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<span class="kw">let </span><font color="Maroon" title="c1">I</font> be   <a href="scmpds_4.html#V3" title="SCMPDS_4:attr.3">shiftable</a>  <a href="compos_1.html#NM5" title="COMPOS_1:NM.5">Program</a> of ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <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><br/>  for <font color="Olive" title="b3">a</font> being    <a href="scmpds_2.html#M1" title="SCMPDS_2:mode.1">Int_position</a>   st <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">I</font></span>)</span> <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>  holds <br/><font color="Maroon" title="c1">I</font> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p1">(<span class="default">(<font color="Olive" title="b3">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></span>)</span> is  <a href="scmpds_4.html#V3" title="SCMPDS_4:attr.3">shiftable</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">k1</font>, <font color="Maroon" title="c3">k2</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being    <a href="scmpds_2.html#M1" title="SCMPDS_2:mode.1">Int_position</a>   st <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">I</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">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>  holds <br/><font color="Maroon" title="c1">I</font> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p1">(<span class="default">(<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c2">k1</font>) <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c3">k2</font></span>)</span> is  <a href="scmpds_4.html#V3" title="SCMPDS_4:attr.3">shiftable</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">a</font> be    <a href="scmpds_2.html#M1" title="SCMPDS_2:mode.1">Int_position</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">I</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">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>  implies <font color="Maroon" title="c1">I</font> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">k1</font>) <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c3">k2</font></span>)</span> is  <a href="scmpds_4.html#V3" title="SCMPDS_4:attr.3">shiftable</a>  )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c5">ii</font> = (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">k1</font>) <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c3">k2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">J</font> =  <a href="compos_1.html#K14" title="COMPOS_1:func.14">Load</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">k1</font>) <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c3">k2</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">Ig</font> = <font color="Maroon" title="c1">I</font> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">k1</font>) <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c3">k2</font></span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E1:44"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">I</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">I</font> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">k1</font>) <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c3">k2</font></span>)</span> is  <a href="scmpds_4.html#V3" title="SCMPDS_4:attr.3">shiftable</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c8">D</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">I</font></span>)</span></span>)</span> where <font color="Olive" title="b1">l</font> is    <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>  : <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><a href="compos_1.html#K14" title="COMPOS_1:func.14">Load</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">k1</font>) <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c3">k2</font></span>)</span></span>)</span> </span> } </span> ;<br/><span class="kw">let </span><font color="Maroon" title="c9">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font> being   <a href="compos_1.html#NM1" title="COMPOS_1:NM.1">Instruction</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>  st <font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">k1</font>) <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c3">k2</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">k1</font>) <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c3">k2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> holds <br/>(  <a href="compos_1.html#K5" title="COMPOS_1:func.5">InsCode</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp;  <a href="compos_1.html#K5" title="COMPOS_1:func.5">InsCode</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 3 &amp; <font color="Olive">b<sub>3</sub></font> <a href="scmpds_4.html#R1" title="SCMPDS_4:pred.1">valid_at</a> <font color="Olive">b<sub>2</sub></font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c10">i</font> be   <a href="compos_1.html#NM1" title="COMPOS_1:NM.1">Instruction</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">k1</font>) <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c3">k2</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">k1</font>) <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c3">k2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> implies (  <a href="compos_1.html#K5" title="COMPOS_1:func.5">InsCode</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp;  <a href="compos_1.html#K5" title="COMPOS_1:func.5">InsCode</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 3 &amp; <font color="Olive">b<sub>2</sub></font> <a href="scmpds_4.html#R1" title="SCMPDS_4:pred.1">valid_at</a> <font color="Olive">b<sub>1</sub></font> ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:44_1"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">k1</font>) <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c3">k2</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:44_1"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">k1</font>) <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c3">k2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="compos_1.html#K5" title="COMPOS_1:func.5">InsCode</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp;  <a href="compos_1.html#K5" title="COMPOS_1:func.5">InsCode</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 3 &amp; <font color="Olive">b<sub>2</sub></font> <a href="scmpds_4.html#R1" title="SCMPDS_4:pred.1">valid_at</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><a NAME="E3:44_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="valued_1.html#K61" title="VALUED_1:func.61">Shift</a> (<span class="p2">(<span class="default"><a href="compos_1.html#K14" title="COMPOS_1:func.14">Load</a> <span class="p3">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">k1</font>) <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c3">k2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">I</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">I</font></span>)</span></span>)</span> where <font color="Olive" title="b1">l</font> is    <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>  : <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><a href="compos_1.html#K14" title="COMPOS_1:func.14">Load</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">k1</font>) <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c3">k2</font></span>)</span></span>)</span> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D12" title="VALUED_1:def.12">VALUED_1:def 12</a></span>;<br/><a NAME="E4:44_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A4</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">k1</font>) <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c3">k2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c1">I</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">I</font></span>)</span></span>)</span> where <font color="Olive" title="b1">l</font> is    <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>  : <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><a href="compos_1.html#K14" title="COMPOS_1:func.14">Load</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">k1</font>) <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c3">k2</font></span>)</span></span>)</span> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:44_1_1"/>
( <font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c1">I</font> or <font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">I</font></span>)</span></span>)</span> where <font color="Olive" title="b1">l</font> is    <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>  : <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><a href="compos_1.html#K14" title="COMPOS_1:func.14">Load</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">k1</font>) <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c3">k2</font></span>)</span></span>)</span> </span> } </span>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:44_1"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E4:44_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:44_1_1_1"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c1">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="compos_1.html#K5" title="COMPOS_1:func.5">InsCode</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp;  <a href="compos_1.html#K5" title="COMPOS_1:func.5">InsCode</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 3 &amp; <font color="Olive">b<sub>2</sub></font> <a href="scmpds_4.html#R1" title="SCMPDS_4:pred.1">valid_at</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><div class="add"><a NAME="E2:44_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:44_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="afinsq_1.html#D3" title="AFINSQ_1:def.3">AFINSQ_1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:44_1_1_1"/>
(  <a href="compos_1.html#K5" title="COMPOS_1:func.5">InsCode</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp;  <a href="compos_1.html#K5" title="COMPOS_1:func.5">InsCode</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 3 &amp; <font color="Maroon" title="c10">i</font> <a href="scmpds_4.html#R1" title="SCMPDS_4:pred.1">valid_at</a> <font color="Maroon" title="c9">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:44_1_1_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="ref" href="scmpds_4.html#D9" target="_self" title="SCMPDS_4:def.9">Def12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:44_1_1_2"/>
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">I</font></span>)</span></span>)</span> where <font color="Olive" title="b1">l</font> is    <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>  : <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><a href="compos_1.html#K14" title="COMPOS_1:func.14">Load</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">k1</font>) <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c3">k2</font></span>)</span></span>)</span> </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="compos_1.html#K5" title="COMPOS_1:func.5">InsCode</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp;  <a href="compos_1.html#K5" title="COMPOS_1:func.5">InsCode</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 3 &amp; <font color="Olive">b<sub>2</sub></font> <a href="scmpds_4.html#R1" title="SCMPDS_4:pred.1">valid_at</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">l</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E3:44_1_1_2"/><span class="lab"><font color="Green" title="E20">A6</font></span>: 
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">I</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:44_1_1_2"/><span class="lab"><font color="Green" title="E21">A7</font></span>: 
<font color="Maroon" title="c11">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><a href="compos_1.html#K14" title="COMPOS_1:func.14">Load</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">k1</font>) <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c3">k2</font></span>)</span></span>)</span>
 ;<br/><a NAME="E5:44_1_1_2"/>
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><a href="compos_1.html#K14" title="COMPOS_1:func.14">Load</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">k1</font>) <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c3">k2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/><a NAME="E6:44_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A8</font></span>: 
<font color="Maroon" title="c11">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:44_1_1_2"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A9</font></span>: (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">k1</font>) <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c3">k2</font> = 
<span class="p1">(<span class="default"><a href="compos_1.html#K14" title="COMPOS_1:func.14">Load</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">k1</font>) <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c3">k2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">l</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T72" title="FUNCOP_1:th.72">FUNCOP_1:72</a></span>
<br/>.= 
<font color="Maroon" title="c10">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:44_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E3:44_1_1_2"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E4:44_1_1_2"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="ref" href="afinsq_1.html#D3" title="AFINSQ_1:def.3">AFINSQ_1:def 3</a></span>
;<br/><span class="kw">hence </span><a NAME="E8:44_1_1_2"/>
(  <a href="compos_1.html#K5" title="COMPOS_1:func.5">InsCode</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp;  <a href="compos_1.html#K5" title="COMPOS_1:func.5">InsCode</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 3 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_2.html#T16" title="SCMPDS_2:th.16">SCMPDS_2:16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">i</font> <a href="scmpds_4.html#R1" title="SCMPDS_4:pred.1">valid_at</a> <font color="Maroon" title="c9">n</font></span><br/><a NAME="E9:44_1_1_2"/><span class="lab"><font color="Green" title="E24">A10</font></span>: 
 <a href="compos_1.html#K5" title="COMPOS_1:func.5">InsCode</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 6
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:44_1_1_2"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="ref" href="scmpds_2.html#T16" title="SCMPDS_2:th.16">SCMPDS_2:16</a></span>;<br/><a NAME="E10:44_1_1_2"/>
(  <a href="compos_1.html#K5" title="COMPOS_1:func.5">InsCode</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="compos_1.html#K5" title="COMPOS_1:func.5">InsCode</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 5 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:44_1_1_2"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="ref" href="scmpds_2.html#T16" title="SCMPDS_2:th.16">SCMPDS_2:16</a></span>;<br/><span class="kw">hence </span><a NAME="E11:44_1_1_2"/>
<font color="Maroon" title="c10">i</font> <a href="scmpds_4.html#R1" title="SCMPDS_4:pred.1">valid_at</a> <font color="Maroon" title="c9">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:44"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E3:44_1_1_2"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E6:44_1_1_2"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="txt" href="#E7:44_1_1_2"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="txt" href="#E9:44_1_1_2"><span class="lab"><font color="Green" title="E24">A10</font></span></a>, <a class="ref" href="scmpds_4.html#D8" target="_self" title="SCMPDS_4:def.8">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:44"/>
<font color="Maroon" title="c1">I</font> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">k1</font>) <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c3">k2</font></span>)</span> is  <a href="scmpds_4.html#V3" title="SCMPDS_4:attr.3">shiftable</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_4.html#D9" target="_self" title="SCMPDS_4:def.9">Def12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
