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<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a> ] <b>means </b>$1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="dtconstr.html#K14" title="DTCONSTR:func.14">NAT-to-PN</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K12" title="DTCONSTR:func.12">PN-to-NAT</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>;<br/>
<div><i><font color="Green" title="E33">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">s</font> be   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a> ;<br/><b>assume </b><a NAME="E1:49_1"/><i><font color="Green" title="E34">A2</font></i>: 
<font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K6" title="DTCONSTR:func.6">Terminals</a> <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a> 
 ;<br/><b>then </b><a href="dtconstr.html#K14" title="DTCONSTR:func.14">NAT-to-PN</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K12" title="DTCONSTR:func.12">PN-to-NAT</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c8">s</font></span>)</span></span>)</span> = 
<a href="dtconstr.html#K14" title="DTCONSTR:func.14">NAT-to-PN</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="dtconstr.html#D11" target="_self" title="DTCONSTR:def.11">Def11</a></i>
<br/>.= 
 <a href="dtconstr.html#K8" title="DTCONSTR:func.8">root-tree</a> <font color="Maroon" title="c6">O</font>
<b>by </b><i><a class="ref" href="dtconstr.html#D13" target="_self" title="DTCONSTR:def.13">Def13</a></i>
;<br/><b>hence </b><a NAME="E3:49_1"/>
<font color="Maroon">S<sub>2</sub></font>[ <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c8">s</font>]
 <b>by </b><i><a class="txt" href="dtconstr.html#E1:49_1"><i><font color="Green" title="E34">A2</font></i></a>, <a class="txt" href="dtconstr.html#E23"><i><font color="Green" title="E17">Lm9</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E34">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">nt</font> be   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a> ;<br/><b>let </b><font color="Maroon" title="c9">ts</font> be    <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> <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a> ;<br/><b>assume </b><a NAME="E1:49_2"/><i><font color="Green" title="E35">A4</font></i>: 
( <font color="Maroon" title="c8">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="Maroon" title="c9">ts</font> &amp; ( for <font color="Olive" title="b1">t</font> being  <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a>   st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">ts</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">t</font>] ) )
 ;<br/><a NAME="E2:49_2"/><b>then </b><i><font color="Green" title="E36">A5</font></i>: 
<font color="Maroon" title="c8">nt</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">O</font>
 <b>by </b><i><a class="txt" href="dtconstr.html#E21"><i><font color="Green" title="E15">Lm7</font></i></a></i>;<br/><a NAME="E3:49_2"/>
(  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c9">ts</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">O</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> or  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c9">ts</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c7">S</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="dtconstr.html#E1:49_2"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="dtconstr.html#D5" target="_self" title="DTCONSTR:def.5">Def5</a></i>;<br/><a NAME="E4:49_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c9">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">dt</font> being    <a href="trees_3.html#M5" title="TREES_3:mode.5">Element</a> of  <a href="trees_3.html#K5" title="TREES_3:func.5">FinTrees</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a> <b> such that </b><br/><a NAME="E6:49_2"/><i><font color="Green" title="E37">A6</font></i>: 
( <font color="Maroon" title="c9">ts</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c10">dt</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> &amp; <font color="Maroon" title="c10">dt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a>  )
 <b>by </b><i><a class="ref" href="dtconstr.html#T5" target="_self" title="DTCONSTR:th.5">Th5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">dt</font> = <font color="Maroon" title="c10">dt</font> as    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a>  <b>by </b><i><a class="txt" href="dtconstr.html#E6:49_2"><i><font color="Green" title="E37">A6</font></i></a></i>;<br/><a NAME="E8:49_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">ts</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c11">dt</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="dtconstr.html#E6:49_2"><i><font color="Green" title="E37">A6</font></i></a>, <a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></i>;<br/><a NAME="E9:49_2"/><b>then </b>
<font color="Maroon" title="c11">dt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">ts</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E10:49_2"/><b>then </b><i><font color="Green" title="E38">A7</font></i>: 
<font color="Maroon" title="c11">dt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="dtconstr.html#K14" title="DTCONSTR:func.14">NAT-to-PN</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K12" title="DTCONSTR:func.12">PN-to-NAT</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">dt</font></span>)</span>
 <b>by </b><i><a class="txt" href="dtconstr.html#E1:49_2"><i><font color="Green" title="E35">A4</font></i></a></i>;<br/><a NAME="E11:49_2"/><i><font color="Green" title="E39">A8</font></i>: 
<a href="dtconstr.html#K12" title="DTCONSTR:func.12">PN-to-NAT</a>  <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c9">ts</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="dtconstr.html#K12" title="DTCONSTR:func.12">PN-to-NAT</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">dt</font></span>)</span></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="dtconstr.html#E6:49_2"><i><font color="Green" title="E37">A6</font></i></a>, <a class="ref" href="finseq_2.html#T39" title="FINSEQ_2:th.39">FINSEQ_2:39</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">N</font> = <a href="dtconstr.html#K12" title="DTCONSTR:func.12">PN-to-NAT</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">dt</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><b>reconsider </b><font color="Maroon" title="c13">x</font> = <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c12">N</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><b>consider </b><font color="Maroon" title="c14">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E15:49_2"/><i><font color="Green" title="E40">A9</font></i>: 
(  <a href="dtconstr.html#K11" title="DTCONSTR:func.11">plus-one</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c12">N</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> <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="c14">n</font> )
 <b>by </b><i><a class="ref" href="dtconstr.html#D10" target="_self" title="DTCONSTR:def.10">Def10</a></i>;<br/><a NAME="E16:49_2"/>
<font color="Maroon" title="c12">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">n</font>
 <b>by </b><i><a class="txt" href="dtconstr.html#E15:49_2"><i><font color="Green" title="E40">A9</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/><b>then </b><a href="dtconstr.html#K14" title="DTCONSTR:func.14">NAT-to-PN</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
 <a href="dtconstr.html#K13" title="DTCONSTR:func.13">PNsucc</a> <font color="Maroon" title="c11">dt</font>
<b>by </b><i><a class="txt" href="dtconstr.html#E10:49_2"><i><font color="Green" title="E38">A7</font></i></a>, <a class="ref" href="dtconstr.html#D13" target="_self" title="DTCONSTR:def.13">Def13</a></i>
<br/>.= 
<font color="Maroon" title="c8">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c9">ts</font>
<b>by </b><i><a class="txt" href="dtconstr.html#E2:49_2"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="dtconstr.html#E6:49_2"><i><font color="Green" title="E37">A6</font></i></a>, <a class="txt" href="dtconstr.html#E15"><i><font color="Green" title="E10">Lm2</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="E18:49_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c9">ts</font>]
 <b>by </b><i><a class="txt" href="dtconstr.html#E1:49_2"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="dtconstr.html#E11:49_2"><i><font color="Green" title="E39">A8</font></i></a>, <a class="txt" href="dtconstr.html#E15:49_2"><i><font color="Green" title="E40">A9</font></i></a>, <a class="ref" href="dtconstr.html#D11" target="_self" title="DTCONSTR:def.11">Def11</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:49"/>
for <font color="Olive" title="b1">t</font> being  <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a>   st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a>  holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">t</font>]
 
<b>from </b><i><a class="ref" href="dtconstr.html#S7" target="_self" title="DTCONSTR:sch.7">DTCONSTR:sch 7</a>(<a class="txt" href="dtconstr.html#E1:49"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="dtconstr.html#E2:49"><i><font color="Green" title="E34">A3</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:49"/>
for <font color="Olive" title="b1">pn</font> being   <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a>  holds  <font color="Olive" title="b1">pn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="dtconstr.html#K14" title="DTCONSTR:func.14">NAT-to-PN</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K12" title="DTCONSTR:func.12">PN-to-NAT</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">pn</font></span>)</span>
 ;<br/>


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