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<span class="kw">let </span><font color="Maroon" title="c1">T</font>, <font color="Maroon" title="c2">T9</font> be   <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a>; <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">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">T9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T</font></span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">T9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T</font> )</span><br/>





<a NAME="E1:45"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T144" title="RELAT_1:th.144">RELAT_1:144</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:45"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
(  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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">T9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T</font></span><br/>

<a NAME="E3:45"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E17">A2</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E4:45"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
 <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">T9</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>
  <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">T9</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">T9</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"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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> ; <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">T9</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">T9</font></span>)</span> ) )</span><br/>

<a NAME="E1:45_1"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
( <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">T9</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">T9</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> ) ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_4.html#D7" target="_self" title="TREES_4:def.7">Def7</a></span>;<br/>
<span class="kw">thus </span><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">T9</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">T9</font></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E1:45_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</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">let </span><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">T9</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">T9</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/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">p9</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45"><span class="lab"><font color="Green" title="E19">A4</font></span></a></span>;<br/>
<a NAME="E6:45"/><span class="lab"><font color="Green" title="E20">A6</font></span>: 
( <font color="Maroon" title="c5">p9</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">p9</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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></span>;<br/>
<span class="kw">thus </span><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">T9</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E6:45"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="ref" href="trees_4.html#D7" target="_self" title="TREES_4:def.7">Def7</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>
