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<b>let </b><font color="Maroon" title="c2">T1</font>, <font color="Maroon" title="c3">T2</font> be    <a href="trees_3.html#M4" title="TREES_3:mode.4">DTree-set</a> of <font color="Maroon" title="c1">D</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:47_1_2"/><i><font color="Green" title="E19">A2</font></i>: 
for <font color="Olive" title="b1">T</font> being  <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of <font color="Maroon" title="c1">D</font> holds <br/> (  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">T</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> iff <font color="Olive" title="b1">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T1</font> )
 <b>and </b><br/><a NAME="E2:47_1_2"/><i><font color="Green" title="E20">A3</font></i>: 
for <font color="Olive" title="b1">T</font> being  <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of <font color="Maroon" title="c1">D</font> holds <br/> (  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">T</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> iff <font color="Olive" title="b1">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T2</font> )
 ;<br/>

<b>thus </b><a NAME="E3:47_1_2"/>
<font color="Maroon" title="c2">T1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">T2</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:47_1_2_1"/><i><font color="Green" title="E21">A4</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">T</font> = <font color="Maroon" title="c4">x</font> as   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of <font color="Maroon" title="c1">D</font> <b>by </b><i><a class="ref" href="trees_3.html#D6" target="_self" title="TREES_3:def.6">Def6</a></i>;<br/>
<a NAME="E3:47_1_2_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">T</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="txt" href="trees_3.html#E1:47_1_2"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="trees_3.html#E1:47_1_2_1"><i><font color="Green" title="E21">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:47_1_2_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T2</font>
 <b>by </b><i><a class="txt" href="trees_3.html#E2:47_1_2"><i><font color="Green" title="E20">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E4:47_1_2"/><i><font color="Green" title="E21">A5</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T2</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">T</font> = <font color="Maroon" title="c4">x</font> as   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of <font color="Maroon" title="c1">D</font> <b>by </b><i><a class="ref" href="trees_3.html#D6" target="_self" title="TREES_3:def.6">Def6</a></i>;<br/>
<a NAME="E6:47_1_2"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">T</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="txt" href="trees_3.html#E2:47_1_2"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="trees_3.html#E4:47_1_2"><i><font color="Green" title="E21">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:47_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T1</font>
 <b>by </b><i><a class="txt" href="trees_3.html#E1:47_1_2"><i><font color="Green" title="E19">A2</font></i></a></i>;<br/>


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