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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <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> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">T</font> &amp;  <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> );<br/>
<b>consider </b><font color="Maroon" title="c2">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:47_1_1"/><i><font color="Green" title="E19">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_3.html#K4" title="TREES_3:func.4">Trees</a> <font color="Maroon" title="c1">D</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:47_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 ;<br/><a NAME="E2:47_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_3.html#K4" title="TREES_3:func.4">Trees</a> <font color="Maroon" title="c1">D</font>
 <b>by </b><i><a class="txt" href="trees_3.html#E2:47_1_1"><i><font color="Green" title="E19">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E3:47_1_1_1"/>
<font color="Maroon" title="c3">x</font> is   <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/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c3">X</font> = <font color="Maroon" title="c2">X</font> as    <a href="trees_3.html#M4" title="TREES_3:mode.4">DTree-set</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/>
<b>take </b>
<font color="Maroon" title="c3">X</font>
;<br/>

<b>let </b><font color="Maroon" title="c4">T</font> be   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of <font color="Maroon" title="c1">D</font>;<br/>

<a NAME="E5:47_1_1"/>
<font color="Maroon" title="c4">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_3.html#K4" title="TREES_3:func.4">Trees</a> <font color="Maroon" title="c1">D</font>
 
<b>by </b><i><a class="ref" href="trees_3.html#D7" target="_self" title="TREES_3:def.7">Def7</a></i>;<br/>
<b>hence </b><a NAME="E6:47_1_1"/>
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">T</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> implies <font color="Maroon" title="c4">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> )
 <b>by </b><i><a class="txt" href="trees_3.html#E2:47_1_1"><i><font color="Green" title="E19">A1</font></i></a></i>;<br/>

<b>assume </b><a NAME="E7:47_1_1"/>
<font color="Maroon" title="c4">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ;<br/>

<a NAME="E8:47_1_1"/><b>then </b>
 ex <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> st <br/>( <font color="Maroon" title="c4">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> &amp;  <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> )
 
<b>by </b><i><a class="txt" href="trees_3.html#E2:47_1_1"><i><font color="Green" title="E19">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:47_1_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">T</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/>


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