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<div class="add">

<b>thus </b><a NAME="E1:8_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> implies for <font color="Olive">b<sub>1</sub></font> being  <a href="finseq_1.html#NM1">FinSequence</a> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> iff <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) )
 ;<br/>

<b>thus </b><a NAME="E2:8_1_1"/>
( ( for <font color="Olive">b<sub>1</sub></font> being  <a href="finseq_1.html#NM1">FinSequence</a> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> iff <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) ) implies <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_1_1_1"/><i><font color="Green">E2</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="finseq_1.html#NM1">FinSequence</a> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> iff <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><i><font color="Green">E3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:8_1_1_1_1_1"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:8_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> is   <a href="finseq_1.html#NM1">FinSequence</a>
 <b>by </b><i><a class="ref" href="margrel1.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E3:8_1_1_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="margrel1.html#E1:8_1_1_1"><i><font color="Green">E2</font></i></a>, <a class="txt" href="margrel1.html#E1:8_1_1_1_1_1"><i><font color="Green">E4</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:8_1_1_1_1_2"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:8_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> is   <a href="finseq_1.html#NM1">FinSequence</a>
 <b>by </b><i><a class="ref" href="margrel1.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E3:8_1_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="margrel1.html#E1:8_1_1_1"><i><font color="Green">E2</font></i></a>, <a class="txt" href="margrel1.html#E1:8_1_1_1_1_2"><i><font color="Green">E4</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:8_1_1_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> iff <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="margrel1.html#E1:8_1_1_1_1"><i><font color="Green">E3</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:8_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div>


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