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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>( ( not <font color="Maroon" title="c2">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> $1 &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 ) or  ex <font color="Olive" title="b1">r</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">r</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">T1</font> &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b1">r</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font> ) );<br/>
<a NAME="E1:63_1_1"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
for <font color="Olive" title="b1">q</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">T1</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">q</font>,<font color="Olive" title="b2">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">q</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">T1</font></span>)</span> implies  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">q</font>,<font color="Olive" title="b1">x</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:63_1_1_1"/><span class="lab"><font color="Green" title="E31">A3</font></span>: 
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">T1</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">q</font>,<font color="Olive" title="b1">x</font>]</span><br/>

<div><span class="lab"><font color="Green" title="E32">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:63_1_1_1_1_1"/>
( <font color="Maroon" title="c2">p</font> <a href="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <font color="Maroon" title="c4">q</font> or <font color="Maroon" title="c2">p</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c4">q</font> or  not <font color="Maroon" title="c2">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c4">q</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:63_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E33">A5</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <font color="Maroon" title="c4">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">q</font>,<font color="Olive" title="b1">x</font>]</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c5">r</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:63_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E34">A6</font></span>: 
( <font color="Maroon" title="c5">r</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">T1</font> &amp; <font color="Maroon" title="c4">q</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">r</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E1:63_1_1_1"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="txt" href="#E1:63_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="ref" href="trees_1.html#D12" title="TREES_1:def.12">TREES_1:def 12</a></span>;<br/><span class="kw">thus </span><a NAME="E4:63_1_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">q</font>,<font color="Olive" title="b1">x</font>]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span>
<font color="Maroon" title="c3">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">r</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">r</font>]</span><br/>

<span class="kw">thus </span><a NAME="E1:63_1_1_1_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">r</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E34">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:63_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E33">A7</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c4">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">q</font>,<font color="Olive" title="b1">x</font>]</span><br/><div class="add"><span class="kw">thus </span><a NAME="E2:63_1_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">q</font>,<font color="Olive" title="b1">x</font>]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span>
<font color="Maroon" title="c3">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span>]</span><br/>

<a NAME="E1:63_1_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E34">A8</font></span>: 
(  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <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">T1</font> &amp; <font color="Maroon" title="c4">q</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a>, <a class="ref" href="trees_1.html#T47" title="TREES_1:th.47">TREES_1:47</a></span>;<br/>
<span class="kw">thus </span><a NAME="E2:63_1_1_1_1_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E34">A8</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:63_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E33">A9</font></span>: 
not <font color="Maroon" title="c2">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c4">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">q</font>,<font color="Olive" title="b1">x</font>]</span><br/><div class="add"><span class="kw">thus </span><a NAME="E2:63_1_1_1_1_1_3"/>
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">q</font>,<font color="Olive" title="b1">x</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E33">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E3:63_1_1_1"/>
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">q</font>,<font color="Olive" title="b1">x</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63_1_1_1"><span class="lab"><font color="Green" title="E32">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E2:63_1_1"/>
 ex <font color="Olive" title="b1">TT</font> being  <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a> st <br/>(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">TT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">T1</font></span>)</span> &amp; ( for <font color="Olive" title="b2">q</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">T1</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">q</font>,<font color="Olive" title="b1">TT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">q</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="trees_2.html#S6" target="_self" title="TREES_2:sch.6">TREES_2:sch 6</a>(<a class="txt" href="#E1:63_1_1"><span class="lab"><font color="Green" title="E30">A2</font></span></a>);</span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
