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<b>set </b><font color="Maroon" title="c3">F</font> =  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">D</font>;<br/>
<a NAME="E1:43_1_1"/>
 <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">D</font> is <a href="trees_3.html#V3" title="TREES_3:attr.3">constituted-DTrees</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_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="trees_3.html#D5" target="_self" title="TREES_3:def.5">TREES_3:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">D</font> implies <font color="Maroon" title="c4">x</font> is   <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a> )</span><br/>

<b>assume </b><a NAME="E1:43_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">D</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> is   <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a></span><br/>

<b>then consider </b><font color="Maroon" title="c5">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:43_1_1_1"/><i><font color="Green" title="E17">A1</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">f</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="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<b>thus </b><a NAME="E4:43_1_1_1"/>
<font color="Maroon" title="c4">x</font> is   <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a>
 <b>by </b><i><a class="txt" href="#E3:43_1_1_1"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="trees_2.html#D8" title="TREES_2:def.8">TREES_2:def 8</a></i>; <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>then reconsider </b><font color="Maroon" title="c4">F</font> =  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">D</font> as  <a href="trees_3.html#V3" title="TREES_3:attr.3">constituted-DTrees</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<a NAME="E3:43_1_1"/>
<font color="Maroon" title="c4">F</font> is    <a href="trees_3.html#M4" title="TREES_3:mode.4">DTree-set</a> of <font color="Maroon" title="c2">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">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="trees_3.html#D6" target="_self" title="TREES_3:def.6">TREES_3:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> implies <font color="Maroon" title="c5">x</font> is   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of <font color="Maroon" title="c2">D</font> )</span><br/>

<b>assume </b><a NAME="E1:43_1_1_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">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="c5">x</font> is   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of <font color="Maroon" title="c2">D</font></span><br/>

<b>then consider </b><font color="Maroon" title="c6">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:43_1_1_2"/><i><font color="Green" title="E17">A2</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">f</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="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<b>thus </b><a NAME="E4:43_1_1_2"/>
<font color="Maroon" title="c5">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>by </b><i><a class="txt" href="#E3:43_1_1_2"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="trees_2.html#D8" title="TREES_2:def.8">TREES_2:def 8</a>, <a class="ref" href="trees_2.html#D9" title="TREES_2:def.9">TREES_2:def 9</a></i>; <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>then reconsider </b><font color="Maroon" title="c5">F</font> = <font color="Maroon" title="c4">F</font> as    <a href="trees_3.html#M4" title="TREES_3:mode.4">DTree-set</a> of <font color="Maroon" title="c2">D</font> ;<br/>
<b>consider </b><font color="Maroon" title="c6">d</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">D</font>;<br/>
<a NAME="E6:43_1_1"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">d</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="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:43_1_1"/><b>then </b>
<font color="Maroon" title="c6">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<b>hence </b><a NAME="E8:43_1_1"/>
 <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">D</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="trees_3.html#M4" title="TREES_3:mode.4">DTree-set</a> of <font color="Maroon" title="c2">D</font>
 ; <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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