Olive" title="b3">s</font> <a href="ami_1.html#R3" title="AMI_1:pred.3">halts_at</a>  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Olive" title="b3">s</font>,<font color="Maroon" title="c2">j</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">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"> ( <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">S</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="ami_1.html#V2" title="AMI_1:attr.2">stored-program</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#V6" title="AMI_1:attr.6">steady-programmed</a> <a href="ami_1.html#V7" title="AMI_1:attr.7">definite</a>  <a href="ami_1.html#L1" title="AMI_1:struct.1">AMI-Struct</a> of <font color="Maroon" title="c3">IL</font>,<font color="Maroon" title="c4">N</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="Olive" title="b1">S</font>  st <font color="Olive" title="b2">s</font> <a href="ami_1.html#R3" title="AMI_1:pred.3">halts_at</a>  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Olive" title="b2">s</font>,<font color="Maroon" title="c1">i</font></span>)</span> holds <br/><font color="Olive" title="b2">s</font> <a href="ami_1.html#R3" title="AMI_1:pred.3">halts_at</a>  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Olive" title="b2">s</font>,<font color="Maroon" title="c2">j</font></span>)</span> )</span><br/>







<b>assume </b><a NAME="E1:135"/><i><font color="Green" title="E59">A1</font></i>: 
<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">S</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="ami_1.html#V2" title="AMI_1:attr.2">stored-program</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#V6" title="AMI_1:attr.6">steady-programmed</a> <a href="ami_1.html#V7" title="AMI_1:attr.7">definite</a>  <a href="ami_1.html#L1" title="AMI_1:struct.1">AMI-Struct</a> of <font color="Maroon" title="c3">IL</font>,<font color="Maroon" title="c4">N</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="Olive" title="b1">S</font>  st <font color="Olive" title="b2">s</font> <a href="ami_1.html#R3" title="AMI_1:pred.3">halts_at</a>  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Olive" title="b2">s</font>,<font color="Maroon" title="c1">i</font></span>)</span> holds <br/><font color="Olive" title="b2">s</font> <a href="ami_1.html#R3" title="AMI_1:pred.3">halts_at</a>  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Olive" title="b2">s</font>,<font color="Maroon" title="c2">j</font></span>)</span></span><br/>

<b>let </b><font color="Maroon" title="c5">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="ami_1.html#V2" title="AMI_1:attr.2">stored-program</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#V6" title="AMI_1:attr.6">steady-programmed</a> <a href="ami_1.html#V7" title="AMI_1:attr.7">definite</a>  <a href="ami_1.html#L1" title="AMI_1:struct.1">AMI-Struct</a> of <font color="Maroon" title="c3">IL</font>,<font color="Maroon" title="c4">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">s</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <font color="Maroon" title="c5">S</font>  st <font color="Olive" title="b1">s</font> <a href="ami_1.html#R3" title="AMI_1:pred.3">halts_at</a>  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Olive" title="b1">s</font>,<font color="Maroon" title="c1">i</font></span>)</span> holds <br/><font color="Olive" title="b1">s</font> <a href="ami_1.html#R3" title="AMI_1:pred.3">halts_at</a>  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Olive" title="b1">s</font>,<font color="Maroon" title="c2">j</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c6">s</font> be   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <font color="Maroon" title="c5">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="c6">s</font> <a href="ami_1.html#R3" title="AMI_1:pred.3">halts_at</a>  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c1">i</font></span>)</span> implies <font color="Maroon" title="c6">s</font> <a href="ami_1.html#R3" title="AMI_1:pred.3">halts_at</a>  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c2">j</font></span>)</span> )</span><br/>



<b>assume </b><a NAME="E2:135"/><i><font color="Green" title="E60">A2</font></i>: 
<font color="Maroon" title="c6">s</font> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c1">i</font></span>)</span></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="c5">S</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="ami_1.html#D42" target="_self" title="AMI_1:def.42">AMI_1:def 42</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">s</font> <a href="ami_1.html#R3" title="AMI_1:pred.3">halts_at</a>  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c2">j</font></span>)</span></span><br/>

<a NAME="E3:135"/><b>then </b>
 <a href="ami_1.html#K8" title="AMI_1:func.8">CurInstr</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c6">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="c5">S</font>
 
<b>by </b><i><a class="ref" href="ami_1.html#T54" target="_self" title="AMI_1:th.54">Th54</a></i>;<br/>
<b>hence </b><a NAME="E4:135"/>
<font color="Maroon" title="c6">s</font> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c2">j</font></span>)</span></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="c5">S</font>
 <b>by </b><i><a class="txt" href="ami_1.html#E1:135"><i><font color="Green" title="E59">A1</font></i></a>, <a class="txt" href="ami_1.html#E2:135"><i><font color="Green" title="E60">A2</font></i></a>, <a class="ref" href="ami_1.html#T52" target="_self" title="AMI_1:th.52">Th52</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="ami_1.html#D42" target="_self" title="AMI_1:def.42">AMI_1:def 42</a> <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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