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<b>let </b><font color="Maroon" title="c1">F</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">t</font>, <font color="Olive" title="b2">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>  st <font color="Olive" title="b1">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">t'</font> &amp; <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="Olive" title="b1">t</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="Olive" title="b2">t'</font> holds <br/> not <font color="Olive" title="b1">t</font>,<font color="Olive" title="b2">t'</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> </span><br/><b>let </b><font color="Maroon" title="c2">t</font>, <font color="Maroon" title="c3">t'</font> be    <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">t'</font> &amp; <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="c2">t</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="c3">t'</font> implies  not <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c3">t'</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a>  )</span><br/>





<b>assume </b><a NAME="E1:46"/><i><font color="Green" title="E40">A1</font></i>: 
( <font color="Maroon" title="c2">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">t'</font> &amp; <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="c2">t</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="c3">t'</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c3">t'</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> </span><br/>

<b>assume </b><a NAME="E2:46"/><i><font color="Green" title="E41">A2</font></i>: 
<font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c3">t'</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_1"/>
( <font color="Maroon" title="c2">t</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c3">t'</font> or <font color="Maroon" title="c3">t'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c2">t</font> )
 </span><b>by </b><i><a class="txt" href="qc_lang4.html#E2:46"><i><font color="Green" title="E41">A2</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="46_1_1"><b>suppose </b></a><a NAME="E1:46_1_1"/>
<font color="Maroon" title="c2">t</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c3">t'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:46_1_1"/><b>then </b>
<font color="Maroon" title="c2">t</font> <a href="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <font color="Maroon" title="c3">t'</font>
 <b>by </b><i><a class="txt" href="qc_lang4.html#E1:46"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/><b>hence </b><a NAME="E3:46_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="qc_lang4.html#E1:46"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="qc_lang4.html#T45" target="_self" title="QC_LANG4:th.45">Th45</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="46_1_2"><b>suppose </b></a><a NAME="E1:46_1_2"/>
<font color="Maroon" title="c3">t'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c2">t</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:46_1_2"/><b>then </b>
<font color="Maroon" title="c3">t'</font> <a href="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <font color="Maroon" title="c2">t</font>
 <b>by </b><i><a class="txt" href="qc_lang4.html#E1:46"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/><b>hence </b><a NAME="E3:46_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="qc_lang4.html#E1:46"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="qc_lang4.html#T45" target="_self" title="QC_LANG4:th.45">Th45</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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