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<b>consider </b><font color="Maroon" title="c2">W</font> being   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a>;<br/>
<b>consider </b><font color="Maroon" title="c3">d</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>;<br/>
<b>set </b><font color="Maroon" title="c4">f</font> = <font color="Maroon" title="c2">W</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c3">d</font>;<br/>
<a NAME="E3:53_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">f</font> = <font color="Maroon" title="c2">W</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c3">d</font> as   <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a> <b>by </b><i><a class="ref" href="trees_2.html#D8" target="_self" title="TREES_2:def.8">Def8</a></i>;<br/>
<a NAME="E5:53_1_1"/><i><font color="Green" title="E26">A1</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T14" title="FUNCOP_1:th.14">FUNCOP_1:14</a></i>;<br/>
<a NAME="E6:53_1_1"/>
<font color="Maroon" title="c5">f</font> is    <a href="trees_2.html#M3" title="TREES_2:mode.3">ParametrizedSubset</a> of <font color="Maroon" title="c1">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">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>,<a class="ref" href="trees_2.html#D9" target="_self" title="TREES_2:def.9">TREES_2:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK4" title="RELAT_1:NK.4">rng</a> <font color="Maroon" title="c5">f</font> or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">D</font> )</span><br/>

<b>assume </b><a NAME="E1:53_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">D</font></span><br/>

<a NAME="E2:53_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font>
 
<b>by </b><i><a class="txt" href="#E5:53_1_1"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:53_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">D</font>
 ; <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>
<b>hence </b><a NAME="E7:53_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="trees_2.html#M3" title="TREES_2:mode.3">ParametrizedSubset</a> of <font color="Maroon" title="c1">D</font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="trees_2.html#V3" title="TREES_2:attr.3">DecoratedTree-like</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="funct_1.html#V1" title="FUNCT_1:attr.1">Function-like</a> )
 ; <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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