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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>(),  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>,   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon">F<sub>2</sub></font>()) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() = <font color="Maroon">F<sub>4</sub></font>($1,<span class="p1">(<span class="default">$2 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<span class="p1">(<span class="default">$2 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span>,<span class="p1">(<span class="default">$3 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<span class="p1">(<span class="default">$3 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span>);<br/>
<b>consider </b><font color="Maroon" title="c1">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E2:21_1"/><i><font color="Green" title="E10">A1</font></i>: 
for <font color="Olive" title="b1">t</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K6" title="DTCONSTR:func.6">Terminals</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Olive" title="b1">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">t</font>)
 <b>and </b><br/><a NAME="E3:21_1"/><i><font color="Green" title="E11">A2</font></i>: 
for <font color="Olive" title="b1">nt</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b2">ts</font> being   <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b1">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b2">ts</font> holds <br/><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Olive" title="b2">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">nt</font>, <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b2">ts</font>,<font color="Maroon" title="c1">f</font> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Olive" title="b2">ts</font>)
 <b>from </b><i><a class="ref" href="dtconstr.html#S8" title="DTCONSTR:sch.8">DTCONSTR:sch 8</a>();<br/></i>
<b>take </b>
<font color="Maroon" title="c1">f</font>
;<br/>

<a NAME="E4:21_1"/><i><font color="Green" title="E12">A3</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>thus </b><a NAME="E5:21_1"/>
for <font color="Olive" title="b1">t</font> being  <a href="dtconstr.html#NM2" title="DTCONSTR:NM.2">Terminal</a> of <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon" title="c1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K8" title="DTCONSTR:func.8">root-tree</a> <font color="Olive" title="b1">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">t</font>)
 <b>by </b><i><a class="txt" href="bintree1.html#E2:21_1"><i><font color="Green" title="E10">A1</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c2">nt</font> be   <a href="dtconstr.html#NM3" title="DTCONSTR:NM.3">NonTerminal</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>let </b><font color="Maroon" title="c3">tl</font>, <font color="Maroon" title="c4">tr</font> be    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <font color="Maroon">F<sub>1</sub></font>();<br/><b>let </b><font color="Maroon" title="c5">rtl</font>, <font color="Maroon" title="c6">rtr</font> be   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>





<b>assume </b><a NAME="E6:21_1"/><i><font color="Green" title="E13">A4</font></i>: 
( <font color="Maroon" title="c5">rtl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bintree1.html#K1" title="BINTREE1:func.1">root-label</a> <font color="Maroon" title="c3">tl</font> &amp; <font color="Maroon" title="c6">rtr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bintree1.html#K1" title="BINTREE1:func.1">root-label</a> <font color="Maroon" title="c4">tr</font> &amp; <font color="Maroon" title="c2">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c5">rtl</font>,<font color="Maroon" title="c6">rtr</font></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span> )
 ;<br/>

<b>let </b><font color="Maroon" title="c7">xl</font>, <font color="Maroon" title="c8">xr</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>();<br/>

<b>assume </b><a NAME="E7:21_1"/><i><font color="Green" title="E14">A5</font></i>: 
( <font color="Maroon" title="c7">xl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">tl</font> &amp; <font color="Maroon" title="c8">xr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">tr</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c9">ts</font> = <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">tl</font>,<font color="Maroon" title="c4">tr</font></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span> as    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <font color="Maroon">F<sub>1</sub></font>() ;<br/>
<b>reconsider </b><font color="Maroon" title="c10">rts</font> = <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c5">rtl</font>,<font color="Maroon" title="c6">rtr</font></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> ;<br/>
<a NAME="E10:21_1"/><i><font color="Green" title="E15">A6</font></i>: 
( <font color="Maroon" title="c9">ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">tl</font> &amp; <font color="Maroon" title="c9">ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">tr</font> &amp; <font color="Maroon" title="c10">rts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">rtl</font> &amp; <font color="Maroon" title="c10">rts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">rtr</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E11:21_1"/><i><font color="Green" title="E16">A7</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">rts</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,2</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></i>;<br/>
<a NAME="E12:21_1"/><i><font color="Green" title="E17">A8</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">ts</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,2</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:21_1_1"/>
<font color="Maroon" title="c11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">ts</font>
 ;<br/><a NAME="E2:21_1_1"/><b>then </b>
<font color="Maroon" title="c11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">ts</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:21_1_1"/><b>then </b><i><font color="Green" title="E18">A9</font></i>: 
<font color="Maroon" title="c11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2
 <b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1_1_1"/>
( <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 </span><b>by </b><i><a class="txt" href="bintree1.html#E3:21_1_1"><i><font color="Green" title="E18">A9</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1"><b>suppose </b></a><a NAME="E1:21_1_1_1_1"/>
<font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:21_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c3">tl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c10">rts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">tl</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>  )
 <b>by </b><i><a class="txt" href="bintree1.html#E6:21_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="txt" href="bintree1.html#E10:21_1"><i><font color="Green" title="E15">A6</font></i></a></i>;<br/><b>hence </b><a NAME="E3:21_1_1_1_1"/>
 ex <font color="Olive" title="b1">T</font> being  <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a> st <br/>( <font color="Olive" title="b1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c10">rts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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>  )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_2"><b>suppose </b></a><a NAME="E1:21_1_1_1_2"/>
<font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:21_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c4">tr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c10">rts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">tr</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>  )
 <b>by </b><i><a class="txt" href="bintree1.html#E6:21_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="txt" href="bintree1.html#E10:21_1"><i><font color="Green" title="E15">A6</font></i></a></i>;<br/><b>hence </b><a NAME="E3:21_1_1_1_2"/>
 ex <font color="Olive" title="b1">T</font> being  <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a> st <br/>( <font color="Olive" title="b1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c10">rts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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>  )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E14:21_1"/><b>then </b><i><font color="Green" title="E18">A10</font></i>: 
<font color="Maroon" title="c10">rts</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c9">ts</font>
 
<b>by </b><i><a class="txt" href="bintree1.html#E11:21_1"><i><font color="Green" title="E16">A7</font></i></a>, <a class="txt" href="bintree1.html#E12:21_1"><i><font color="Green" title="E17">A8</font></i></a>, <a class="ref" href="trees_3.html#D18" title="TREES_3:def.18">TREES_3:def 18</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">x</font> = <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c7">xl</font>,<font color="Maroon" title="c8">xr</font></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon">F<sub>2</sub></font>() ;<br/>
<a NAME="E16:21_1"/><i><font color="Green" title="E19">A11</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,2</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></i>;<br/>
<div><i><font color="Green" title="E20">A12</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><i><font color="Green" title="E21">A13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_1_2_1"/><i><font color="Green" title="E22">A14</font></i>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">x</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1_2_1_1"/>
( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 </span><b>by </b><i><a class="txt" href="bintree1.html#E16:21_1"><i><font color="Green" title="E19">A11</font></i></a>, <a class="txt" href="bintree1.html#E1:21_1_2_1"><i><font color="Green" title="E22">A14</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:21_1_2_1_1_1"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_2_1_1_1"/>
( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">ts</font> &amp; <font color="Maroon" title="c9">ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font> )
 <b>by </b><i><a class="txt" href="bintree1.html#E4:21_1"><i><font color="Green" title="E12">A3</font></i></a>, <a class="txt" href="bintree1.html#E10:21_1"><i><font color="Green" title="E15">A6</font></i></a>, <a class="txt" href="bintree1.html#E12:21_1"><i><font color="Green" title="E17">A8</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:21_1_2_1_1_2"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_2_1_1_2"/>
( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">ts</font> &amp; <font color="Maroon" title="c9">ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font> )
 <b>by </b><i><a class="txt" href="bintree1.html#E4:21_1"><i><font color="Green" title="E12">A3</font></i></a>, <a class="txt" href="bintree1.html#E10:21_1"><i><font color="Green" title="E15">A6</font></i></a>, <a class="txt" href="bintree1.html#E12:21_1"><i><font color="Green" title="E17">A8</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_1_2_2"/><i><font color="Green" title="E22">A15</font></i>: 
( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">ts</font> &amp; <font color="Maroon" title="c9">ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1_2_2_1"/>
( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 </span><b>by </b><i><a class="txt" href="bintree1.html#E12:21_1"><i><font color="Green" title="E17">A8</font></i></a>, <a class="txt" href="bintree1.html#E1:21_1_2_2"><i><font color="Green" title="E22">A15</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:21_1_2_2_1_1"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_2_2_1_1"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">x</font>
 <b>by </b><i><a class="txt" href="bintree1.html#E16:21_1"><i><font color="Green" title="E19">A11</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:21_1_2_2_1_2"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_2_2_1_2"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">x</font>
 <b>by </b><i><a class="txt" href="bintree1.html#E16:21_1"><i><font color="Green" title="E19">A11</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:21_1_2"/>
( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">x</font> iff ( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">ts</font> &amp; <font color="Maroon" title="c9">ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font> ) )
 <b>by </b><i><a class="txt" href="bintree1.html#E1:21_1_2"><i><font color="Green" title="E21">A13</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:21_1_3"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">x</font>
 ;<br/><a NAME="E2:21_1_3"/><b>then </b>
( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <b>by </b><i><a class="txt" href="bintree1.html#E16:21_1"><i><font color="Green" title="E19">A11</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E3:21_1_3"/>
<font color="Maroon" title="c11">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="bintree1.html#E7:21_1"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="bintree1.html#E10:21_1"><i><font color="Green" title="E15">A6</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/></div><b>end;</b></div>
<a NAME="E19:21_1"/><b>then </b>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c9">ts</font>
 
<b>by </b><i><a class="txt" href="bintree1.html#E17:21_1"><i><font color="Green" title="E20">A12</font></i></a>, <a class="ref" href="funct_1.html#T20" title="FUNCT_1:th.20">FUNCT_1:20</a></i>;<br/>
<a NAME="E20:21_1"/><b>then </b><i><font color="Green" title="E21">A16</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c9">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c2">nt</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">rts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">rts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span>)
 
<b>by </b><i><a class="txt" href="bintree1.html#E3:21_1"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="bintree1.html#E6:21_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="txt" href="bintree1.html#E14:21_1"><i><font color="Green" title="E18">A10</font></i></a></i>;<br/>
<a NAME="E21:21_1"/>
( <font color="Maroon" title="c10">rts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">rtl</font> &amp; <font color="Maroon" title="c10">rts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">rtr</font> &amp; <font color="Maroon" title="c11">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">xl</font> &amp; <font color="Maroon" title="c11">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">xr</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/>
<b>hence </b><a NAME="E22:21_1"/>
<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">nt</font> <a href="trees_4.html#K10" title="TREES_4:func.10">-tree</a> <font color="Maroon" title="c3">tl</font>,<font color="Maroon" title="c4">tr</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c2">nt</font>,<font color="Maroon" title="c5">rtl</font>,<font color="Maroon" title="c6">rtr</font>,<font color="Maroon" title="c7">xl</font>,<font color="Maroon" title="c8">xr</font>)
 <b>by </b><i><a class="txt" href="bintree1.html#E20:21_1"><i><font color="Green" title="E21">A16</font></i></a>, <a class="ref" href="trees_4.html#D6" title="TREES_4:def.6">TREES_4:def 6</a></i>;<br/>


</div>
