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<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="142_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:142_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="142_1_1_1"><b>suppose </b></a><a NAME="E1:142_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>()
 ;<br/><div class="add"><b>hence </b><a NAME="E2:142_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>()
 <b>by </b><i><a class="ref" href="aofa_000.html#T83" target="_self" title="AOFA_000:th.83">Th83</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="142_1_1_2"><b>suppose </b></a><a NAME="E1:142_1_1_2"/><i><font color="Green" title="E92">A3</font></i>: 
<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>()
 ;<br/><div class="add"><b>deffunc </b><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> ) <b>-&gt; </b>   <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/><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="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon">F<sub>3</sub></font>()<b> such that </b><br/><a NAME="E3:142_1_1_2"/><i><font color="Green" title="E93">A4</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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>()
 <b>and </b><br/><a NAME="E4:142_1_1_2"/><i><font color="Green" title="E94">A5</font></i>: 
for <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>  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"><font color="Olive" title="b1">i</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">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>)
 <b>from </b><i><a class="ref" href="recdef_1.html#S4" title="RECDEF_1:sch.4">RECDEF_1:sch 4</a>();<br/></i><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b> 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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1;<br/><a NAME="E5:142_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="funct_2.html#K3" title="FUNCT_2:func.3">.</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> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E6:142_1_1_2"/><b>then </b><i><font color="Green" title="E95">A6</font></i>: 
 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/><b>consider </b><font color="Maroon" title="c2">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E8:142_1_1_2"/><i><font color="Green" title="E96">A7</font></i>: 
( <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> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="aofa_000.html#E6:142_1_1_2"><i><font color="Green" title="E95">A6</font></i></a>);<br/></i><b>consider </b><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> <b> such that </b><br/><a NAME="E10:142_1_1_2"/><i><font color="Green" title="E97">A8</font></i>: 
<font color="Maroon">F<sub>7</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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>
 <b>by </b><i><a class="txt" href="aofa_000.html#E8:142_1_1_2"><i><font color="Green" title="E96">A7</font></i></a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1];<br/><a NAME="E11:142_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="funct_2.html#K3" title="FUNCT_2:func.3">.</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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">i0</font>
 <b>by </b><i><a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a></i>;<br/><a NAME="E12:142_1_1_2"/><b>then </b>
<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"><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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">i0</font></span>)</span>,<font color="Maroon">F<sub>2</sub></font>()
 <b>by </b><i><a class="txt" href="aofa_000.html#E4:142_1_1_2"><i><font color="Green" title="E94">A5</font></i></a></i>;<br/><a NAME="E13:142_1_1_2"/><b>then </b>
( <font color="Maroon">F<sub>7</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <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="funct_2.html#K3" title="FUNCT_2:func.3">.</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> ) )
 <b>by </b><i><a class="txt" href="aofa_000.html#E2:142"><i><font color="Green" title="E91">A2</font></i></a>, <a class="txt" href="aofa_000.html#E10:142_1_1_2"><i><font color="Green" title="E97">A8</font></i></a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E14:142_1_1_2"/><b>then </b><i><font color="Green" title="E98">A9</font></i>: 
 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>]
 <b>by </b><i><a class="txt" href="aofa_000.html#E8:142_1_1_2"><i><font color="Green" title="E96">A7</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c4">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E16:142_1_1_2"/><i><font color="Green" title="E99">A10</font></i>: 
( <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> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="aofa_000.html#E14:142_1_1_2"><i><font color="Green" title="E98">A9</font></i></a>);<br/></i><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b>   <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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">$1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>;<br/><b>consider </b><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>()<b> such that </b><br/><a NAME="E18:142_1_1_2"/><i><font color="Green" title="E100">A11</font></i>: 
(  <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#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> 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>) ) )
 <b>from </b><i><a class="ref" href="finseq_2.html#S1" title="FINSEQ_2:sch.1">FINSEQ_2:sch 1</a>();<br/></i><b>reconsider </b><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>() <b>by </b><i><a class="txt" href="aofa_000.html#E18:142_1_1_2"><i><font color="Green" title="E100">A11</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/><b>take </b>
<font color="Maroon" title="c6">p</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="aofa_000.html#D33" target="_self" title="AOFA_000:def.33">AOFA_000:def 33</a><br/><a NAME="E20:142_1_1_2"/><i><font color="Green" title="E101">A12</font></i>: 
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
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E21:142_1_1_2"/><b>then </b>
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/><b>hence </b><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)
<b>by </b><i><a class="txt" href="aofa_000.html#E18:142_1_1_2"><i><font color="Green" title="E100">A11</font></i></a></i>
<br/>.= 
<font color="Maroon">F<sub>4</sub></font>()
<b>by </b><i><a class="txt" href="aofa_000.html#E3:142_1_1_2"><i><font color="Green" title="E93">A4</font></i></a>, <a class="ref" href="binarith.html#T51" title="BINARITH:th.51">BINARITH:51</a></i>
;<br/><br/><a NAME="E23:142_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>
 <b>by </b><i><a class="txt" href="aofa_000.html#E18:142_1_1_2"><i><font color="Green" title="E100">A11</font></i></a>, <a class="txt" href="aofa_000.html#E20:142_1_1_2"><i><font color="Green" title="E101">A12</font></i></a></i>;<br/><b>then </b><i><font color="Green" title="E102">A13</font></i>: <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)
<b>by </b><i><a class="txt" href="aofa_000.html#E18:142_1_1_2"><i><font color="Green" title="E100">A11</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">j</font>
<b>by </b><i><a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>
;<br/><a NAME="E25:142_1_1_2"/>
<font color="Maroon" title="c4">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="aofa_000.html#E1:142"><i><font color="Green" title="E90">A1</font></i></a>, <a class="txt" href="aofa_000.html#E1:142_1_1_2"><i><font color="Green" title="E92">A3</font></i></a>, <a class="txt" href="aofa_000.html#E3:142_1_1_2"><i><font color="Green" title="E93">A4</font></i></a>, <a class="txt" href="aofa_000.html#E16:142_1_1_2"><i><font color="Green" title="E99">A10</font></i></a></i>;<br/><a NAME="E26:142_1_1_2"/><b>then </b>
<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> 
 ;<br/><a NAME="E27:142_1_1_2"/><b>then </b>
<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
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">j'</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E29:142_1_1_2"/><i><font color="Green" title="E103">A14</font></i>: 
<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">j'</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">j'</font> = <font color="Maroon" title="c7">j'</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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E31:142_1_1_2"/><i><font color="Green" title="E104">A15</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">j'</font></span>)</span>,<font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Maroon">F<sub>2</sub></font>()
 <b>by </b><i><a class="txt" href="aofa_000.html#E4:142_1_1_2"><i><font color="Green" title="E94">A5</font></i></a>, <a class="txt" href="aofa_000.html#E29:142_1_1_2"><i><font color="Green" title="E103">A14</font></i></a></i>;<br/><a NAME="E32:142_1_1_2"/>
<font color="Maroon" title="c8">j'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">j</font>
 <b>by </b><i><a class="txt" href="aofa_000.html#E29:142_1_1_2"><i><font color="Green" title="E103">A14</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E33:142_1_1_2"/><b>then </b>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">j'</font>] &amp;  <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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">j'</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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">j'</font> )
 <b>by </b><i><a class="txt" href="aofa_000.html#E16:142_1_1_2"><i><font color="Green" title="E99">A10</font></i></a>, <a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a></i>;<br/><b>hence </b><a NAME="E34:142_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>()
 <b>by </b><i><a class="txt" href="aofa_000.html#E2:142"><i><font color="Green" title="E91">A2</font></i></a>, <a class="txt" href="aofa_000.html#E16:142_1_1_2"><i><font color="Green" title="E99">A10</font></i></a>, <a class="txt" href="aofa_000.html#E24:142_1_1_2"><i><font color="Green" title="E102">A13</font></i></a>, <a class="txt" href="aofa_000.html#E31:142_1_1_2"><i><font color="Green" title="E104">A15</font></i></a></i>;<br/><b>let </b><font color="Maroon" title="c9">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E35:142_1_1_2"/><i><font color="Green" title="E105">A16</font></i>: 
( 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> )
 ;<br/><a NAME="E36:142_1_1_2"/><b>then </b>
( <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 &amp; <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> )
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E37:142_1_1_2"/><b>then </b>
( <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> &amp; <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> )
 <b>by </b><i><a class="txt" href="aofa_000.html#E18:142_1_1_2"><i><font color="Green" title="E100">A11</font></i></a>, <a class="txt" href="aofa_000.html#E35:142_1_1_2"><i><font color="Green" title="E105">A16</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E38:142_1_1_2"/><b>then </b><i><font color="Green" title="E106">A17</font></i>: 
( <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>) &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">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) )
 <b>by </b><i><a class="txt" href="aofa_000.html#E18:142_1_1_2"><i><font color="Green" title="E100">A11</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c10">i'</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E40:142_1_1_2"/><i><font color="Green" title="E107">A18</font></i>: 
<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">i'</font>
 <b>by </b><i><a class="txt" href="aofa_000.html#E35:142_1_1_2"><i><font color="Green" title="E105">A16</font></i></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">i'</font> = <font color="Maroon" title="c10">i'</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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E42:142_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>
 <b>by </b><i><a class="txt" href="aofa_000.html#E18:142_1_1_2"><i><font color="Green" title="E100">A11</font></i></a>, <a class="txt" href="aofa_000.html#E35:142_1_1_2"><i><font color="Green" title="E105">A16</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E43:142_1_1_2"/><b>then </b><i><font color="Green" title="E108">A19</font></i>: 
<font color="Maroon" title="c11">i'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">j</font>
 <b>by </b><i><a class="txt" href="aofa_000.html#E40:142_1_1_2"><i><font color="Green" title="E107">A18</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E44:142_1_1_2"/><b>then </b><i><font color="Green" title="E109">A20</font></i>: 
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">i'</font>] &amp; <font color="Maroon" title="c9">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i'</font> &amp; <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i</font> &amp;  <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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">i'</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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">i'</font> )
 <b>by </b><i><a class="txt" href="aofa_000.html#E16:142_1_1_2"><i><font color="Green" title="E99">A10</font></i></a>, <a class="txt" href="aofa_000.html#E40:142_1_1_2"><i><font color="Green" title="E107">A18</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a>, <a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="142_1_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:142_1_1_2_3"/>
<font color="Maroon" title="c11">i'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E2:142_1_1_2_3"/><b>then </b>
<font color="Maroon" title="c11">i'</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> 
 ;<br/><a NAME="E3:142_1_1_2_3"/><b>then </b>
<font color="Maroon" title="c11">i'</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
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c12">i''</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E5:142_1_1_2_3"/><i><font color="Green" title="E110">A21</font></i>: 
<font color="Maroon" title="c11">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="c12">i''</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">i''</font> = <font color="Maroon" title="c12">i''</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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E7:142_1_1_2_3"/>
<font color="Maroon" title="c13">i''</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i'</font>
 <b>by </b><i><a class="txt" href="aofa_000.html#E5:142_1_1_2_3"><i><font color="Green" title="E110">A21</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E8:142_1_1_2_3"/><b>then </b>
<font color="Maroon" title="c13">i''</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">j</font>
 <b>by </b><i><a class="txt" href="aofa_000.html#E43:142_1_1_2"><i><font color="Green" title="E108">A19</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E9:142_1_1_2_3"/><b>then </b>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i''</font>] &amp;  <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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i''</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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i''</font> &amp; <font color="Maroon" title="c1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">i''</font>,<font color="Maroon" title="c1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i''</font>) )
 <b>by </b><i><a class="txt" href="aofa_000.html#E4:142_1_1_2"><i><font color="Green" title="E94">A5</font></i></a>, <a class="txt" href="aofa_000.html#E16:142_1_1_2"><i><font color="Green" title="E99">A10</font></i></a>, <a class="txt" href="aofa_000.html#E5:142_1_1_2_3"><i><font color="Green" title="E110">A21</font></i></a>, <a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a></i>;<br/><b>hence </b><a NAME="E10:142_1_1_2_3"/>
<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>()
 <b>by </b><i><a class="txt" href="aofa_000.html#E2:142"><i><font color="Green" title="E91">A2</font></i></a>, <a class="txt" href="aofa_000.html#E38:142_1_1_2"><i><font color="Green" title="E106">A17</font></i></a>, <a class="txt" href="aofa_000.html#E44:142_1_1_2"><i><font color="Green" title="E109">A20</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E46:142_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>() )
 <b>by </b><i><a class="txt" href="aofa_000.html#E1:142_1_1_2"><i><font color="Green" title="E92">A3</font></i></a>, <a class="txt" href="aofa_000.html#E3:142_1_1_2"><i><font color="Green" title="E93">A4</font></i></a>, <a class="txt" href="aofa_000.html#E4:142_1_1_2"><i><font color="Green" title="E94">A5</font></i></a>, <a class="txt" href="aofa_000.html#E38:142_1_1_2"><i><font color="Green" title="E106">A17</font></i></a>, <a class="txt" href="aofa_000.html#E40:142_1_1_2"><i><font color="Green" title="E107">A18</font></i></a>, <a class="txt" href="aofa_000.html#E44:142_1_1_2"><i><font color="Green" title="E109">A20</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
