<?xml version="1.0"?>
<div class="add">

<span class="kw">set </span><font color="Maroon" title="c2">A</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="trees_3.html#K3" title="TREES_3:func.3">Funcs</a> (<font color="Olive" title="b1">T</font>,<font color="Maroon" title="c1">D</font>)</span>)</span> where <font color="Olive" title="b1">T</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="trees_3.html#K1" title="TREES_3:func.1">Trees</a>  : verum </span> } </span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">TT</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="trees_3.html#K3" title="TREES_3:func.3">Funcs</a> (<font color="Olive" title="b1">T</font>,<font color="Maroon" title="c1">D</font>)</span>)</span> where <font color="Olive" title="b1">T</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="trees_3.html#K1" title="TREES_3:func.1">Trees</a>  : verum </span> } </span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">f</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="trees_3.html#K1" title="TREES_3:func.1">Trees</a>  <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>;<br/>
<a NAME="E1:43_1_1"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="trees_3.html#K1" title="TREES_3:func.1">Trees</a>  <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="trees_3.html#K3" title="TREES_3:func.3">Funcs</a> ( the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="trees_3.html#K1" title="TREES_3:func.1">Trees</a> ,<font color="Maroon" title="c1">D</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T8" onmouseover="rs('funct_2/T8')" onmouseout="rh()">FUNCT_2:8</a></span>;<br/>
<a NAME="E2:43_1_1"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
 <a href="trees_3.html#K3" title="TREES_3:func.3">Funcs</a> ( the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="trees_3.html#K1" title="TREES_3:func.1">Trees</a> ,<font color="Maroon" title="c1">D</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="trees_3.html#K3" title="TREES_3:func.3">Funcs</a> (<font color="Olive" title="b1">T</font>,<font color="Maroon" title="c1">D</font>)</span>)</span> where <font color="Olive" title="b1">T</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="trees_3.html#K1" title="TREES_3:func.1">Trees</a>  : verum </span> } </span> 
 
;<br/>
<a NAME="E3:43_1_1"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="trees_3.html#K3" title="TREES_3:func.3">Funcs</a> (<font color="Olive" title="b1">T</font>,<font color="Maroon" title="c1">D</font>)</span>)</span> where <font color="Olive" title="b1">T</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="trees_3.html#K1" title="TREES_3:func.1">Trees</a>  : verum </span> } </span>  is  <a href="trees_3.html#V3" title="TREES_3:attr.3">constituted-DTrees</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="trees_3.html#D5" target="_self" onmouseover="rs('trees_3/D5')" onmouseout="rh()">TREES_3:def 5</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="trees_3.html#K3" title="TREES_3:func.3">Funcs</a> (<font color="Olive" title="b1">T</font>,<font color="Maroon" title="c1">D</font>)</span>)</span> where <font color="Olive" title="b1">T</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="trees_3.html#K1" title="TREES_3:func.1">Trees</a>  : verum </span> } </span>  implies <font color="Maroon" title="c5">x</font> is   <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a> )</span><br/>

<span class="kw">assume </span><a NAME="E1:43_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="trees_3.html#K3" title="TREES_3:func.3">Funcs</a> (<font color="Olive" title="b1">T</font>,<font color="Maroon" title="c1">D</font>)</span>)</span> where <font color="Olive" title="b1">T</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="trees_3.html#K1" title="TREES_3:func.1">Trees</a>  : verum </span> } </span> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font> is   <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">X</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:43_1_1_1"/><span class="lab"><font color="Green" title="E19">A3</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font>
 <span class="kw">and </span><br/><a NAME="E4:43_1_1_1"/><span class="lab"><font color="Green" title="E20">A4</font></span>: 
<font color="Maroon" title="c6">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="trees_3.html#K3" title="TREES_3:func.3">Funcs</a> (<font color="Olive" title="b1">T</font>,<font color="Maroon" title="c1">D</font>)</span>)</span> where <font color="Olive" title="b1">T</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="trees_3.html#K1" title="TREES_3:func.1">Trees</a>  : verum </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/>
<a NAME="E5:43_1_1_1"/>
 ex <font color="Olive" title="b1">T</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="trees_3.html#K1" title="TREES_3:func.1">Trees</a>  st <font color="Maroon" title="c6">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_3.html#K3" title="TREES_3:func.3">Funcs</a> (<font color="Olive" title="b1">T</font>,<font color="Maroon" title="c1">D</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_1_1_1"><span class="lab"><font color="Green" title="E20">A4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:43_1_1_1"/>
<font color="Maroon" title="c5">x</font> is   <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1_1_1"><span class="lab"><font color="Green" title="E19">A3</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">TT</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="trees_3.html#K3" title="TREES_3:func.3">Funcs</a> (<font color="Olive" title="b1">T</font>,<font color="Maroon" title="c1">D</font>)</span>)</span> where <font color="Olive" title="b1">T</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="trees_3.html#K1" title="TREES_3:func.1">Trees</a>  : verum </span> } </span>  as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="trees_3.html#V3" title="TREES_3:attr.3">constituted-DTrees</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_1_1"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E1:43_1_1"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/>
<a NAME="E5:43_1_1"/>
<font color="Maroon" title="c5">TT</font> is    <a href="trees_3.html#M1" title="TREES_3:mode.1">DTree-set</a> of <font color="Maroon" title="c1">D</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="trees_3.html#D6" target="_self" onmouseover="rs('trees_3/D6')" onmouseout="rh()">TREES_3:def 6</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c5">TT</font> implies <font color="Maroon" title="c6">x</font> is   <a href="trees_2.html#NM5" title="TREES_2:NM.5">DecoratedTree</a> of <font color="Maroon" title="c1">D</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:43_1_1_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">TT</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">x</font> is   <a href="trees_2.html#NM5" title="TREES_2:NM.5">DecoratedTree</a> of <font color="Maroon" title="c1">D</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">X</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:43_1_1_2"/><span class="lab"><font color="Green" title="E19">A5</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font>
 <span class="kw">and </span><br/><a NAME="E4:43_1_1_2"/><span class="lab"><font color="Green" title="E20">A6</font></span>: 
<font color="Maroon" title="c7">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="trees_3.html#K3" title="TREES_3:func.3">Funcs</a> (<font color="Olive" title="b1">T</font>,<font color="Maroon" title="c1">D</font>)</span>)</span> where <font color="Olive" title="b1">T</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="trees_3.html#K1" title="TREES_3:func.1">Trees</a>  : verum </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/>
<a NAME="E5:43_1_1_2"/>
 ex <font color="Olive" title="b1">T</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="trees_3.html#K1" title="TREES_3:func.1">Trees</a>  st <font color="Maroon" title="c7">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_3.html#K3" title="TREES_3:func.3">Funcs</a> (<font color="Olive" title="b1">T</font>,<font color="Maroon" title="c1">D</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_1_1_2"><span class="lab"><font color="Green" title="E20">A6</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:43_1_1_2"/>
<font color="Maroon" title="c6">x</font> is   <a href="trees_2.html#NM5" title="TREES_2:NM.5">DecoratedTree</a> of <font color="Maroon" title="c1">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1_1_2"><span class="lab"><font color="Green" title="E19">A5</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">TT</font> = <font color="Maroon" title="c5">TT</font> as    <a href="trees_3.html#M1" title="TREES_3:mode.1">DTree-set</a> of <font color="Maroon" title="c1">D</font> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c6">TT</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">T</font> being   <a href="trees_2.html#NM5" title="TREES_2:NM.5">DecoratedTree</a> of <font color="Maroon" title="c1">D</font> holds  <font color="Olive" title="b1">T</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">TT</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c7">T</font> be   <a href="trees_2.html#NM5" title="TREES_2:NM.5">DecoratedTree</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">T</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">TT</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">t</font> =  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c7">T</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="trees_3.html#K1" title="TREES_3:func.1">Trees</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_3.html#D1" target="_self" onmouseover="rs('trees_3/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<a NAME="E8:43_1_1"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font>
 
;<br/>
<a NAME="E9:43_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A7</font></span>: 
<font color="Maroon" title="c7">T</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="trees_3.html#K3" title="TREES_3:func.3">Funcs</a> (<font color="Maroon" title="c8">t</font>,<font color="Maroon" title="c1">D</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D2" onmouseover="rs('funct_2/D2')" onmouseout="rh()">FUNCT_2:def 2</a></span>;<br/>
<a NAME="E10:43_1_1"/>
 <a href="trees_3.html#K3" title="TREES_3:func.3">Funcs</a> (<font color="Maroon" title="c8">t</font>,<font color="Maroon" title="c1">D</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="trees_3.html#K3" title="TREES_3:func.3">Funcs</a> (<font color="Olive" title="b1">W</font>,<font color="Maroon" title="c1">D</font>)</span>)</span> where <font color="Olive" title="b1">W</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="trees_3.html#K1" title="TREES_3:func.1">Trees</a>  : verum </span> } </span> 
 
;<br/>
<span class="kw">hence </span><a NAME="E11:43_1_1"/>
<font color="Maroon" title="c7">T</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">TT</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:43_1_1"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
