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<div class="add">

<b>consider </b><font color="Maroon" title="c5">T2</font> being   <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a><b> such that </b><br/><a NAME="E2:21_1_1"/><i><font color="Green" title="E18">A2</font></i>: 
<font color="Maroon" title="c5">T2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_a.html#K3" title="TREES_A:func.3">tree</a> <font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">T1</font>
 ;<br/>
<a NAME="E3:21_1_1"/>
<font color="Maroon" title="c5">T2</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="21_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:21_1_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">T2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="trees_2.html#D9" title="TREES_2:def.9">TREES_2:def 9</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">y</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="E1:21_1_1_1_1"/>
<font color="Maroon" title="c6">y</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">T2</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:21_1_1_1_1"/><i><font color="Green" title="E19">A3</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">T2</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">T2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</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="E4:21_1_1_1_1"/>
<font color="Maroon" title="c7">x</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">T2</font>
 
<b>by </b><i><a class="txt" href="trees_a.html#E3:21_1_1_1_1"><i><font color="Green" title="E19">A3</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">q</font> = <font color="Maroon" title="c7">x</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E6:21_1_1_1_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">T2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_a.html#K1" title="TREES_A:func.1">tree</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">T</font></span>)</span>,<font color="Maroon" title="c3">P</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">T1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="trees_a.html#E1:21"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="trees_a.html#E2:21_1_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="trees_a.html#D2" target="_self" title="TREES_A:def.2">Def2</a></i>;<br/>
<a NAME="E7:21_1_1_1_1"/><b>then </b><i><font color="Green" title="E20">A4</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">T2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">t1</font> where <font color="Olive" title="b1">t1</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">T</font> : for <font color="Olive" title="b2">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> holds <br/> not <font color="Olive" title="b2">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Olive" title="b1">t1</font> </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">p</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">T</font>, <font color="Olive" title="b2">s</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">T1</font> : <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="trees_a.html#E1:21"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="trees_a.html#T7" target="_self" title="TREES_A:th.7">Th7</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1_1_1_1_1"/>
( <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">t1</font> where <font color="Olive" title="b1">t1</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">T</font> : for <font color="Olive" title="b2">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> holds <br/> not <font color="Olive" title="b2">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Olive" title="b1">t1</font> </span>}</span>  or <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">p</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">T</font>, <font color="Olive" title="b2">s</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">T1</font> : <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="trees_a.html#E3:21_1_1_1_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="trees_a.html#E7:21_1_1_1_1"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:21_1_1_1_1_1_1"/><i><font color="Green" title="E21">A5</font></i>: 
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">t1</font> where <font color="Olive" title="b1">t1</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">T</font> : for <font color="Olive" title="b2">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> holds <br/> not <font color="Olive" title="b2">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Olive" title="b1">t1</font> </span>}</span> 
 ;<br/><div class="add"><a NAME="E2:21_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E22">A6</font></i>: 
<span class="p1">(<span class="default"><a href="trees_a.html#K3" title="TREES_A:func.3">tree</a> <font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">T1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">q</font>
 <b>by </b><i><a class="txt" href="trees_a.html#E1:21"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="trees_a.html#E2:21_1_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="trees_a.html#E3:21_1_1_1_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="trees_a.html#T15" target="_self" title="TREES_A:th.15">Th15</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon" title="c9">t'</font> being    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">T</font><b> such that </b><br/><a NAME="E2:21_1_1_1_1_1_1_1"/><i><font color="Green" title="E23">A7</font></i>: 
( <font color="Maroon" title="c8">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">t'</font> &amp; ( for <font color="Olive" title="b1">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> holds <br/> not <font color="Olive" title="b1">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c9">t'</font> ) )
 <b>by </b><i><a class="txt" href="trees_a.html#E1:21_1_1_1_1_1_1"><i><font color="Green" title="E21">A5</font></i></a></i>;<br/><b>thus </b><a NAME="E3:21_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="trees_a.html#E2:21_1_1_1_1_1_1_1"><i><font color="Green" title="E23">A7</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E4:21_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E23">A8</font></i>: 
<font color="Maroon" title="c6">y</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="c2">T</font>
 <b>by </b><i><a class="txt" href="trees_a.html#E2:21_1_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="trees_a.html#E3:21_1_1_1_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="trees_a.html#E2:21_1_1_1_1_1_1"><i><font color="Green" title="E22">A6</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:21_1_1_1_1_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font>
 <b>by </b><i><a class="ref" href="trees_2.html#D9" title="TREES_2:def.9">TREES_2:def 9</a></i>;<br/><b>hence </b><a NAME="E6:21_1_1_1_1_1_1"/>
<font color="Maroon" title="c6">y</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="txt" href="trees_a.html#E4:21_1_1_1_1_1_1"><i><font color="Green" title="E23">A8</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:21_1_1_1_1_1_2"/>
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">p</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">T</font>, <font color="Olive" title="b2">s</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">T1</font> : <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c9">p</font> being    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">T</font>, <font color="Maroon" title="c10">r</font> being    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">T1</font><b> such that </b><br/><a NAME="E3:21_1_1_1_1_1_2"/><i><font color="Green" title="E21">A9</font></i>: 
( <font color="Maroon" title="c8">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c10">r</font> &amp; <font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <span class="p1">(<span class="default"><a href="trees_a.html#K3" title="TREES_A:func.3">tree</a> <font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">T1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">T1</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c10">r</font> )
 <b>by </b><i><a class="txt" href="trees_a.html#E2:21_1_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="trees_a.html#E3:21_1_1_1_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="trees_a.html#T17" target="_self" title="TREES_A:th.17">Th17</a></i>;<br/><b>thus </b><a NAME="E4:21_1_1_1_1_1_2"/>
<font color="Maroon" title="c6">y</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="txt" href="trees_a.html#E2:21_1_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="trees_a.html#E3:21_1_1_1_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="trees_a.html#E3:21_1_1_1_1_1_2"><i><font color="Green" title="E21">A9</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:21_1_1"/>
 <a href="trees_a.html#K3" title="TREES_A:func.3">tree</a> <font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">T1</font> is   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of <font color="Maroon" title="c1">D</font>
 <b>by </b><i><a class="txt" href="trees_a.html#E2:21_1_1"><i><font color="Green" title="E18">A2</font></i></a></i>;<br/>


</div>
