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<b>let </b><font color="Maroon" title="c1">T1</font>, <font color="Maroon" title="c2">T2</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>  holds <br/> ( ( <font color="Maroon" title="c1">T1</font> is <a href="bintree1.html#V2" title="BINTREE1:attr.2">binary</a> &amp; <font color="Maroon" title="c2">T2</font> is <a href="bintree1.html#V2" title="BINTREE1:attr.2">binary</a> ) iff <font color="Olive" title="b1">x</font> <a href="trees_4.html#K6" title="TREES_4:func.6">-tree</a> <font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font> is <a href="bintree1.html#V2" title="BINTREE1:attr.2">binary</a> )</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"> ( ( <font color="Maroon" title="c1">T1</font> is <a href="bintree1.html#V2" title="BINTREE1:attr.2">binary</a> &amp; <font color="Maroon" title="c2">T2</font> is <a href="bintree1.html#V2" title="BINTREE1:attr.2">binary</a> ) iff <font color="Maroon" title="c3">x</font> <a href="trees_4.html#K6" title="TREES_4:func.6">-tree</a> <font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font> is <a href="bintree1.html#V2" title="BINTREE1:attr.2">binary</a> )</span><br/>



<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="trees_4.html#K6" title="TREES_4:func.6">-tree</a> <font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font> is <a href="bintree1.html#V2" title="BINTREE1:attr.2">binary</a> implies ( <font color="Maroon" title="c1">T1</font> is <a href="bintree1.html#V2" title="BINTREE1:attr.2">binary</a> &amp; <font color="Maroon" title="c2">T2</font> is <a href="bintree1.html#V2" title="BINTREE1:attr.2">binary</a> ) )</span>
<div class="add"><b>assume </b><a NAME="E1:14_1"/>
( <font color="Maroon" title="c1">T1</font> is <a href="bintree1.html#V2" title="BINTREE1:attr.2">binary</a> &amp; <font color="Maroon" title="c2">T2</font> is <a href="bintree1.html#V2" title="BINTREE1:attr.2">binary</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="trees_4.html#K6" title="TREES_4:func.6">-tree</a> <font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font> is <a href="bintree1.html#V2" title="BINTREE1:attr.2">binary</a></span><br/><a NAME="E2:14_1"/><b>then </b>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">T1</font> is <a href="bintree1.html#V1" title="BINTREE1:attr.1">binary</a> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">T2</font> is <a href="bintree1.html#V1" title="BINTREE1:attr.1">binary</a> )
 <b>by </b><i><a class="ref" href="bintree1.html#D3" target="_self" title="BINTREE1:def.3">Def3</a></i>;<br/><a NAME="E3:14_1"/><b>then </b><i><font color="Green" title="E8">A1</font></i>: 
 <a href="trees_3.html#K15" title="TREES_3:func.15">tree</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">T1</font></span>)</span>,<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">T2</font></span>)</span> is <a href="bintree1.html#V1" title="BINTREE1:attr.1">binary</a>
 <b>by </b><i><a class="ref" href="bintree1.html#T10" target="_self" title="BINTREE1:th.10">Th10</a></i>;<br/><a NAME="E4:14_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="trees_4.html#K6" title="TREES_4:func.6">-tree</a> <font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_3.html#K15" title="TREES_3:func.15">tree</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">T1</font></span>)</span>,<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">T2</font></span>)</span>
 <b>by </b><i><a class="ref" href="trees_4.html#T14" title="TREES_4:th.14">TREES_4:14</a></i>;<br/><b>hence </b><a NAME="E5:14_1"/>
<font color="Maroon" title="c3">x</font> <a href="trees_4.html#K6" title="TREES_4:func.6">-tree</a> <font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font> is <a href="bintree1.html#V2" title="BINTREE1:attr.2">binary</a>
 <b>by </b><i><a class="txt" href="bintree1.html#E3:14_1"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="bintree1.html#D3" target="_self" title="BINTREE1:def.3">Def3</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>assume </b><a NAME="E2:14"/>
<font color="Maroon" title="c3">x</font> <a href="trees_4.html#K6" title="TREES_4:func.6">-tree</a> <font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font> is <a href="bintree1.html#V2" title="BINTREE1:attr.2">binary</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">T1</font> is <a href="bintree1.html#V2" title="BINTREE1:attr.2">binary</a> &amp; <font color="Maroon" title="c2">T2</font> is <a href="bintree1.html#V2" title="BINTREE1:attr.2">binary</a> )</span><br/>

<a NAME="E3:14"/><b>then </b><i><font color="Green" title="E8">A2</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="c3">x</font> <a href="trees_4.html#K6" title="TREES_4:func.6">-tree</a> <font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span>)</span> is <a href="bintree1.html#V1" title="BINTREE1:attr.1">binary</a>
 
<b>by </b><i><a class="ref" href="bintree1.html#D3" target="_self" title="BINTREE1:def.3">Def3</a></i>;<br/>
<a NAME="E4:14"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="trees_4.html#K6" title="TREES_4:func.6">-tree</a> <font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_3.html#K15" title="TREES_3:func.15">tree</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">T1</font></span>)</span>,<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">T2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="trees_4.html#T14" title="TREES_4:th.14">TREES_4:14</a></i>;<br/>
<a NAME="E5:14"/><b>then </b>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">T1</font> is <a href="bintree1.html#V1" title="BINTREE1:attr.1">binary</a> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">T2</font> is <a href="bintree1.html#V1" title="BINTREE1:attr.1">binary</a> )
 
<b>by </b><i><a class="txt" href="bintree1.html#E3:14"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="bintree1.html#T10" target="_self" title="BINTREE1:th.10">Th10</a></i>;<br/>
<b>hence </b><a NAME="E6:14"/>
( <font color="Maroon" title="c1">T1</font> is <a href="bintree1.html#V2" title="BINTREE1:attr.2">binary</a> &amp; <font color="Maroon" title="c2">T2</font> is <a href="bintree1.html#V2" title="BINTREE1:attr.2">binary</a> )
 <b>by </b><i><a class="ref" href="bintree1.html#D3" target="_self" title="BINTREE1:def.3">Def3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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