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<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</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"> for <font color="Olive" title="b1">p</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> is <a href="trees_3.html#V6" title="TREES_3:attr.6">DTree-yielding</a> holds <br/>( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/><b>let </b><font color="Maroon" title="c3">p</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c3">p</font> &amp; <font color="Maroon" title="c3">p</font> is <a href="trees_3.html#V6" title="TREES_3:attr.6">DTree-yielding</a> implies ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) )</span><br/>





<b>assume </b><a NAME="E1:28"/><i><font color="Green" title="E15">A1</font></i>: 
(  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c3">p</font> &amp; <font color="Maroon" title="c3">p</font> is <a href="trees_3.html#V6" title="TREES_3:attr.6">DTree-yielding</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">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>

<b>then reconsider </b><font color="Maroon" title="c4">p'</font> = <font color="Maroon" title="c3">p</font> as  <a href="trees_3.html#V6" title="TREES_3:attr.6">DTree-yielding</a> <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> ;<br/>
<b>thus </b><font color="Maroon" title="c1">x</font> = 
<span class="p1">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="ref" href="trees_4.html#T3" target="_self" title="TREES_4:th.3">Th3</a></i>
<br/>.= 
<font color="Maroon" title="c2">y</font>
<b>by </b><i><a class="txt" href="#E1:28"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="trees_4.html#D4" target="_self" title="TREES_4:def.4">Def4</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<a NAME="E4:28"/>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c3">p</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="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c4">p'</font></span>)</span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="trees_4.html#T3" target="_self" title="TREES_4:th.3">Th3</a>, <a class="ref" href="trees_4.html#T10" target="_self" title="TREES_4:th.10">Th10</a></i>;<br/>
<a NAME="E5:28"/><b>then </b>
(  <a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">p</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="#E1:28"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="trees_3.html#T39" title="TREES_3:th.39">TREES_3:39</a>, <a class="ref" href="trees_3.html#T53" title="TREES_3:th.53">TREES_3:53</a>, <a class="ref" href="trees_3.html#T55" title="TREES_3:th.55">TREES_3:55</a></i>;<br/>
<b>hence </b><a NAME="E6:28"/>
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <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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