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<b>let </b><font color="Maroon" title="c4">A1</font>, <font color="Maroon" title="c5">A2</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:83_1_2"/><i><font color="Green" title="E39">A2</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">X</font> &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">X</font> holds <br/><font color="Maroon" title="c4">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="roughs_1.html#K6" title="ROUGHS_1:func.6">MemberFunc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="roughs_1.html#K1" title="ROUGHS_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font> ) )
 <b>and </b><br/><a NAME="E2:83_1_2"/><i><font color="Green" title="E40">A3</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">X</font> &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">X</font> holds <br/><font color="Maroon" title="c5">A2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="roughs_1.html#K6" title="ROUGHS_1:func.6">MemberFunc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="roughs_1.html#K1" title="ROUGHS_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font> ) )
 ;<br/>

<a NAME="E3:83_1_2"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">A1</font> holds <br/><font color="Maroon" title="c4">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:83_1_2_1"/><i><font color="Green" title="E41">A4</font></i>: 
<font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">A1</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c7">n</font> = <font color="Maroon" title="c6">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<font color="Maroon" title="c4">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> = 
<span class="p1">(<span class="default"><a href="roughs_1.html#K6" title="ROUGHS_1:func.6">MemberFunc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="roughs_1.html#K1" title="ROUGHS_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span>,<font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font>
<b>by </b><i><a class="txt" href="roughs_1.html#E1:83_1_2"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="roughs_1.html#E1:83_1_2_1"><i><font color="Green" title="E41">A4</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">A2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font>
<b>by </b><i><a class="txt" href="roughs_1.html#E1:83_1_2"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="roughs_1.html#E2:83_1_2"><i><font color="Green" title="E40">A3</font></i></a>, <a class="txt" href="roughs_1.html#E1:83_1_2_1"><i><font color="Green" title="E41">A4</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E4:83_1_2_1"/>
<font color="Maroon" title="c4">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:83_1_2"/>
<font color="Maroon" title="c4">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A2</font>
 <b>by </b><i><a class="txt" href="roughs_1.html#E1:83_1_2"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="roughs_1.html#E2:83_1_2"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></i>;<br/>


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