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<b>thus </b><a NAME="E1:23_1_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> implies for <font color="Olive">a</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font> holds <br/> ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> iff <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> ) )
 ;<br/>

<b>thus </b><a NAME="E2:23_1_1"/>
( ( for <font color="Olive">a</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font> holds <br/> ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> iff <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> ) ) implies <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_1_1_1"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">a</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font> holds <br/> ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> iff <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_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/><div><i><font color="Green">E36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_1_1"><b>now </b></a><div class="add"><a NAME="E1:23_1_1_1_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">p</font> is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="finseq_1.html#K13">*</a> </span>)</span>
 <b>by </b><i/>;<br/><b>assume </b><a NAME="E2:23_1_1_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font>
 ;<br/><a NAME="E3:23_1_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> is    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/><b>hence </b><a NAME="E4:23_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_1_2"><b>now </b></a><div class="add"><a NAME="E1:23_1_1_1_1_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">r</font> is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="finseq_1.html#K13">*</a> </span>)</span>
 <b>by </b><i/>;<br/><b>assume </b><a NAME="E2:23_1_1_1_1_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font>
 ;<br/><a NAME="E3:23_1_1_1_1_2"/><b>then </b>
<font color="Maroon">x</font> is    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/><b>hence </b><a NAME="E4:23_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:23_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> iff <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:23_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</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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