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<b>let </b><font color="Maroon">P</font> be  <a href="ami_3.html#V1">programmed</a>  <a href="ami_1.html#M1">FinPartState</a> of  <a href="scmpds_2.html#K1">SCMPDS</a> , <font color="Maroon">Q</font> be  <a href="ami_3.html#V1">programmed</a>  <a href="ami_1.html#M1">FinPartState</a> of  <a href="scmpds_2.html#K1">SCMPDS</a> ;<br/><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:3"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font>
 ;<br/>

<a NAME="E2:3"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Q</font>
 
<b>by </b><i><a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/>
<a NAME="E3:3"/><i><font color="Green">E51</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K6">Shift</a> <font color="Maroon">P</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmpds_3.html#K2">inspos</a> <span class="p3">(<span class="default"><font color="Olive">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> where <font color="Olive">m</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="scmpds_3.html#K2">inspos</a> <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="scmpds_3.html#D7">SCMPDS_3:def 7</a></i>;<br/>
<a NAME="E4:3"/><i><font color="Green">E54</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K6">Shift</a> <font color="Maroon">Q</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmpds_3.html#K2">inspos</a> <span class="p3">(<span class="default"><font color="Olive">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> where <font color="Olive">m</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="scmpds_3.html#K2">inspos</a> <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Q</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="scmpds_3.html#D7">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K6">Shift</a> <font color="Maroon">P</font>,<font color="Maroon">k</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">m1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:3_1"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_3.html#K2">inspos</a> <span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> &amp;  <a href="scmpds_3.html#K2">inspos</a> <font color="Maroon">m1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font> )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E4:3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K6">Shift</a> <font color="Maroon">Q</font>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<a NAME="E6:3"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K6">Shift</a> <font color="Maroon">P</font>,<font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K6">Shift</a> <font color="Maroon">Q</font>,<font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:3_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K6">Shift</a> <font color="Maroon">P</font>,<font color="Maroon">k</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">m1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:3_2"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_3.html#K2">inspos</a> <span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> &amp;  <a href="scmpds_3.html#K2">inspos</a> <font color="Maroon">m1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font> )
 <b>by </b><i/>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="scmpds_3.html#K6">Shift</a> <font color="Maroon">P</font>,<font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="scmpds_3.html#K6">Shift</a> <font color="Maroon">P</font>,<font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_3.html#K2">inspos</a> <font color="Maroon">m1</font></span>)</span> <a href="scmpds_3.html#K3">+</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, <a class="ref" href="scmpds_3.html#D3">SCMPDS_3:def 3</a></i>
<br/>.= 
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K2">inspos</a> <font color="Maroon">m1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="scmpds_3.html#T37">SCMPDS_3:37</a></i>
<br/>.= 
<font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K2">inspos</a> <font color="Maroon">m1</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="scmpds_3.html#K6">Shift</a> <font color="Maroon">Q</font>,<font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_3.html#K2">inspos</a> <font color="Maroon">m1</font></span>)</span> <a href="scmpds_3.html#K3">+</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="scmpds_3.html#T37">SCMPDS_3:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="scmpds_3.html#K6">Shift</a> <font color="Maroon">Q</font>,<font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="scmpds_3.html#D3">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">Shift</a> <font color="Maroon">P</font>,<font color="Maroon">k</font> <a href="tarski.html#R1">c=</a>  <a href="scmpds_3.html#K6">Shift</a> <font color="Maroon">Q</font>,<font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/>


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