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<b>let </b><font color="Maroon" title="c1">i</font> be   <a href="ami_1.html#NM3" title="AMI_1:NM.3">Instruction</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> ;<br/><b>let </b><font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:81"/><i><font color="Green" title="E52">A1</font></i>: 
( <font color="Maroon" title="c1">i</font> <a href="scmpds_4.html#R1" title="SCMPDS_4:pred.1">valid_at</a> <font color="Maroon" title="c2">m</font> &amp; <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> )
 ;<br/>

<div><i><font color="Green" title="E53">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:81_1"/>
 <a href="ami_5.html#K1" title="AMI_5:func.1">InsCode</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><b>then consider </b><font color="Maroon" title="c4">k1</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E3:81_1"/><i><font color="Green" title="E54">A3</font></i>: 
( <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_2.html#K4" title="SCMPDS_2:func.4">goto</a> <font color="Maroon" title="c4">k1</font> &amp; <font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">k1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 )
 <b>by </b><i><a class="txt" href="scmpds_4.html#E1:81"><i><font color="Green" title="E52">A1</font></i></a>, <a class="ref" href="scmpds_4.html#D11" target="_self" title="SCMPDS_4:def.11">Def11</a></i>;<br/><b>take </b><font color="Maroon" title="c5">k1</font> = <font color="Maroon" title="c4">k1</font>;<br/><b>thus </b><a NAME="E4:81_1"/>
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_2.html#K4" title="SCMPDS_2:func.4">goto</a> <font color="Maroon" title="c5">k1</font>
 <b>by </b><i><a class="txt" href="scmpds_4.html#E3:81_1"><i><font color="Green" title="E54">A3</font></i></a></i>;<br/><b>thus </b><a NAME="E5:81_1"/>
<font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">k1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="scmpds_4.html#E1:81"><i><font color="Green" title="E52">A1</font></i></a>, <a class="txt" href="scmpds_4.html#E3:81_1"><i><font color="Green" title="E54">A3</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E54">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:81_2"/>
 <a href="ami_5.html#K1" title="AMI_5:func.1">InsCode</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 ;<br/><b>then consider </b><font color="Maroon" title="c4">a</font> being    <a href="scmpds_2.html#M1" title="SCMPDS_2:mode.1">Int_position</a> , <font color="Maroon" title="c5">k1</font>, <font color="Maroon" title="c6">k2</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E3:81_2"/><i><font color="Green" title="E55">A5</font></i>: 
( <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">k1</font> <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c6">k2</font> &amp; <font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">k2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 )
 <b>by </b><i><a class="txt" href="scmpds_4.html#E1:81"><i><font color="Green" title="E52">A1</font></i></a>, <a class="ref" href="scmpds_4.html#D11" target="_self" title="SCMPDS_4:def.11">Def11</a></i>;<br/><b>take </b><font color="Maroon" title="c7">a</font> = <font color="Maroon" title="c4">a</font>;<br/><b>take </b><font color="Maroon" title="c8">k1</font> = <font color="Maroon" title="c5">k1</font>;<br/><b>take </b><font color="Maroon" title="c9">k2</font> = <font color="Maroon" title="c6">k2</font>;<br/><b>thus </b><a NAME="E4:81_2"/>
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">k1</font> <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c9">k2</font>
 <b>by </b><i><a class="txt" href="scmpds_4.html#E3:81_2"><i><font color="Green" title="E55">A5</font></i></a></i>;<br/><b>thus </b><a NAME="E5:81_2"/>
<font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">k2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="scmpds_4.html#E1:81"><i><font color="Green" title="E52">A1</font></i></a>, <a class="txt" href="scmpds_4.html#E3:81_2"><i><font color="Green" title="E55">A5</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E55">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="81_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:81_3"/>
 <a href="ami_5.html#K1" title="AMI_5:func.1">InsCode</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5
 ;<br/><b>then consider </b><font color="Maroon" title="c4">a</font> being    <a href="scmpds_2.html#M1" title="SCMPDS_2:mode.1">Int_position</a> , <font color="Maroon" title="c5">k1</font>, <font color="Maroon" title="c6">k2</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E3:81_3"/><i><font color="Green" title="E56">A7</font></i>: 
( <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">k1</font> <a href="scmpds_2.html#K9" title="SCMPDS_2:func.9">&lt;=0_goto</a> <font color="Maroon" title="c6">k2</font> &amp; <font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">k2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 )
 <b>by </b><i><a class="txt" href="scmpds_4.html#E1:81"><i><font color="Green" title="E52">A1</font></i></a>, <a class="ref" href="scmpds_4.html#D11" target="_self" title="SCMPDS_4:def.11">Def11</a></i>;<br/><b>take </b><font color="Maroon" title="c7">a</font> = <font color="Maroon" title="c4">a</font>;<br/><b>take </b><font color="Maroon" title="c8">k1</font> = <font color="Maroon" title="c5">k1</font>;<br/><b>take </b><font color="Maroon" title="c9">k2</font> = <font color="Maroon" title="c6">k2</font>;<br/><b>thus </b><a NAME="E4:81_3"/>
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">k1</font> <a href="scmpds_2.html#K9" title="SCMPDS_2:func.9">&lt;=0_goto</a> <font color="Maroon" title="c9">k2</font>
 <b>by </b><i><a class="txt" href="scmpds_4.html#E3:81_3"><i><font color="Green" title="E56">A7</font></i></a></i>;<br/><b>thus </b><a NAME="E5:81_3"/>
<font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">k2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="scmpds_4.html#E1:81"><i><font color="Green" title="E52">A1</font></i></a>, <a class="txt" href="scmpds_4.html#E3:81_3"><i><font color="Green" title="E56">A7</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:81_4"/>
 <a href="ami_5.html#K1" title="AMI_5:func.1">InsCode</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 6
 ;<br/><b>then consider </b><font color="Maroon" title="c4">a</font> being    <a href="scmpds_2.html#M1" title="SCMPDS_2:mode.1">Int_position</a> , <font color="Maroon" title="c5">k1</font>, <font color="Maroon" title="c6">k2</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E3:81_4"/><i><font color="Green" title="E56">A8</font></i>: 
( <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">k1</font> <a href="scmpds_2.html#K10" title="SCMPDS_2:func.10">&gt;=0_goto</a> <font color="Maroon" title="c6">k2</font> &amp; <font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">k2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 )
 <b>by </b><i><a class="txt" href="scmpds_4.html#E1:81"><i><font color="Green" title="E52">A1</font></i></a>, <a class="ref" href="scmpds_4.html#D11" target="_self" title="SCMPDS_4:def.11">Def11</a></i>;<br/><b>take </b><font color="Maroon" title="c7">a</font> = <font color="Maroon" title="c4">a</font>;<br/><b>take </b><font color="Maroon" title="c8">k1</font> = <font color="Maroon" title="c5">k1</font>;<br/><b>take </b><font color="Maroon" title="c9">k2</font> = <font color="Maroon" title="c6">k2</font>;<br/><b>thus </b><a NAME="E4:81_4"/>
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">k1</font> <a href="scmpds_2.html#K10" title="SCMPDS_2:func.10">&gt;=0_goto</a> <font color="Maroon" title="c9">k2</font>
 <b>by </b><i><a class="txt" href="scmpds_4.html#E3:81_4"><i><font color="Green" title="E56">A8</font></i></a></i>;<br/><b>thus </b><a NAME="E5:81_4"/>
<font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">k2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="scmpds_4.html#E1:81"><i><font color="Green" title="E52">A1</font></i></a>, <a class="txt" href="scmpds_4.html#E3:81_4"><i><font color="Green" title="E56">A8</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:81"/>
<font color="Maroon" title="c1">i</font> <a href="scmpds_4.html#R1" title="SCMPDS_4:pred.1">valid_at</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="txt" href="scmpds_4.html#E2:81"><i><font color="Green" title="E53">A2</font></i></a>, <a class="txt" href="scmpds_4.html#E3:81"><i><font color="Green" title="E54">A4</font></i></a>, <a class="txt" href="scmpds_4.html#E4:81"><i><font color="Green" title="E55">A6</font></i></a>, <a class="ref" href="scmpds_4.html#D11" target="_self" title="SCMPDS_4:def.11">Def11</a></i>;<br/>


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