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<span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">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/><span class="kw">let </span><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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><span class="kw">hereby </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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"><span class="kw">assume </span><a NAME="E1:13_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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E2:13_1"/><span class="kw">then </span>
(  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">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#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">T2</font> is  <a href="bintree1.html#V1" title="BINTREE1:attr.1">binary</a>  )
 ;<br/><a NAME="E3:13_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E6">A1</font></span>: 
 <a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> (<span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c1">T1</font></span>)</span>,<span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">T2</font></span>)</span>) is  <a href="bintree1.html#V1" title="BINTREE1:attr.1">binary</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bintree1.html#T8" target="_self" onmouseover="rs('bintree1/T8')" onmouseout="rh()">Th8</a></span>;<br/><a NAME="E4:13_1"/>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">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#K13" title="TREES_3:func.13">tree</a> (<span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c1">T1</font></span>)</span>,<span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">T2</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_4.html#T14" onmouseover="rs('trees_4/T14')" onmouseout="rh()">TREES_4:14</a></span>;<br/><span class="kw">hence </span><a NAME="E5:13_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:13_1"><span class="lab"><font color="Green" title="E6">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:13"/>
<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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E3:13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E6">A2</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">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> 
 
;<br/>
<a NAME="E4:13"/>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">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#K13" title="TREES_3:func.13">tree</a> (<span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c1">T1</font></span>)</span>,<span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">T2</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_4.html#T14" onmouseover="rs('trees_4/T14')" onmouseout="rh()">TREES_4:14</a></span>;<br/>
<a NAME="E5:13"/><span class="kw">then </span>
(  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">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#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">T2</font> is  <a href="bintree1.html#V1" title="BINTREE1:attr.1">binary</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:13"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="bintree1.html#T8" target="_self" onmouseover="rs('bintree1/T8')" onmouseout="rh()">Th8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:13"/>
( <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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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