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<b>consider </b><font color="Maroon">D</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>
<b>reconsider </b><font color="Maroon">E</font> = <font color="Maroon">D</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">q</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> &amp; <font color="Olive">q</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">p</font> );<br/>
<b>consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:20_1_1"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> <a href="finseq_1.html#K13">*</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<b>take </b>
<font color="Maroon">X</font>
;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E5:20_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> implies  ex <font color="Olive">q</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> &amp; <font color="Olive">q</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">p</font> ) )
 <b>by </b><i/>;<br/>

<b>given </b><font color="Maroon">q</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><a NAME="E6:20_1_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> &amp; <font color="Maroon">q</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">p</font> )
 ;<br/>

<a NAME="E7:20_1_1"/>
<font color="Maroon">q</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<a NAME="E8:20_1_1"/><b>then </b>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Olive">n</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E9:20_1_1"/><b>then </b>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">E</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T99">RELAT_1:99</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E10:20_1_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">E</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">E</font> <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E12:20_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> <a href="finseq_1.html#K13">*</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<b>hence </b><a NAME="E13:20_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, </i>;<br/>


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