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<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="143_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:143_1_1"/>
( <font color="Maroon">F<sub>4</sub></font>() <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon">F<sub>5</sub></font>() or <font color="Maroon">F<sub>4</sub></font>() <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>5</sub></font>() )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="143_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:143_1_1_1"/>
<font color="Maroon">F<sub>4</sub></font>() <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon">F<sub>5</sub></font>()
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">F<sub>6</sub></font>() <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:143_1_1_1"/>
<font color="Maroon">F<sub>6</sub></font>() <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_000.html#T83" target="_self" title="AOFA_000:th.83">Th83</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="143_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:143_1_1_2"/><span class="lab"><font color="Green" title="E93">A4</font></span>: 
<font color="Maroon">F<sub>4</sub></font>() <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>5</sub></font>()
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">F<sub>6</sub></font>() <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()</span><br/><div class="add"><span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>3</sub></font>() = <font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<span class="p1">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> ($2,<font color="Maroon">F<sub>3</sub></font>())</span>)</span>,<font color="Maroon">F<sub>2</sub></font>());<br/><span class="kw">consider </span><font color="Maroon" title="c1">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<font color="Maroon">F<sub>3</sub></font>()<span class="kw"> such that </span><br/><a NAME="E3:143_1_1_2"/><span class="lab"><font color="Green" title="E94">A5</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>()
 <span class="kw">and </span><br/><a NAME="E4:143_1_1_2"/><span class="lab"><font color="Green" title="E95">A6</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c1">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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">i</font>,<font color="Maroon" title="c1">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S12" title="NAT_1:sch.12">NAT_1:sch 12</a>();<br/></span><span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span> ex <font color="Olive" title="b1">i</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>  st <font color="Maroon">F<sub>7</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1;<br/><a NAME="E5:143_1_1_2"/>
<font color="Maroon">F<sub>7</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><a NAME="E6:143_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A7</font></span>: 
 ex <font color="Olive" title="b1">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>]
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c2">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E8:143_1_1_2"/><span class="lab"><font color="Green" title="E97">A8</font></span>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">j</font>] &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Maroon" title="c2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="#E6:143_1_1_2"><span class="lab"><font color="Green" title="E96">A7</font></span></a>);<br/></span><span class="kw">consider </span><font color="Maroon" title="c3">i0</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> <span class="kw"> such that </span><br/><a NAME="E10:143_1_1_2"/><span class="lab"><font color="Green" title="E98">A9</font></span>: 
<font color="Maroon">F<sub>7</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">i0</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:143_1_1_2"><span class="lab"><font color="Green" title="E97">A8</font></span></a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>(  not <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1] or  not <font color="Maroon" title="c1">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>5</sub></font>() or  not <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c1">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1] );<br/><a NAME="E11:143_1_1_2"/>
 <a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">i0</font></span>)</span>,<font color="Maroon">F<sub>3</sub></font>()) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">i0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a></span>;<br/><a NAME="E12:143_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">i0</font></span>)</span>,<font color="Maroon">F<sub>2</sub></font>())
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:143_1_1_2"><span class="lab"><font color="Green" title="E95">A6</font></span></a></span>;<br/><a NAME="E13:143_1_1_2"/><span class="kw">then </span>
(  not <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c3">i0</font>] implies <font color="Maroon">F<sub>7</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">j</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:143"><span class="lab"><font color="Green" title="E92">A3</font></span></a>, <a class="txt" href="#E10:143_1_1_2"><span class="lab"><font color="Green" title="E98">A9</font></span></a></span>;<br/><a NAME="E14:143_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A10</font></span>: 
 ex <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:143_1_1_2"><span class="lab"><font color="Green" title="E97">A8</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c4">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E16:143_1_1_2"/><span class="lab"><font color="Green" title="E100">A11</font></span>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">j</font>] &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="#E14:143_1_1_2"><span class="lab"><font color="Green" title="E99">A10</font></span></a>);<br/></span><span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>3</sub></font>() = <font color="Maroon" title="c1">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default">$1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>;<br/><span class="kw">consider </span><font color="Maroon" title="c5">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon">F<sub>3</sub></font>()<span class="kw"> such that </span><br/><a NAME="E18:143_1_1_2"/><span class="lab"><font color="Green" title="E101">A12</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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="c5">p</font> holds <br/><font color="Maroon" title="c5">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">i</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_2.html#S1" title="FINSEQ_2:sch.1">FINSEQ_2:sch 1</a>();<br/></span><a NAME="E19:143_1_1_2"/><span class="lab"><font color="Green" title="E102">A13</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:143_1_1_2"><span class="lab"><font color="Green" title="E101">A12</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c6">p</font> = <font color="Maroon" title="c5">p</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon">F<sub>3</sub></font>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:143_1_1_2"><span class="lab"><font color="Green" title="E101">A12</font></span></a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c6">p</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="aofa_000.html#D33" target="_self" title="AOFA_000:def.33">AOFA_000:def 33</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">p</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">F<sub>4</sub></font>() &amp; <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon">F<sub>5</sub></font>() &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font> holds <br/>( <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>5</sub></font>() &amp; <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<font color="Maroon">F<sub>2</sub></font>()) ) ) )</span><br/><a NAME="E21:143_1_1_2"/><span class="lab"><font color="Green" title="E103">A14</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E22:143_1_1_2"/><span class="kw">then </span>
1 <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"><font color="Maroon" title="c4">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/><span class="kw">hence </span><font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<font color="Maroon">H<sub>2</sub></font>(1)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:143_1_1_2"><span class="lab"><font color="Green" title="E101">A12</font></span></a>, <a class="txt" href="#E19:143_1_1_2"><span class="lab"><font color="Green" title="E102">A13</font></span></a></span>
<br/>.= 
<font color="Maroon">F<sub>4</sub></font>()
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:143_1_1_2"><span class="lab"><font color="Green" title="E94">A5</font></span></a>, <a class="ref" href="xreal_1.html#T232" title="XREAL_1:th.232">XREAL_1:232</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon">F<sub>5</sub></font>() &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font> holds <br/>( <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>5</sub></font>() &amp; <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<font color="Maroon">F<sub>2</sub></font>()) ) ) )</span><br/><a NAME="E24:143_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</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"><font color="Maroon" title="c4">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:143_1_1_2"><span class="lab"><font color="Green" title="E101">A12</font></span></a>, <a class="txt" href="#E21:143_1_1_2"><span class="lab"><font color="Green" title="E103">A14</font></span></a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A15</font></span>: <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font></span>)</span> = 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c4">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:143_1_1_2"><span class="lab"><font color="Green" title="E101">A12</font></span></a>, <a class="txt" href="#E19:143_1_1_2"><span class="lab"><font color="Green" title="E102">A13</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c1">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">j</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>
;<br/><a NAME="E26:143_1_1_2"/>
<font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:143"><span class="lab"><font color="Green" title="E90">A1</font></span></a>, <a class="txt" href="#E2:143"><span class="lab"><font color="Green" title="E91">A2</font></span></a>, <a class="txt" href="#E1:143_1_1_2"><span class="lab"><font color="Green" title="E93">A4</font></span></a>, <a class="txt" href="#E3:143_1_1_2"><span class="lab"><font color="Green" title="E94">A5</font></span></a>, <a class="txt" href="#E16:143_1_1_2"><span class="lab"><font color="Green" title="E100">A11</font></span></a></span>;<br/><a NAME="E27:143_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">j9</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E29:143_1_1_2"/><span class="lab"><font color="Green" title="E105">A16</font></span>: 
<font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">j9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">j9</font> = <font color="Maroon" title="c7">j9</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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><a NAME="E31:143_1_1_2"/><span class="lab"><font color="Green" title="E106">A17</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<span class="p1">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">j9</font></span>)</span>,<font color="Maroon">F<sub>3</sub></font>())</span>)</span>,<font color="Maroon">F<sub>2</sub></font>())
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:143_1_1_2"><span class="lab"><font color="Green" title="E95">A6</font></span></a>, <a class="txt" href="#E29:143_1_1_2"><span class="lab"><font color="Green" title="E105">A16</font></span></a></span>;<br/><a NAME="E32:143_1_1_2"/><span class="lab"><font color="Green" title="E107">A18</font></span>: 
<font color="Maroon" title="c8">j9</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:143_1_1_2"><span class="lab"><font color="Green" title="E105">A16</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E33:143_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E108">A19</font></span>: 
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c1">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">j9</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:143_1_1_2"><span class="lab"><font color="Green" title="E100">A11</font></span></a></span>;<br/><a NAME="E34:143_1_1_2"/><span class="lab"><font color="Green" title="E109">A20</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">j9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>5</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:143_1_1_2"><span class="lab"><font color="Green" title="E100">A11</font></span></a>, <a class="txt" href="#E32:143_1_1_2"><span class="lab"><font color="Green" title="E107">A18</font></span></a></span>;<br/><a NAME="E35:143_1_1_2"/><span class="lab"><font color="Green" title="E110">A21</font></span>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">j9</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:143_1_1_2"><span class="lab"><font color="Green" title="E100">A11</font></span></a>, <a class="txt" href="#E32:143_1_1_2"><span class="lab"><font color="Green" title="E107">A18</font></span></a></span>;<br/><a NAME="E36:143_1_1_2"/>
 <a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">j9</font></span>)</span>,<font color="Maroon">F<sub>3</sub></font>()) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">j9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E37:143_1_1_2"/>
<font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon">F<sub>5</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:143"><span class="lab"><font color="Green" title="E92">A3</font></span></a>, <a class="txt" href="#E16:143_1_1_2"><span class="lab"><font color="Green" title="E100">A11</font></span></a>, <a class="txt" href="#E25:143_1_1_2"><span class="lab"><font color="Green" title="E104">A15</font></span></a>, <a class="txt" href="#E31:143_1_1_2"><span class="lab"><font color="Green" title="E106">A17</font></span></a>, <a class="txt" href="#E33:143_1_1_2"><span class="lab"><font color="Green" title="E108">A19</font></span></a>, <a class="txt" href="#E34:143_1_1_2"><span class="lab"><font color="Green" title="E109">A20</font></span></a>, <a class="txt" href="#E35:143_1_1_2"><span class="lab"><font color="Green" title="E110">A21</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font> holds <br/>( <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>5</sub></font>() &amp; <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<font color="Maroon">F<sub>2</sub></font>()) )</span><br/><span class="kw">let </span><font color="Maroon" title="c9">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i</font> &amp; <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font> implies ( <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>5</sub></font>() &amp; <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span>,<font color="Maroon">F<sub>2</sub></font>()) ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E38:143_1_1_2"/><span class="lab"><font color="Green" title="E111">A22</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i</font>
 <span class="kw">and </span><br/><a NAME="E39:143_1_1_2"/><span class="lab"><font color="Green" title="E112">A23</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>5</sub></font>() &amp; <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span>,<font color="Maroon">F<sub>2</sub></font>()) )</span><br/><a NAME="E40:143_1_1_2"/><span class="lab"><font color="Green" title="E113">A24</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:143_1_1_2"><span class="lab"><font color="Green" title="E111">A22</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E41:143_1_1_2"/><span class="lab"><font color="Green" title="E114">A25</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:143_1_1_2"><span class="lab"><font color="Green" title="E112">A23</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E42:143_1_1_2"/><span class="lab"><font color="Green" title="E115">A26</font></span>: 
<font color="Maroon" title="c9">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"><font color="Maroon" title="c4">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:143_1_1_2"><span class="lab"><font color="Green" title="E101">A12</font></span></a>, <a class="txt" href="#E38:143_1_1_2"><span class="lab"><font color="Green" title="E111">A22</font></span></a>, <a class="txt" href="#E39:143_1_1_2"><span class="lab"><font color="Green" title="E112">A23</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E43:143_1_1_2"/><span class="lab"><font color="Green" title="E116">A27</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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"><font color="Maroon" title="c4">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:143_1_1_2"><span class="lab"><font color="Green" title="E101">A12</font></span></a>, <a class="txt" href="#E40:143_1_1_2"><span class="lab"><font color="Green" title="E113">A24</font></span></a>, <a class="txt" href="#E41:143_1_1_2"><span class="lab"><font color="Green" title="E114">A25</font></span></a></span>;<br/><a NAME="E44:143_1_1_2"/><span class="lab"><font color="Green" title="E117">A28</font></span>: 
<font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c9">i</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:143_1_1_2"><span class="lab"><font color="Green" title="E101">A12</font></span></a>, <a class="txt" href="#E19:143_1_1_2"><span class="lab"><font color="Green" title="E102">A13</font></span></a>, <a class="txt" href="#E42:143_1_1_2"><span class="lab"><font color="Green" title="E115">A26</font></span></a></span>;<br/><a NAME="E45:143_1_1_2"/><span class="lab"><font color="Green" title="E118">A29</font></span>: 
<font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:143_1_1_2"><span class="lab"><font color="Green" title="E101">A12</font></span></a>, <a class="txt" href="#E19:143_1_1_2"><span class="lab"><font color="Green" title="E102">A13</font></span></a>, <a class="txt" href="#E43:143_1_1_2"><span class="lab"><font color="Green" title="E116">A27</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">i9</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E47:143_1_1_2"/><span class="lab"><font color="Green" title="E119">A30</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">i9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:143_1_1_2"><span class="lab"><font color="Green" title="E111">A22</font></span></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">i9</font> = <font color="Maroon" title="c10">i9</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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><a NAME="E49:143_1_1_2"/>
<font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:143_1_1_2"><span class="lab"><font color="Green" title="E101">A12</font></span></a>, <a class="txt" href="#E39:143_1_1_2"><span class="lab"><font color="Green" title="E112">A23</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E50:143_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E120">A31</font></span>: 
<font color="Maroon" title="c11">i9</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:143_1_1_2"><span class="lab"><font color="Green" title="E119">A30</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E51:143_1_1_2"/><span class="lab"><font color="Green" title="E121">A32</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:143_1_1_2"><span class="lab"><font color="Green" title="E119">A30</font></span></a>, <a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/><a NAME="E52:143_1_1_2"/><span class="lab"><font color="Green" title="E122">A33</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/><a NAME="E53:143_1_1_2"/>
 <a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">i9</font></span>)</span>,<font color="Maroon">F<sub>3</sub></font>()) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">i9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E54:143_1_1_2"/>
( <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>5</sub></font>() &amp; <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span>,<font color="Maroon">F<sub>2</sub></font>()) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:143_1_1_2"><span class="lab"><font color="Green" title="E95">A6</font></span></a>, <a class="txt" href="#E16:143_1_1_2"><span class="lab"><font color="Green" title="E100">A11</font></span></a>, <a class="txt" href="#E44:143_1_1_2"><span class="lab"><font color="Green" title="E117">A28</font></span></a>, <a class="txt" href="#E45:143_1_1_2"><span class="lab"><font color="Green" title="E118">A29</font></span></a>, <a class="txt" href="#E47:143_1_1_2"><span class="lab"><font color="Green" title="E119">A30</font></span></a>, <a class="txt" href="#E50:143_1_1_2"><span class="lab"><font color="Green" title="E120">A31</font></span></a>, <a class="txt" href="#E51:143_1_1_2"><span class="lab"><font color="Green" title="E121">A32</font></span></a>, <a class="txt" href="#E52:143_1_1_2"><span class="lab"><font color="Green" title="E122">A33</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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