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<span class="kw">let </span><font color="Maroon" title="c1">W</font> be   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font>, <font color="Olive" title="b3">v</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><br/> for <font color="Olive" title="b4">C</font> being   <a href="trees_2.html#M1" title="TREES_2:mode.1">Chain</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b1">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">C</font> &amp; <font color="Olive" title="b2">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">C</font> &amp; <font color="Olive" title="b3">v</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Olive" title="b2">v2</font> &amp; not <font color="Olive" title="b1">v1</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="Olive" title="b3">v</font> holds <br/><font color="Olive" title="b3">v</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Olive" title="b1">v1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">v1</font>, <font color="Maroon" title="c3">v2</font>, <font color="Maroon" title="c4">v</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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</font> being   <a href="trees_2.html#M1" title="TREES_2:mode.1">Chain</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon" title="c2">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c3">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c4">v</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c3">v2</font> &amp; not <font color="Maroon" title="c2">v1</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="c4">v</font> holds <br/><font color="Maroon" title="c4">v</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c2">v1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">C</font> be    <a href="trees_2.html#M1" title="TREES_2:mode.1">Chain</a> of <font color="Maroon" title="c1">W</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" title="c2">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> &amp; <font color="Maroon" title="c3">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> &amp; <font color="Maroon" title="c4">v</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c3">v2</font> &amp; not <font color="Maroon" title="c2">v1</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="c4">v</font> implies <font color="Maroon" title="c4">v</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c2">v1</font> )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:37"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
( <font color="Maroon" title="c2">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> &amp; <font color="Maroon" title="c3">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> )
 <span class="kw">and </span><br/><a NAME="E2:37"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
<font color="Maroon" title="c4">v</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c3">v2</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" title="c2">v1</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="c4">v</font> or <font color="Maroon" title="c4">v</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c2">v1</font> )</span><br/>

<a NAME="E3:37"/><span class="lab"><font color="Green" title="E19">A3</font></span>: 
( <font color="Maroon" title="c2">v1</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="c3">v2</font> or <font color="Maroon" title="c3">v2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c2">v1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="trees_2.html#T23" target="_self" title="TREES_2:th.23">Th23</a></span>;<br/>
<a NAME="E4:37"/><span class="lab"><font color="Green" title="E20">A4</font></span>: 
( <font color="Maroon" title="c2">v1</font> <a href="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <font color="Maroon" title="c3">v2</font> or <font color="Maroon" title="c4">v</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c2">v1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E3:37"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="trees_1.html#T36" title="TREES_1:th.36">TREES_1:36</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E5:37"/><span class="lab"><font color="Green" title="E21">A5</font></span>: 
<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c2">v1</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="#E2:37"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E4:37"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="trees_2.html#T2" target="_self" title="TREES_2:th.2">Th2</a>, <a class="ref" href="xboole_0.html#D9" title="XBOOLE_0:def.9">XBOOLE_0:def 9</a></span>;<br/>
<a NAME="E6:37"/><span class="lab"><font color="Green" title="E22">A6</font></span>: 
( <font color="Maroon" title="c4">v</font> <a href="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <font color="Maroon" title="c2">v1</font> or <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v1</font> or <font color="Maroon" title="c2">v1</font> <a href="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <font color="Maroon" title="c4">v</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="ref" href="xboole_1.html#T104" title="XBOOLE_1:th.104">XBOOLE_1:104</a></span>;<br/>
<span class="kw">thus </span><a NAME="E7:37"/>
( <font color="Maroon" title="c2">v1</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="c4">v</font> or <font color="Maroon" title="c4">v</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c2">v1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="ref" href="trees_1.html#T36" title="TREES_1:th.36">TREES_1:36</a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</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/>


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