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<span class="kw">let </span><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font>, <font color="Maroon" title="c4">r</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K1" title="TREES_1:func.1">ProperPrefixes</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Maroon" title="c4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K1" title="TREES_1:func.1">ProperPrefixes</a> <font color="Maroon" title="c2">p</font> implies <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">r</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:23"/>
<font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K1" title="TREES_1:func.1">ProperPrefixes</a> <font color="Maroon" title="c2">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K1" title="TREES_1:func.1">ProperPrefixes</a> <font color="Maroon" title="c2">p</font> or <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">r</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a>  )</span><br/>

<a NAME="E2:23"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A1</font></span>: 
 ex <font color="Olive" title="b1">q1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q1</font> &amp; <font color="Olive" title="b1">q1</font> <a href="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <font color="Maroon" title="c2">p</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#D2" target="_self" title="TREES_1:def.2">Def4</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:23"/>
<font color="Maroon" title="c4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K1" title="TREES_1:func.1">ProperPrefixes</a> <font color="Maroon" title="c2">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">r</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> </span><br/>

<a NAME="E4:23"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A2</font></span>: 
 ex <font color="Olive" title="b1">r1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r1</font> &amp; <font color="Olive" title="b1">r1</font> <a href="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <font color="Maroon" title="c2">p</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#D2" target="_self" title="TREES_1:def.2">Def4</a></span>;<br/>
<a NAME="E5:23"/>
<font color="Maroon" title="c3">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c2">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">n</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E7:23"/><span class="lab"><font color="Green" title="E21">A3</font></span>: 
<font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c5">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#D1" target="_self" title="TREES_1:def.1">Def1</a></span>;<br/>
<a NAME="E8:23"/>
<font color="Maroon" title="c4">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c2">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">k</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E10:23"/><span class="lab"><font color="Green" title="E22">A4</font></span>: 
<font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#D1" target="_self" title="TREES_1:def.1">Def1</a></span>;<br/>
<a NAME="E11:23"/><span class="lab"><font color="Green" title="E23">A5</font></span>: 
( <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> implies <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">r</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:23_1"/>
<font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">r</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> </span><br/>

<a NAME="E2:23_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c5">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></span>;<br/>
<a NAME="E3:23_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c5">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:23"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E10:23"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="ref" href="funct_1.html#T51" title="FUNCT_1:th.51">FUNCT_1:51</a></span>;<br/>
<a NAME="E4:23_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c4">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#D1" target="_self" title="TREES_1:def.1">Def1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:23_1"/>
<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">r</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D9" title="XBOOLE_0:def.9">XBOOLE_0:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E12:23"/>
( <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font> implies <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">r</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:23_2"/>
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">r</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> </span><br/>

<a NAME="E2:23_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c5">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></span>;<br/>
<a NAME="E3:23_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:23"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E10:23"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="ref" href="funct_1.html#T51" title="FUNCT_1:th.51">FUNCT_1:51</a></span>;<br/>
<a NAME="E4:23_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c3">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#D1" target="_self" title="TREES_1:def.1">Def1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:23_2"/>
<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">r</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D9" title="XBOOLE_0:def.9">XBOOLE_0:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E13:23"/>
<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">r</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:23"><span class="lab"><font color="Green" title="E23">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
