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<b>let </b><font color="Maroon" title="c1">D'</font>, <font color="Maroon" title="c2">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:68"/><i><font color="Green" title="E33">A1</font></i>: 
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">D'</font> holds <br/><font color="Olive" title="b1">x</font> is   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of <font color="Maroon" title="c2">D</font>
 <b>and </b><br/><a NAME="E2:68"/><i><font color="Green" title="E34">A2</font></i>: 
<font color="Maroon" title="c1">D'</font> is <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>
 ;<br/>

<a NAME="E3:68"/>
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">D'</font> holds <br/><font color="Olive" title="b1">x</font> is   <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a>
 
<b>by </b><i><a class="txt" href="trees_2.html#E1:68"><i><font color="Green" title="E33">A1</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">T</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">D'</font> as   <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a> <b>by </b><i><a class="txt" href="trees_2.html#E2:68"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="trees_2.html#T37" target="_self" title="TREES_2:th.37">Th37</a></i>;<br/>
<a NAME="E5:68"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_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="E1:68_1"/>
<font color="Maroon" title="c4">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="c3">T</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:68_1"/><i><font color="Green" title="E35">A3</font></i>: 
( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">T</font> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:68_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T</font> &amp; <font color="Maroon" title="c3">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">T</font> )
 
<b>by </b><i><a class="txt" href="trees_2.html#E3:68_1"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:68_1"/><i><font color="Green" title="E36">A4</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font> &amp; <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> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">X</font> = <font color="Maroon" title="c6">X</font> as   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of <font color="Maroon" title="c2">D</font> <b>by </b><i><a class="txt" href="trees_2.html#E1:68"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="trees_2.html#E6:68_1"><i><font color="Green" title="E36">A4</font></i></a></i>;<br/>
<a NAME="E8:68_1"/>
( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">X</font> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> )
 
<b>by </b><i><a class="txt" href="trees_2.html#E6:68_1"><i><font color="Green" title="E36">A4</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/>
<a NAME="E9:68_1"/><b>then </b>
( <font color="Maroon" title="c4">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="c7">X</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">D</font> )
 
<b>by </b><i><a class="ref" href="trees_2.html#D9" target="_self" title="TREES_2:def.9">Def9</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E10:68_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:68"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">D'</font> is   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of <font color="Maroon" title="c2">D</font>
 <b>by </b><i><a class="ref" href="trees_2.html#D9" target="_self" title="TREES_2:def.9">Def9</a></i>;<br/>


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