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<b>let </b><font color="Maroon" title="c3">P</font>, <font color="Maroon" title="c4">Q</font> be  <a href="ami_1.html#V15" title="AMI_1:attr.15">programmed</a> <a href="ami_1.html#NM5" title="AMI_1:NM.5">FinPartState</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> ;<br/><b>let </b><font color="Maroon" title="c5">k</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="E1:3"/><i><font color="Green" title="E2">A1</font></i>: 
<font color="Maroon" title="c3">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Q</font>
 ;<br/>

<a NAME="E2:3"/><b>then </b><i><font color="Green" title="E3">A2</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">Q</font>
 
<b>by </b><i><a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>;<br/>
<a NAME="E3:3"/><i><font color="Green" title="E4">A3</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K6" title="SCMPDS_3:func.6">Shift</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c5">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmpds_3.html#K2" title="SCMPDS_3:func.2">inspos</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span> where <font color="Olive" title="b1">m</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>  :  <a href="scmpds_3.html#K2" title="SCMPDS_3:func.2">inspos</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">P</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="scmpds_3.html#D7" title="SCMPDS_3:def.7">SCMPDS_3:def 7</a></i>;<br/>
<a NAME="E4:3"/><i><font color="Green" title="E5">A4</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K6" title="SCMPDS_3:func.6">Shift</a> <font color="Maroon" title="c4">Q</font>,<font color="Maroon" title="c5">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmpds_3.html#K2" title="SCMPDS_3:func.2">inspos</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span> where <font color="Olive" title="b1">m</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>  :  <a href="scmpds_3.html#K2" title="SCMPDS_3:func.2">inspos</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">Q</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="scmpds_3.html#D7" title="SCMPDS_3:def.7">SCMPDS_3:def 7</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:3_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K6" title="SCMPDS_3:func.6">Shift</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c5">k</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c7">m1</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> <b> such that </b><br/><a NAME="E3:3_1"/><i><font color="Green" title="E6">A5</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_3.html#K2" title="SCMPDS_3:func.2">inspos</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">k</font></span>)</span> &amp;  <a href="scmpds_3.html#K2" title="SCMPDS_3:func.2">inspos</a> <font color="Maroon" title="c7">m1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">P</font> )
 <b>by </b><i><a class="txt" href="scmpds_7.html#E3:3"><i><font color="Green" title="E4">A3</font></i></a></i>;<br/><b>thus </b><a NAME="E4:3_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K6" title="SCMPDS_3:func.6">Shift</a> <font color="Maroon" title="c4">Q</font>,<font color="Maroon" title="c5">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="scmpds_7.html#E2:3"><i><font color="Green" title="E3">A2</font></i></a>, <a class="txt" href="scmpds_7.html#E4:3"><i><font color="Green" title="E5">A4</font></i></a>, <a class="txt" href="scmpds_7.html#E3:3_1"><i><font color="Green" title="E6">A5</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:3"/><b>then </b><i><font color="Green" title="E6">A6</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K6" title="SCMPDS_3:func.6">Shift</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c5">k</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K6" title="SCMPDS_3:func.6">Shift</a> <font color="Maroon" title="c4">Q</font>,<font color="Maroon" title="c5">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:3_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K6" title="SCMPDS_3:func.6">Shift</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c5">k</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c7">m1</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> <b> such that </b><br/><a NAME="E3:3_2"/><i><font color="Green" title="E7">A7</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_3.html#K2" title="SCMPDS_3:func.2">inspos</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">k</font></span>)</span> &amp;  <a href="scmpds_3.html#K2" title="SCMPDS_3:func.2">inspos</a> <font color="Maroon" title="c7">m1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">P</font> )
 <b>by </b><i><a class="txt" href="scmpds_7.html#E3:3"><i><font color="Green" title="E4">A3</font></i></a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="scmpds_3.html#K6" title="SCMPDS_3:func.6">Shift</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c5">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> = 
<span class="p1">(<span class="default"><a href="scmpds_3.html#K6" title="SCMPDS_3:func.6">Shift</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c5">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_3.html#K2" title="SCMPDS_3:func.2">inspos</a> <font color="Maroon" title="c7">m1</font></span>)</span> <a href="scmpds_3.html#K3" title="SCMPDS_3:func.3">+</a> <font color="Maroon" title="c5">k</font></span>)</span>
<b>by </b><i><a class="txt" href="scmpds_7.html#E3:3_2"><i><font color="Green" title="E7">A7</font></i></a>, <a class="ref" href="scmpds_3.html#D3" title="SCMPDS_3:def.3">SCMPDS_3:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K2" title="SCMPDS_3:func.2">inspos</a> <font color="Maroon" title="c7">m1</font></span>)</span>
<b>by </b><i><a class="txt" href="scmpds_7.html#E3:3_2"><i><font color="Green" title="E7">A7</font></i></a>, <a class="ref" href="scmpds_3.html#T37" title="SCMPDS_3:th.37">SCMPDS_3:37</a></i>
<br/>.= 
<font color="Maroon" title="c4">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K2" title="SCMPDS_3:func.2">inspos</a> <font color="Maroon" title="c7">m1</font></span>)</span>
<b>by </b><i><a class="txt" href="scmpds_7.html#E1:3"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="scmpds_7.html#E3:3_2"><i><font color="Green" title="E7">A7</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="scmpds_3.html#K6" title="SCMPDS_3:func.6">Shift</a> <font color="Maroon" title="c4">Q</font>,<font color="Maroon" title="c5">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_3.html#K2" title="SCMPDS_3:func.2">inspos</a> <font color="Maroon" title="c7">m1</font></span>)</span> <a href="scmpds_3.html#K3" title="SCMPDS_3:func.3">+</a> <font color="Maroon" title="c5">k</font></span>)</span>
<b>by </b><i><a class="txt" href="scmpds_7.html#E2:3"><i><font color="Green" title="E3">A2</font></i></a>, <a class="txt" href="scmpds_7.html#E3:3_2"><i><font color="Green" title="E7">A7</font></i></a>, <a class="ref" href="scmpds_3.html#T37" title="SCMPDS_3:th.37">SCMPDS_3:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="scmpds_3.html#K6" title="SCMPDS_3:func.6">Shift</a> <font color="Maroon" title="c4">Q</font>,<font color="Maroon" title="c5">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
<b>by </b><i><a class="txt" href="scmpds_7.html#E3:3_2"><i><font color="Green" title="E7">A7</font></i></a>, <a class="ref" href="scmpds_3.html#D3" title="SCMPDS_3:def.3">SCMPDS_3:def 3</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:3"/>
 <a href="scmpds_3.html#K6" title="SCMPDS_3:func.6">Shift</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c5">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="scmpds_3.html#K6" title="SCMPDS_3:func.6">Shift</a> <font color="Maroon" title="c4">Q</font>,<font color="Maroon" title="c5">k</font>
 <b>by </b><i><a class="txt" href="scmpds_7.html#E6:3"><i><font color="Green" title="E6">A6</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>;<br/>


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