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<span class="kw">let </span><font color="Maroon" title="c1">S</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c1">S</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies ( <font color="Maroon" title="c1">S</font> is  <a href="trees_3.html#V1" title="TREES_3:attr.1">constituted-Trees</a>  &amp; <font color="Maroon" title="c1">S</font> is  <a href="trees_3.html#V2" title="TREES_3:attr.2">constituted-FinTrees</a>  &amp; <font color="Maroon" title="c1">S</font> is  <a href="trees_3.html#V3" title="TREES_3:attr.3">constituted-DTrees</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:18_1_1"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
<font color="Maroon" title="c1">S</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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="c1">S</font> is  <a href="trees_3.html#V1" title="TREES_3:attr.1">constituted-Trees</a>  &amp; <font color="Maroon" title="c1">S</font> is  <a href="trees_3.html#V2" title="TREES_3:attr.2">constituted-FinTrees</a>  &amp; <font color="Maroon" title="c1">S</font> is  <a href="trees_3.html#V3" title="TREES_3:attr.3">constituted-DTrees</a>  )</span><br/>

<span class="kw">hence </span><a NAME="E2:18_1_1"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font> holds <br/><font color="Olive" title="b1">x</font> is   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="trees_3.html#D3" target="_self" title="TREES_3:def.3">TREES_3:def 3</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="c1">S</font> is  <a href="trees_3.html#V2" title="TREES_3:attr.2">constituted-FinTrees</a>  &amp; <font color="Maroon" title="c1">S</font> is  <a href="trees_3.html#V3" title="TREES_3:attr.3">constituted-DTrees</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E3:18_1_1"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font> holds <br/><font color="Olive" title="b1">x</font> is   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_1_1"><span class="lab"><font color="Green" title="E10">A1</font></span></a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="trees_3.html#D4" target="_self" title="TREES_3:def.4">TREES_3:def 4</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="c1">S</font> is  <a href="trees_3.html#V3" title="TREES_3:attr.3">constituted-DTrees</a> </span><br/>

<span class="kw">thus </span><a NAME="E4:18_1_1"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font> holds <br/><font color="Olive" title="b1">x</font> is   <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_1_1"><span class="lab"><font color="Green" title="E10">A1</font></span></a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="trees_3.html#D5" target="_self" title="TREES_3:def.5">TREES_3:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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