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<b>let </b><font color="Maroon" title="c2">TSG1</font>, <font color="Maroon" title="c3">TSG2</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="trees_3.html#K5" title="TREES_3:func.5">FinTrees</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font></span>)</span>; <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">d</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lang1.html#K5" title="LANG1:func.5">Terminals</a> <font color="Maroon" title="c1">G</font> holds <br/> <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">TSG1</font> ) &amp; ( for <font color="Olive" title="b1">o</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c1">G</font><br/> for <font color="Olive" title="b2">p</font> being   <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of <font color="Maroon" title="c2">TSG1</font>  st <font color="Olive" title="b1">o</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b2">p</font> holds <br/><font color="Olive" title="b1">o</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">TSG1</font> ) &amp; ( for <font color="Olive" title="b1">F</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="trees_3.html#K5" title="TREES_3:func.5">FinTrees</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font></span>)</span>  st ( for <font color="Olive" title="b2">d</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lang1.html#K5" title="LANG1:func.5">Terminals</a> <font color="Maroon" title="c1">G</font> holds <br/> <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Olive" title="b2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> ) &amp; ( for <font color="Olive" title="b2">o</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c1">G</font><br/> for <font color="Olive" title="b3">p</font> being   <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of <font color="Olive" title="b1">F</font>  st <font color="Olive" title="b2">o</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b3">p</font> holds <br/><font color="Olive" title="b2">o</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Olive" title="b3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> ) holds <br/><font color="Maroon" title="c2">TSG1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">F</font> ) &amp; ( for <font color="Olive" title="b1">d</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lang1.html#K5" title="LANG1:func.5">Terminals</a> <font color="Maroon" title="c1">G</font> holds <br/> <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">TSG2</font> ) &amp; ( for <font color="Olive" title="b1">o</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c1">G</font><br/> for <font color="Olive" title="b2">p</font> being   <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of <font color="Maroon" title="c3">TSG2</font>  st <font color="Olive" title="b1">o</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b2">p</font> holds <br/><font color="Olive" title="b1">o</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">TSG2</font> ) &amp; ( for <font color="Olive" title="b1">F</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="trees_3.html#K5" title="TREES_3:func.5">FinTrees</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font></span>)</span>  st ( for <font color="Olive" title="b2">d</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lang1.html#K5" title="LANG1:func.5">Terminals</a> <font color="Maroon" title="c1">G</font> holds <br/> <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Olive" title="b2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> ) &amp; ( for <font color="Olive" title="b2">o</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c1">G</font><br/> for <font color="Olive" title="b3">p</font> being   <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of <font color="Olive" title="b1">F</font>  st <font color="Olive" title="b2">o</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b3">p</font> holds <br/><font color="Olive" title="b2">o</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Olive" title="b3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> ) holds <br/><font color="Maroon" title="c3">TSG2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">F</font> ) implies <font color="Maroon" title="c2">TSG1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">TSG2</font> )</span><br/>

<b>assume </b><a NAME="E1:21_1_2"/><i><font color="Green" title="E8">A5</font></i>: 
( ( for <font color="Olive" title="b1">d</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lang1.html#K5" title="LANG1:func.5">Terminals</a> <font color="Maroon" title="c1">G</font> holds <br/> <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">TSG1</font> ) &amp; ( for <font color="Olive" title="b1">o</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c1">G</font><br/> for <font color="Olive" title="b2">p</font> being   <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of <font color="Maroon" title="c2">TSG1</font>  st <font color="Olive" title="b1">o</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b2">p</font> holds <br/><font color="Olive" title="b1">o</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">TSG1</font> ) &amp; ( for <font color="Olive" title="b1">F</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="trees_3.html#K5" title="TREES_3:func.5">FinTrees</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font></span>)</span>  st ( for <font color="Olive" title="b2">d</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lang1.html#K5" title="LANG1:func.5">Terminals</a> <font color="Maroon" title="c1">G</font> holds <br/> <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Olive" title="b2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> ) &amp; ( for <font color="Olive" title="b2">o</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c1">G</font><br/> for <font color="Olive" title="b3">p</font> being   <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of <font color="Olive" title="b1">F</font>  st <font color="Olive" title="b2">o</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b3">p</font> holds <br/><font color="Olive" title="b2">o</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Olive" title="b3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> ) holds <br/><font color="Maroon" title="c2">TSG1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">F</font> ) &amp; ( for <font color="Olive" title="b1">d</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lang1.html#K5" title="LANG1:func.5">Terminals</a> <font color="Maroon" title="c1">G</font> holds <br/> <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">TSG2</font> ) &amp; ( for <font color="Olive" title="b1">o</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c1">G</font><br/> for <font color="Olive" title="b2">p</font> being   <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of <font color="Maroon" title="c3">TSG2</font>  st <font color="Olive" title="b1">o</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b2">p</font> holds <br/><font color="Olive" title="b1">o</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">TSG2</font> ) &amp; ( for <font color="Olive" title="b1">F</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="trees_3.html#K5" title="TREES_3:func.5">FinTrees</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font></span>)</span>  st ( for <font color="Olive" title="b2">d</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lang1.html#K5" title="LANG1:func.5">Terminals</a> <font color="Maroon" title="c1">G</font> holds <br/> <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Olive" title="b2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> ) &amp; ( for <font color="Olive" title="b2">o</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c1">G</font><br/> for <font color="Olive" title="b3">p</font> being   <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of <font color="Olive" title="b1">F</font>  st <font color="Olive" title="b2">o</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b3">p</font> holds <br/><font color="Olive" title="b2">o</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Olive" title="b3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> ) holds <br/><font color="Maroon" title="c3">TSG2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">F</font> ) &amp; not <font color="Maroon" title="c2">TSG1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">TSG2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:21_1_2"/><b>then </b>
( <font color="Maroon" title="c2">TSG1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">TSG2</font> &amp; <font color="Maroon" title="c3">TSG2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">TSG1</font> )
 
;<br/>
<b>hence </b><a NAME="E3:21_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="dtconstr.html#E1:21_1_2"><i><font color="Green" title="E8">A5</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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