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<b>let </b><font color="Maroon" title="c1">T</font>, <font color="Maroon" title="c2">T'</font> be   <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st (  not <font color="Olive" title="b1">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="c1">T</font> or  not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_2.html#K4" title="TREES_2:func.4">Leaves</a> <font color="Maroon" title="c1">T</font> ) holds <br/><font color="Maroon" title="c1">T</font>,<font color="Olive" title="b1">x</font> <a href="trees_4.html#K13" title="TREES_4:func.13">&lt;-</a> <font color="Maroon" title="c2">T'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T</font></span><br/><b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( (  not <font color="Maroon" title="c3">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="c1">T</font> or  not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_2.html#K4" title="TREES_2:func.4">Leaves</a> <font color="Maroon" title="c1">T</font> ) implies <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c3">x</font> <a href="trees_4.html#K13" title="TREES_4:func.13">&lt;-</a> <font color="Maroon" title="c2">T'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T</font> )</span><br/>





<a NAME="E1:45"/><i><font color="Green" title="E16">A1</font></i>: 
 <a href="trees_2.html#K4" title="TREES_2:func.4">Leaves</a> <font color="Maroon" title="c1">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T144" title="RELAT_1:th.144">RELAT_1:144</a></i>;<br/>
<b>assume </b><a NAME="E2:45"/>
(  not <font color="Maroon" title="c3">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="c1">T</font> or  not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_2.html#K4" title="TREES_2:func.4">Leaves</a> <font color="Maroon" title="c1">T</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c3">x</font> <a href="trees_4.html#K13" title="TREES_4:func.13">&lt;-</a> <font color="Maroon" title="c2">T'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T</font></span><br/>

<a NAME="E3:45"/><b>then </b><i><font color="Green" title="E17">A2</font></i>: 
not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_2.html#K4" title="TREES_2:func.4">Leaves</a> <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="txt" href="#E1:45"><i><font color="Green" title="E16">A1</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:45"/><i><font color="Green" title="E18">A3</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c3">x</font> <a href="trees_4.html#K13" title="TREES_4:func.13">&lt;-</a> <font color="Maroon" title="c2">T'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">T</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="trees_4.html#D1" target="_self" title="TREES_4:def.1">TREES_4:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font> being  <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c3">x</font> <a href="trees_4.html#K13" title="TREES_4:func.13">&lt;-</a> <font color="Maroon" title="c2">T'</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c3">x</font> <a href="trees_4.html#K13" title="TREES_4:func.13">&lt;-</a> <font color="Maroon" title="c2">T'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">p</font> be    <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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="trees_2.html#D1" title="TREES_2:def.1">TREES_2:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( (  not <font color="Maroon" title="c4">p</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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c3">x</font> <a href="trees_4.html#K13" title="TREES_4:func.13">&lt;-</a> <font color="Maroon" title="c2">T'</font></span>)</span> or <font color="Maroon" title="c4">p</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="c1">T</font> ) &amp; (  not <font color="Maroon" title="c4">p</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="c1">T</font> or <font color="Maroon" title="c4">p</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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c3">x</font> <a href="trees_4.html#K13" title="TREES_4:func.13">&lt;-</a> <font color="Maroon" title="c2">T'</font></span>)</span> ) )</span><br/>

<a NAME="E1:45_1"/>
( <font color="Maroon" title="c4">p</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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c3">x</font> <a href="trees_4.html#K13" title="TREES_4:func.13">&lt;-</a> <font color="Maroon" title="c2">T'</font></span>)</span> iff ( <font color="Maroon" title="c4">p</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="c1">T</font> or  ex <font color="Olive" title="b1">q</font> being  <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c1">T</font> ex <font color="Olive" title="b2">r</font> being  <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c2">T'</font> st <br/>( <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K3" title="TREES_1:func.3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">T</font></span>)</span> &amp; <font color="Maroon" title="c1">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">r</font> ) ) )
 
<b>by </b><i><a class="ref" href="trees_4.html#D7" target="_self" title="TREES_4:def.7">Def7</a></i>;<br/>
<b>hence </b><a NAME="E2:45_1"/>
( (  not <font color="Maroon" title="c4">p</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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c3">x</font> <a href="trees_4.html#K13" title="TREES_4:func.13">&lt;-</a> <font color="Maroon" title="c2">T'</font></span>)</span> or <font color="Maroon" title="c4">p</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="c1">T</font> ) &amp; (  not <font color="Maroon" title="c4">p</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="c1">T</font> or <font color="Maroon" title="c4">p</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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c3">x</font> <a href="trees_4.html#K13" title="TREES_4:func.13">&lt;-</a> <font color="Maroon" title="c2">T'</font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="#E3:45"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</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>let </b><font color="Maroon" title="c4">p</font> be   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c3">x</font> <a href="trees_4.html#K13" title="TREES_4:func.13">&lt;-</a> <font color="Maroon" title="c2">T'</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c3">x</font> <a href="trees_4.html#K13" title="TREES_4:func.13">&lt;-</a> <font color="Maroon" title="c2">T'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">p</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c5">p'</font> = <font color="Maroon" title="c4">p</font> as   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="#E4:45"><i><font color="Green" title="E18">A3</font></i></a></i>;<br/>
<a NAME="E6:45"/>
( <font color="Maroon" title="c5">p'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K3" title="TREES_1:func.3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">T</font></span>)</span> implies <font color="Maroon" title="c1">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">p'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_2.html#K4" title="TREES_2:func.4">Leaves</a> <font color="Maroon" title="c1">T</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E7:45"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c3">x</font> <a href="trees_4.html#K13" title="TREES_4:func.13">&lt;-</a> <font color="Maroon" title="c2">T'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="txt" href="#E3:45"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="trees_4.html#D7" target="_self" title="TREES_4:def.7">Def7</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>
