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<b>let </b><font color="Maroon">i</font> be   <a href="ami_1.html#NM3">Instruction</a> of <a href="scmpds_2.html#K1">SCMPDS</a> ;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:81"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">i</font> <a href="scmpds_4.html#R1" target="_self">valid_at</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 ;<br/>

<div><i><font color="Green">E31</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">InsCode</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>then consider </b><font color="Maroon">k1</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E3:81_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K4">goto</a> <font color="Maroon">k1</font> &amp; <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i>, </i>;<br/><b>take </b><font color="Maroon">k1</font> = <font color="Maroon">k1</font>;<br/><b>thus </b><a NAME="E4:81_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K4">goto</a> <font color="Maroon">k1</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E5:81_1"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E38</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">InsCode</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 4
 ;<br/><b>then consider </b><font color="Maroon">a</font> being    <a href="scmpds_2.html#M1">Int_position</a> , <font color="Maroon">k1</font> being   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">k2</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E3:81_2"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>,<font color="Maroon">k1</font> <a href="scmpds_2.html#K8">&lt;&gt;0_goto</a> <font color="Maroon">k2</font> &amp; <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i>, </i>;<br/><b>take </b><font color="Maroon">a</font> = <font color="Maroon">a</font>;<br/><b>take </b><font color="Maroon">k1</font> = <font color="Maroon">k1</font>;<br/><b>take </b><font color="Maroon">k2</font> = <font color="Maroon">k2</font>;<br/><b>thus </b><a NAME="E4:81_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>,<font color="Maroon">k1</font> <a href="scmpds_2.html#K8">&lt;&gt;0_goto</a> <font color="Maroon">k2</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E5:81_2"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E41</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">InsCode</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 5
 ;<br/><b>then consider </b><font color="Maroon">a</font> being    <a href="scmpds_2.html#M1">Int_position</a> , <font color="Maroon">k1</font> being   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">k2</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E3:81_3"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>,<font color="Maroon">k1</font> <a href="scmpds_2.html#K9">&lt;=0_goto</a> <font color="Maroon">k2</font> &amp; <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i>, </i>;<br/><b>take </b><font color="Maroon">a</font> = <font color="Maroon">a</font>;<br/><b>take </b><font color="Maroon">k1</font> = <font color="Maroon">k1</font>;<br/><b>take </b><font color="Maroon">k2</font> = <font color="Maroon">k2</font>;<br/><b>thus </b><a NAME="E4:81_3"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>,<font color="Maroon">k1</font> <a href="scmpds_2.html#K9">&lt;=0_goto</a> <font color="Maroon">k2</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E5:81_3"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T8">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">InsCode</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 6
 ;<br/><b>then consider </b><font color="Maroon">a</font> being    <a href="scmpds_2.html#M1">Int_position</a> , <font color="Maroon">k1</font> being   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">k2</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E3:81_4"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>,<font color="Maroon">k1</font> <a href="scmpds_2.html#K10">&gt;=0_goto</a> <font color="Maroon">k2</font> &amp; <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i>, </i>;<br/><b>take </b><font color="Maroon">a</font> = <font color="Maroon">a</font>;<br/><b>take </b><font color="Maroon">k1</font> = <font color="Maroon">k1</font>;<br/><b>take </b><font color="Maroon">k2</font> = <font color="Maroon">k2</font>;<br/><b>thus </b><a NAME="E4:81_4"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>,<font color="Maroon">k1</font> <a href="scmpds_2.html#K10">&gt;=0_goto</a> <font color="Maroon">k2</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E5:81_4"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:81"/>
<font color="Maroon">i</font> <a href="scmpds_4.html#R1" target="_self">valid_at</a> <font color="Maroon">n</font>
 <b>by </b><i>, , , </i>;<br/>


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