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<b>let </b><font color="Maroon" title="c1">F</font>, <font color="Maroon" title="c2">G</font>, <font color="Maroon" title="c3">H</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c1">F</font></span>)</span>  st <font color="Maroon" title="c2">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b2">t</font></span>)</span> where <font color="Olive" title="b2">t</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c1">F</font></span>)</span> : <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> </span>}</span>  &amp; <font color="Maroon" title="c3">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b2">t</font></span>)</span> where <font color="Olive" title="b2">t</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c1">F</font></span>)</span> : <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> </span>}</span>  &amp; not <font color="Maroon" title="c2">G</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a> <font color="Maroon" title="c3">H</font> holds <br/><font color="Maroon" title="c3">H</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a> <font color="Maroon" title="c2">G</font></span><br/><b>let </b><font color="Maroon" title="c4">C</font> be    <a href="trees_2.html#M1" title="TREES_2:mode.1">Chain</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c1">F</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b1">t</font></span>)</span> where <font color="Olive" title="b1">t</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c1">F</font></span>)</span> : <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> </span>}</span>  &amp; <font color="Maroon" title="c3">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b1">t</font></span>)</span> where <font color="Olive" title="b1">t</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c1">F</font></span>)</span> : <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> </span>}</span>  &amp; not <font color="Maroon" title="c2">G</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a> <font color="Maroon" title="c3">H</font> implies <font color="Maroon" title="c3">H</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a> <font color="Maroon" title="c2">G</font> )</span><br/>







<b>assume </b><a NAME="E1:63"/><i><font color="Green" title="E55">A1</font></i>: 
( <font color="Maroon" title="c2">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b1">t</font></span>)</span> where <font color="Olive" title="b1">t</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c1">F</font></span>)</span> : <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> </span>}</span>  &amp; <font color="Maroon" title="c3">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b1">t</font></span>)</span> where <font color="Olive" title="b1">t</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c1">F</font></span>)</span> : <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> </span>}</span>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">G</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a> <font color="Maroon" title="c3">H</font> or <font color="Maroon" title="c3">H</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a> <font color="Maroon" title="c2">G</font> )</span><br/>

<b>then consider </b><font color="Maroon" title="c5">t'</font> being    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c1">F</font></span>)</span><b> such that </b><br/><a NAME="E3:63"/><i><font color="Green" title="E56">A2</font></i>: 
( <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c5">t'</font> &amp; <font color="Maroon" title="c5">t'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c6">t''</font> being    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c1">F</font></span>)</span><b> such that </b><br/><a NAME="E5:63"/><i><font color="Green" title="E57">A3</font></i>: 
( <font color="Maroon" title="c3">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c6">t''</font> &amp; <font color="Maroon" title="c6">t''</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> )
 <b>by </b><i><a class="txt" href="#E1:63"><i><font color="Green" title="E55">A1</font></i></a></i>;<br/>
<a NAME="E6:63"/><i><font color="Green" title="E58">A4</font></i>: 
<font color="Maroon" title="c5">t'</font>,<font color="Maroon" title="c6">t''</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> 
 
<b>by </b><i><a class="txt" href="#E3:63"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="#E5:63"><i><font color="Green" title="E57">A3</font></i></a>, <a class="ref" href="trees_2.html#D3" title="TREES_2:def.3">TREES_2:def 3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:63_1"/>
( <font color="Maroon" title="c5">t'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c6">t''</font> or <font color="Maroon" title="c6">t''</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c5">t'</font> )
 </span><b>by </b><i><a class="txt" href="#E6:63"><i><font color="Green" title="E58">A4</font></i></a>, <a class="ref" href="xboole_0.html#D9" title="XBOOLE_0:def.9">XBOOLE_0:def 9</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1"><b>suppose </b></a><a NAME="E1:63_1_1"/>
<font color="Maroon" title="c5">t'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c6">t''</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">G</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a> <font color="Maroon" title="c3">H</font> or <font color="Maroon" title="c3">H</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a> <font color="Maroon" title="c2">G</font> )</span><br/><div class="add"><b>hence </b><a NAME="E2:63_1_1"/>
( <font color="Maroon" title="c2">G</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a> <font color="Maroon" title="c3">H</font> or <font color="Maroon" title="c3">H</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a> <font color="Maroon" title="c2">G</font> )
 <b>by </b><i><a class="txt" href="#E3:63"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="#E5:63"><i><font color="Green" title="E57">A3</font></i></a>, <a class="ref" href="qc_lang4.html#T42" target="_self" title="QC_LANG4:th.42">Th42</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2"><b>suppose </b></a><a NAME="E1:63_1_2"/>
<font color="Maroon" title="c6">t''</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c5">t'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">G</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a> <font color="Maroon" title="c3">H</font> or <font color="Maroon" title="c3">H</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a> <font color="Maroon" title="c2">G</font> )</span><br/><div class="add"><b>hence </b><a NAME="E2:63_1_2"/>
( <font color="Maroon" title="c2">G</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a> <font color="Maroon" title="c3">H</font> or <font color="Maroon" title="c3">H</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a> <font color="Maroon" title="c2">G</font> )
 <b>by </b><i><a class="txt" href="#E3:63"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="#E5:63"><i><font color="Green" title="E57">A3</font></i></a>, <a class="ref" href="qc_lang4.html#T42" target="_self" title="QC_LANG4:th.42">Th42</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

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