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<span class="kw">let </span><font color="Maroon" title="c1">X1</font>, <font color="Maroon" title="c2">X2</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="trees_3.html#V3" title="TREES_3:attr.3">constituted-DTrees</a>   <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"> ( ( for <font color="Olive" title="b1">t</font> being   <a href="trees_3.html#M3" title="TREES_3:mode.3">Element</a> of <font color="Maroon" title="c1">X1</font> holds  <font color="Olive" title="b1">t</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  ) &amp; ( for <font color="Olive" title="b1">t</font> being   <a href="trees_3.html#M3" title="TREES_3:mode.3">Element</a> of <font color="Maroon" title="c2">X2</font> holds  <font color="Olive" title="b1">t</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  ) implies for <font color="Olive" title="b1">C</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds  <font color="Olive" title="b1">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">X2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <font color="Maroon" title="c1">X1</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <font color="Maroon" title="c2">X2</font></span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:18"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
for <font color="Olive" title="b1">t</font> being   <a href="trees_3.html#M3" title="TREES_3:mode.3">Element</a> of <font color="Maroon" title="c1">X1</font> holds  <font color="Olive" title="b1">t</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">and </span><br/><a NAME="E2:18"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
for <font color="Olive" title="b1">t</font> being   <a href="trees_3.html#M3" title="TREES_3:mode.3">Element</a> of <font color="Maroon" title="c2">X2</font> holds  <font color="Olive" title="b1">t</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</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">C</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds  <font color="Olive" title="b1">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">X2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <font color="Maroon" title="c1">X1</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <font color="Maroon" title="c2">X2</font></span>)</span></span><br/>

<div><span class="lab"><font color="Green" title="E11">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">t</font> be    <a href="trees_3.html#M3" title="TREES_3:mode.3">Element</a> of <font color="Maroon" title="c1">X1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">X2</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="c3">t</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span><br/><a NAME="E1:18_1"/>
( <font color="Maroon" title="c3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X1</font> or <font color="Maroon" title="c3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E2:18_1"/>
<font color="Maroon" title="c3">t</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E10">A2</font></span></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="c3">C</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"> <font color="Maroon" title="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">X2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <font color="Maroon" title="c1">X1</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <font color="Maroon" title="c2">X2</font></span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">X</font> = <font color="Maroon" title="c1">X1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">X2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">f</font> = <font color="Maroon" title="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">X2</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">f1</font> = <font color="Maroon" title="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <font color="Maroon" title="c1">X1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">f2</font> = <font color="Maroon" title="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <font color="Maroon" title="c2">X2</font>;<br/>
<a NAME="E4:18"/><span class="lab"><font color="Green" title="E12">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="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">X2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">C</font> <a href="trees_9.html#K12" title="TREES_9:func.12">-Subtrees</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">X2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E5:18"/><span class="lab"><font color="Green" title="E13">A5</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="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <font color="Maroon" title="c1">X1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">C</font> <a href="trees_9.html#K12" title="TREES_9:func.12">-Subtrees</a> <font color="Maroon" title="c1">X1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E6:18"/><span class="lab"><font color="Green" title="E14">A6</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="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <font color="Maroon" title="c2">X2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">C</font> <a href="trees_9.html#K12" title="TREES_9:func.12">-Subtrees</a> <font color="Maroon" title="c2">X2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E7:18"/><span class="lab"><font color="Green" title="E15">A7</font></span>: 
<font color="Maroon" title="c3">C</font> <a href="trees_9.html#K12" title="TREES_9:func.12">-Subtrees</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">X2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="trees_9.html#K12" title="TREES_9:func.12">-Subtrees</a> <font color="Maroon" title="c1">X1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="trees_9.html#K12" title="TREES_9:func.12">-Subtrees</a> <font color="Maroon" title="c2">X2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="circtrm1.html#T9" target="_self" title="CIRCTRM1:th.9">Th9</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">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"> ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <font color="Maroon" title="c1">X1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <font color="Maroon" title="c2">X2</font></span>)</span></span>)</span> implies ( ( <font color="Maroon" title="c8">x</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="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <font color="Maroon" title="c2">X2</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">X2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <font color="Maroon" title="c2">X2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> ) &amp; ( not <font color="Maroon" title="c8">x</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="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <font color="Maroon" title="c2">X2</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">X2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <font color="Maroon" title="c1">X1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> ) ) )</span><br/><span class="kw">assume </span><a NAME="E1:18_2"/><span class="lab"><font color="Green" title="E16">A8</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <font color="Maroon" title="c1">X1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <font color="Maroon" title="c2">X2</font></span>)</span></span>)</span>
 ; <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="c8">x</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="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <font color="Maroon" title="c2">X2</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">X2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <font color="Maroon" title="c2">X2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> ) &amp; ( not <font color="Maroon" title="c8">x</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="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <font color="Maroon" title="c2">X2</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">X2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <font color="Maroon" title="c1">X1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> ) )</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">t</font> = <font color="Maroon" title="c8">x</font> as    <a href="trees_3.html#M3" title="TREES_3:mode.3">Element</a> of  <a href="trees_9.html#K9" title="TREES_9:func.9">Subtrees</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">X2</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:18"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E6:18"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E7:18"><span class="lab"><font color="Green" title="E15">A7</font></span></a></span>;<br/><a NAME="E3:18_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">X2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="trees_9.html#K9" title="TREES_9:func.9">Subtrees</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">X2</font></span>)</span></span>)</span> <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:18"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E6:18"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E7:18"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E1:18_2"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">p</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">X2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="trees_9.html#K9" title="TREES_9:func.9">Subtrees</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">X2</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/><div><span class="kw">hereby </span> <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="c8">x</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="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <font color="Maroon" title="c2">X2</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">X2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <font color="Maroon" title="c1">X1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:18_2_1"/><span class="lab"><font color="Green" title="E17">A9</font></span>: 
<font color="Maroon" title="c8">x</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="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <font color="Maroon" title="c2">X2</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="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">X2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <font color="Maroon" title="c2">X2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span><br/><a NAME="E2:18_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <font color="Maroon" title="c2">X2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="trees_9.html#K9" title="TREES_9:func.9">Subtrees</a> <font color="Maroon" title="c2">X2</font></span>)</span> <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:18"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">p2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <font color="Maroon" title="c2">X2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="trees_9.html#K9" title="TREES_9:func.9">Subtrees</a> <font color="Maroon" title="c2">X2</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/><a NAME="E4:18_2_1"/><span class="lab"><font color="Green" title="E18">A10</font></span>: 
<font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c10">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E5:18"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E6:18"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E7:18"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E1:18_2"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="ref" href="trees_9.html#D13" title="TREES_9:def.13">TREES_9:def 13</a></span>;<br/><a NAME="E5:18_2_1"/>
<font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c11">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E6:18"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E1:18_2_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="ref" href="trees_9.html#D13" title="TREES_9:def.13">TREES_9:def 13</a></span>;<br/><span class="kw">hence </span><a NAME="E6:18_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">X2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <font color="Maroon" title="c2">X2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18_2_1"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="ref" href="trees_4.html#T15" title="TREES_4:th.15">TREES_4:15</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">assume </span><a NAME="E6:18_2"/>
not <font color="Maroon" title="c8">x</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="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <font color="Maroon" title="c2">X2</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="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">X2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <font color="Maroon" title="c1">X1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span><br/><a NAME="E7:18_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A11</font></span>: 
<font color="Maroon" title="c8">x</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="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <font color="Maroon" title="c1">X1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_2"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E8:18_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <font color="Maroon" title="c1">X1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="trees_9.html#K9" title="TREES_9:func.9">Subtrees</a> <font color="Maroon" title="c1">X1</font></span>)</span> <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:18"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">p1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <font color="Maroon" title="c1">X1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="trees_9.html#K9" title="TREES_9:func.9">Subtrees</a> <font color="Maroon" title="c1">X1</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/><a NAME="E10:18_2"/><span class="lab"><font color="Green" title="E18">A12</font></span>: 
<font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c10">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E5:18"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E6:18"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E7:18"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E1:18_2"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="ref" href="trees_9.html#D13" title="TREES_9:def.13">TREES_9:def 13</a></span>;<br/><a NAME="E11:18_2"/>
<font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c11">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E5:18"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E7:18_2"><span class="lab"><font color="Green" title="E17">A11</font></span></a>, <a class="ref" href="trees_9.html#D13" title="TREES_9:def.13">TREES_9:def 13</a></span>;<br/><span class="kw">hence </span><a NAME="E12:18_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">X2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <font color="Maroon" title="c1">X1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:18_2"><span class="lab"><font color="Green" title="E18">A12</font></span></a>, <a class="ref" href="trees_4.html#T15" title="TREES_4:th.15">TREES_4:15</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">hence </span><a NAME="E9:18"/>
<font color="Maroon" title="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">X2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <font color="Maroon" title="c1">X1</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <font color="Maroon" title="c2">X2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E5:18"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E6:18"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E7:18"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</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/>


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