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<span class="kw">set </span><font color="Maroon" title="c2">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="trees_3.html#M1" title="TREES_3:mode.1">Element</a> of  <a href="trees_3.html#K1" title="TREES_3:func.1">Trees</a>  : verum </span>}</span> ;<br/>
<a NAME="E1:46_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="trees_3.html#M1" title="TREES_3: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" onclick="hs2(this)" href="javascript:()" title="46_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">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="trees_3.html#M1" title="TREES_3: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="c3">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="E1:46_1_1_1"/>
<font color="Maroon" title="c3">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="trees_3.html#M1" title="TREES_3:mode.1">Element</a> of  <a href="trees_3.html#K1" title="TREES_3:func.1">Trees</a>  : verum </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">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="c4">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:46_1_1_1"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">and </span><br/><a NAME="E4:46_1_1_1"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
<font color="Maroon" title="c4">X</font> <a href="hidden.html#R2" title="HIDDEN: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="trees_3.html#M1" title="TREES_3: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" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<a NAME="E5:46_1_1_1"/>
 ex <font color="Olive" title="b1">T</font> being   <a href="trees_3.html#M1" title="TREES_3:mode.1">Element</a> of  <a href="trees_3.html#K1" title="TREES_3:func.1">Trees</a>  st <font color="Maroon" title="c4">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="#E4:46_1_1_1"><span class="lab"><font color="Green" title="E18">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:46_1_1_1"/>
<font color="Maroon" title="c3">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="#E3:46_1_1_1"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="trees_3.html#D6" target="_self" title="TREES_3:def.6">Def6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="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="trees_3.html#M1" title="TREES_3:mode.1">Element</a> of  <a href="trees_3.html#K1" title="TREES_3:func.1">Trees</a>  : verum </span>}</span>  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="E3:46_1_1"/>
<font color="Maroon" title="c3">TT</font> is    <a href="trees_3.html#M4" title="TREES_3:mode.4">DTree-set</a> of <font color="Maroon" title="c1">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">TT</font> implies <font color="Maroon" title="c4">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="E1:46_1_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">TT</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">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="c5">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:46_1_1_2"/><span class="lab"><font color="Green" title="E17">A3</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font>
 <span class="kw">and </span><br/><a NAME="E4:46_1_1_2"/><span class="lab"><font color="Green" title="E18">A4</font></span>: 
<font color="Maroon" title="c5">X</font> <a href="hidden.html#R2" title="HIDDEN: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="trees_3.html#M1" title="TREES_3: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" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<a NAME="E5:46_1_1_2"/>
 ex <font color="Olive" title="b1">T</font> being   <a href="trees_3.html#M1" title="TREES_3:mode.1">Element</a> of  <a href="trees_3.html#K1" title="TREES_3:func.1">Trees</a>  st <font color="Maroon" title="c5">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="#E4:46_1_1_2"><span class="lab"><font color="Green" title="E18">A4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:46_1_1_2"/>
<font color="Maroon" title="c4">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="#E3:46_1_1_2"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="ref" href="trees_3.html#D6" target="_self" title="TREES_3:def.6">Def6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c4">TT</font> = <font color="Maroon" title="c3">TT</font> as    <a href="trees_3.html#M4" title="TREES_3:mode.4">DTree-set</a> of <font color="Maroon" title="c1">D</font> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c4">TT</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">TT</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">TT</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">t</font> =  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">T</font> as    <a href="trees_3.html#M1" title="TREES_3: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" title="TREES_3:def.1">Def1</a></span>;<br/>
<a NAME="E6:46_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<a NAME="E7:46_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A5</font></span>: 
<font color="Maroon" title="c5">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_3.html#K3" title="TREES_3:func.3">Funcs</a> <font color="Maroon" title="c6">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" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/>
<a NAME="E8:46_1_1"/>
 <a href="trees_3.html#K3" title="TREES_3:func.3">Funcs</a> <font color="Maroon" title="c6">t</font>,<font color="Maroon" title="c1">D</font> <a href="hidden.html#R2" title="HIDDEN: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="trees_3.html#M1" title="TREES_3: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="E9:46_1_1"/>
<font color="Maroon" title="c5">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">TT</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:46_1_1"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
