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<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="trees_2.html#NM6">DecoratedTree</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="dtconstr.html#K5" target="_self">PeanoNat</a> ] means <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <a href="dtconstr.html#K14" target="_self">NAT-to-PN</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K12" target="_self">PN-to-NAT</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<div><i><font color="Green">E33</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">c<sub>8</sub></font> be   <a href="lang1.html#NM1">Symbol</a> of <a href="dtconstr.html#K5" target="_self">PeanoNat</a> ;<br/><b>assume </b><a NAME="E1:49_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K6" target="_self">Terminals</a> <a href="dtconstr.html#K5" target="_self">PeanoNat</a> 
 ;<br/><b>then </b><a href="dtconstr.html#K14" target="_self">NAT-to-PN</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K12" target="_self">PN-to-NAT</a>  <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> = 
<a href="dtconstr.html#K14" target="_self">NAT-to-PN</a>  <a href="funct_2.html#K3">.</a> 0
<b>by </b><i><a class="ref" href="dtconstr.html#D11" target="_self">Def11</a></i>
<br/>.= 
 <a href="dtconstr.html#K8" target="_self">root-tree</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="dtconstr.html#D13" target="_self">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">root-tree</a> <font color="Maroon">c<sub>8</sub></font>]
 <b>by </b><i><a class="txt" href="dtconstr.html#E1:49_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="dtconstr.html#E23"><i><font color="Green">Lemma17</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E34</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">c<sub>8</sub></font> be   <a href="lang1.html#NM1">Symbol</a> of <a href="dtconstr.html#K5" target="_self">PeanoNat</a> ;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="trees_4.html#M1">FinSequence</a> of  <a href="dtconstr.html#K4" target="_self">TS</a> <a href="dtconstr.html#K5" target="_self">PeanoNat</a> ;<br/><b>assume </b><a NAME="E1:49_2"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K1" target="_self">roots</a> <font color="Maroon">c<sub>9</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="trees_2.html#NM6">DecoratedTree</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="dtconstr.html#K5" target="_self">PeanoNat</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) )
 ;<br/><a NAME="E2:49_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="dtconstr.html#E21"><i><font color="Green">Lemma15</font></i></a></i>;<br/><a NAME="E3:49_2"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="dtconstr.html#E15"><i><font color="Green">Lemma10</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E4:49_2"/>
(  <a href="dtconstr.html#K1" target="_self">roots</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="lang1.html#K3">*&gt;</a></span> or  <a href="dtconstr.html#K1" target="_self">roots</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="lang1.html#K3">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="dtconstr.html#E1:49_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="dtconstr.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E5:49_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K1" target="_self">roots</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="trees_3.html#M5">Element</a> of  <a href="trees_3.html#K5">FinTrees</a> the <a href="struct_0.html#U1">carrier</a> of <a href="dtconstr.html#K5" target="_self">PeanoNat</a> <b> such that </b><br/><a NAME="E7:49_2"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="lang1.html#K3">*&gt;</a></span> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K4" target="_self">TS</a> <a href="dtconstr.html#K5" target="_self">PeanoNat</a>  )
 <b>by </b><i><a class="ref" href="dtconstr.html#T5" target="_self">Th5</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as    <a href="dtconstr.html#M1" target="_self">Element</a> of  <a href="dtconstr.html#K4" target="_self">TS</a> <a href="dtconstr.html#K5" target="_self">PeanoNat</a>  <b>by </b><i><a class="txt" href="dtconstr.html#E7:49_2"><i><font color="Green">E37</font></i></a></i>;<br/><a NAME="E9:49_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="dtconstr.html#E7:49_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/><a NAME="E10:49_2"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E11:49_2"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <a href="dtconstr.html#K14" target="_self">NAT-to-PN</a>  <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K12" target="_self">PN-to-NAT</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="dtconstr.html#E1:49_2"><i><font color="Green">E35</font></i></a></i>;<br/><a NAME="E12:49_2"/><i><font color="Green">E39</font></i>: 
<a href="dtconstr.html#K12" target="_self">PN-to-NAT</a>  <a href="finseqop.html#K5">*</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="dtconstr.html#K12" target="_self">PN-to-NAT</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="lang1.html#K3">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="dtconstr.html#E7:49_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_2.html#T39">FINSEQ_2:39</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <a href="dtconstr.html#K12" target="_self">PN-to-NAT</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>12</sub></font></span><a href="lang1.html#K3">*&gt;</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E16:49_2"/><i><font color="Green">E40</font></i>: 
(  <a href="dtconstr.html#K11" target="_self">plus-one</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>12</sub></font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="ref" href="dtconstr.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E17:49_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="dtconstr.html#E16:49_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><b>then </b><a href="dtconstr.html#K14" target="_self">NAT-to-PN</a>  <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
 <a href="dtconstr.html#K13" target="_self">PNsucc</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="dtconstr.html#E11:49_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="dtconstr.html#D13" target="_self">Def13</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="dtconstr.html#E3:49_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="dtconstr.html#E7:49_2"><i><font color="Green">E37</font></i></a></i>
;<br/><b>hence </b><a NAME="E19:49_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>8</sub></font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">c<sub>9</sub></font>]
 <b>by </b><i><a class="txt" href="dtconstr.html#E1:49_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="dtconstr.html#E12:49_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="dtconstr.html#E16:49_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="dtconstr.html#D11" target="_self">Def11</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:49"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="trees_2.html#NM6">DecoratedTree</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="dtconstr.html#K5" target="_self">PeanoNat</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K4" target="_self">TS</a> <a href="dtconstr.html#K5" target="_self">PeanoNat</a>  holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="dtconstr.html#S7" target="_self">DTCONSTR:sch 7</a>(<a class="txt" href="dtconstr.html#E1:49"><i><font color="Green">E33</font></i></a>, <a class="txt" href="dtconstr.html#E2:49"><i><font color="Green">E34</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:49"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="dtconstr.html#M1" target="_self">Element</a> of  <a href="dtconstr.html#K4" target="_self">TS</a> <a href="dtconstr.html#K5" target="_self">PeanoNat</a>  holds <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <a href="dtconstr.html#K14" target="_self">NAT-to-PN</a>  <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K12" target="_self">PN-to-NAT</a>  <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 ;<br/>


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