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<b>let </b><font color="Maroon" title="c1">i</font> be   <a href="ami_1.html#NM4" title="AMI_1:NM.4">Instruction</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</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">m</font>, <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Maroon" title="c1">i</font> <a href="scmpds_4.html#R1" title="SCMPDS_4:pred.1">valid_at</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> holds <br/><font color="Maroon" title="c1">i</font> <a href="scmpds_4.html#R1" title="SCMPDS_4:pred.1">valid_at</a> <font color="Olive" title="b2">n</font></span><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="ami_2.html#K5" title="AMI_2:func.5">NAT</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="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> implies <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> )</span><br/>



<b>assume </b><a NAME="E1:69"/><i><font color="Green" title="E44">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> )
 ; <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="scmpds_4.html#R1" title="SCMPDS_4:pred.1">valid_at</a> <font color="Maroon" title="c3">n</font></span><br/>

<div><i><font color="Green" title="E45">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:69_1"/>
 <a href="ami_1.html#K21" title="AMI_1:func.21">InsCode</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">k1</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <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="Olive" title="b1">k1</font> &amp; <font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b1">k1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><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="E3:69_1"/><i><font color="Green" title="E46">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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="scmpds_4.html#E1:69"><i><font color="Green" title="E44">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>; <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="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> &amp; <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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><b>thus </b><a NAME="E4:69_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#E3:69_1"><i><font color="Green" title="E46">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><b>thus </b><a NAME="E5:69_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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="scmpds_4.html#E1:69"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="scmpds_4.html#E3:69_1"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E46">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="69_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:69_2"/>
 <a href="ami_1.html#K21" title="AMI_1:func.21">InsCode</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">a</font> being   <a href="scmpds_2.html#M1" title="SCMPDS_2:mode.1">Int_position</a>  ex <font color="Olive" title="b2">k1</font>, <font color="Olive" title="b3">k2</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">k1</font> <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Olive" title="b3">k2</font> &amp; <font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b3">k2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><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="E3:69_2"/><i><font color="Green" title="E47">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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="scmpds_4.html#E1:69"><i><font color="Green" title="E44">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">k1</font>, <font color="Olive" title="b2">k2</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <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="Olive" title="b1">k1</font> <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Olive" title="b2">k2</font> &amp; <font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">k2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><b>take </b><font color="Maroon" title="c8">k1</font> = <font color="Maroon" title="c5">k1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">k2</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <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="Olive" title="b1">k2</font> &amp; <font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b1">k2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><b>take </b><font color="Maroon" title="c9">k2</font> = <font color="Maroon" title="c6">k2</font>; <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="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> &amp; <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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><b>thus </b><a NAME="E4:69_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#E3:69_2"><i><font color="Green" title="E47">A5</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><b>thus </b><a NAME="E5:69_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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="scmpds_4.html#E1:69"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="scmpds_4.html#E3:69_2"><i><font color="Green" title="E47">A5</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E47">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="69_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:69_3"/>
 <a href="ami_1.html#K21" title="AMI_1:func.21">InsCode</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">a</font> being   <a href="scmpds_2.html#M1" title="SCMPDS_2:mode.1">Int_position</a>  ex <font color="Olive" title="b2">k1</font>, <font color="Olive" title="b3">k2</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">k1</font> <a href="scmpds_2.html#K9" title="SCMPDS_2:func.9">&lt;=0_goto</a> <font color="Olive" title="b3">k2</font> &amp; <font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b3">k2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><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="E3:69_3"/><i><font color="Green" title="E48">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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="scmpds_4.html#E1:69"><i><font color="Green" title="E44">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">k1</font>, <font color="Olive" title="b2">k2</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <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="Olive" title="b1">k1</font> <a href="scmpds_2.html#K9" title="SCMPDS_2:func.9">&lt;=0_goto</a> <font color="Olive" title="b2">k2</font> &amp; <font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">k2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><b>take </b><font color="Maroon" title="c8">k1</font> = <font color="Maroon" title="c5">k1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">k2</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <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="Olive" title="b1">k2</font> &amp; <font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b1">k2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><b>take </b><font color="Maroon" title="c9">k2</font> = <font color="Maroon" title="c6">k2</font>; <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="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> &amp; <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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><b>thus </b><a NAME="E4:69_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#E3:69_3"><i><font color="Green" title="E48">A7</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><b>thus </b><a NAME="E5:69_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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="scmpds_4.html#E1:69"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="scmpds_4.html#E3:69_3"><i><font color="Green" title="E48">A7</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:69_4"/>
 <a href="ami_1.html#K21" title="AMI_1:func.21">InsCode</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 6
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">a</font> being   <a href="scmpds_2.html#M1" title="SCMPDS_2:mode.1">Int_position</a>  ex <font color="Olive" title="b2">k1</font>, <font color="Olive" title="b3">k2</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">k1</font> <a href="scmpds_2.html#K10" title="SCMPDS_2:func.10">&gt;=0_goto</a> <font color="Olive" title="b3">k2</font> &amp; <font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b3">k2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><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="E3:69_4"/><i><font color="Green" title="E48">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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="scmpds_4.html#E1:69"><i><font color="Green" title="E44">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">k1</font>, <font color="Olive" title="b2">k2</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <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="Olive" title="b1">k1</font> <a href="scmpds_2.html#K10" title="SCMPDS_2:func.10">&gt;=0_goto</a> <font color="Olive" title="b2">k2</font> &amp; <font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">k2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><b>take </b><font color="Maroon" title="c8">k1</font> = <font color="Maroon" title="c5">k1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">k2</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <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="Olive" title="b1">k2</font> &amp; <font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b1">k2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><b>take </b><font color="Maroon" title="c9">k2</font> = <font color="Maroon" title="c6">k2</font>; <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="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> &amp; <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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><b>thus </b><a NAME="E4:69_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#E3:69_4"><i><font color="Green" title="E48">A8</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><b>thus </b><a NAME="E5:69_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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="scmpds_4.html#E1:69"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="scmpds_4.html#E3:69_4"><i><font color="Green" title="E48">A8</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:69"/>
<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#E2:69"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="scmpds_4.html#E3:69"><i><font color="Green" title="E46">A4</font></i></a>, <a class="txt" href="scmpds_4.html#E4:69"><i><font color="Green" title="E47">A6</font></i></a>, <a class="ref" href="scmpds_4.html#D11" target="_self" title="SCMPDS_4:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
