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<b>let </b><font color="Maroon" title="c2">g1</font>, <font color="Maroon" title="c3">g2</font> be   <a href="ami_1.html#NM7" title="AMI_1:NM.7">FinPartState</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:4_1_2"/><i><font color="Green" title="E4">A14</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">g1</font> <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">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</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>  : <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">f</font> </span>}</span>  &amp; ( for <font color="Olive" title="b1">k</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>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">g1</font> holds <br/><font color="Maroon" title="c2">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) )
 <b>and </b><br/><a NAME="E2:4_1_2"/><i><font color="Green" title="E5">A15</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">g2</font> <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">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</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>  : <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">f</font> </span>}</span>  &amp; ( for <font color="Olive" title="b1">k</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>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">g2</font> holds <br/><font color="Maroon" title="c3">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:4_1_2_1"/><i><font color="Green" title="E6">A16</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">g1</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c5">k1</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="E3:4_1_2_1"/><i><font color="Green" title="E7">A17</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp; <font color="Maroon" title="c5">k1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">f</font> )
 <b>by </b><i><a class="txt" href="scmfsa_7.html#E1:4_1_2"><i><font color="Green" title="E4">A14</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">k</font> = <font color="Maroon" title="c5">k1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 as    <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/><a NAME="E5:4_1_2_1"/>
<font color="Maroon" title="c3">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="scmfsa_7.html#E1:4_1_2"><i><font color="Green" title="E4">A14</font></i></a>, <a class="txt" href="scmfsa_7.html#E2:4_1_2"><i><font color="Green" title="E5">A15</font></i></a>, <a class="txt" href="scmfsa_7.html#E1:4_1_2_1"><i><font color="Green" title="E6">A16</font></i></a>, <a class="txt" href="scmfsa_7.html#E3:4_1_2_1"><i><font color="Green" title="E7">A17</font></i></a></i>;<br/><b>hence </b><a NAME="E6:4_1_2_1"/>
<font color="Maroon" title="c2">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="scmfsa_7.html#E1:4_1_2"><i><font color="Green" title="E4">A14</font></i></a>, <a class="txt" href="scmfsa_7.html#E1:4_1_2_1"><i><font color="Green" title="E6">A16</font></i></a>, <a class="txt" href="scmfsa_7.html#E3:4_1_2_1"><i><font color="Green" title="E7">A17</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:4_1_2"/>
<font color="Maroon" title="c2">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g2</font>
 <b>by </b><i><a class="txt" href="scmfsa_7.html#E1:4_1_2"><i><font color="Green" title="E4">A14</font></i></a>, <a class="txt" href="scmfsa_7.html#E2:4_1_2"><i><font color="Green" title="E5">A15</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


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