<?xml version="1.0"?>
<div class="add">

<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> ) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> $2</span>)</span></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> $2</span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">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>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a><span class="kw"> such that </span><br/><a NAME="E2:5_1_1"/><span class="lab"><font color="Green" title="E3">A1</font></span>: 
<font color="Maroon" title="c4">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>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <span class="kw">and </span><br/><a NAME="E3:5_1_1"/><span class="lab"><font color="Green" title="E4">A2</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="c4">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="c4">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">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="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">$1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">z</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E5:5_1_1"/><span class="lab"><font color="Green" title="E5">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E6:5_1_1"/><span class="lab"><font color="Green" title="E6">A4</font></span>: 
 for <font color="Olive" title="b1">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">j</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">z</font> holds <br/><font color="Maroon" title="c5">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">j</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_1.html#S2" title="FINSEQ_1:sch.2">FINSEQ_1:sch 2</a>();<br/></span>
<a NAME="E7:5_1_1"/>
<font color="Maroon" title="c5">z</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a>,<a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">z</font> or <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:5_1_1_1"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">z</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">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:5_1_1_1"/><span class="lab"><font color="Green" title="E7">A5</font></span>: 
<font color="Maroon" title="c7">b</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">z</font>
 <span class="kw">and </span><br/><a NAME="E4:5_1_1_1"/><span class="lab"><font color="Green" title="E8">A6</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E5:5_1_1_1"/><span class="lab"><font color="Green" title="E9">A7</font></span>: 
<font color="Maroon" title="c7">b</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> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:5_1_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E3:5_1_1_1"><span class="lab"><font color="Green" title="E7">A5</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="c8">b</font> = <font color="Maroon" title="c7">b</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5_1_1_1"><span class="lab"><font color="Green" title="E7">A5</font></span></a></span>;<br/>
<a NAME="E7:5_1_1_1"/>
<font color="Maroon" title="c8">b</font> <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="#E5:5_1_1_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">c</font> = <font color="Maroon" title="c8">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T5" title="INT_1:th.5">INT_1:5</a></span>;<br/>
<a NAME="E9:5_1_1_1"/>
<font color="Maroon" title="c5">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:5_1_1"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="txt" href="#E3:5_1_1_1"><span class="lab"><font color="Green" title="E7">A5</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:5_1_1_1"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:5_1_1_1"><span class="lab"><font color="Green" title="E8">A6</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>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">z</font> = <font color="Maroon" title="c5">z</font> as   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c1">n</font>, <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:5_1_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c6">z</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">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  &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> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon" title="c6">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> ) )</span><br/>

<a NAME="E9:5_1_1"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<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/>
<a NAME="E10:5_1_1"/><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> <font color="Maroon" title="c1">n</font>
 
;<br/>
<a NAME="E11:5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A8</font></span>: 
1 <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="c6">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:5_1_1"><span class="lab"><font color="Green" title="E5">A3</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">hence </span><font color="Maroon" title="c6">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<font color="Maroon" title="c6">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:5_1_1"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="txt" href="#E11:5_1_1"><span class="lab"><font color="Green" title="E7">A8</font></span></a></span>
<br/>.= 
 <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:5_1_1"><span class="lab"><font color="Green" title="E3">A1</font></span></a></span>
;<br/> <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> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon" title="c6">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c7">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="c7">i</font> &amp; <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c6">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i</font></span>)</span></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E13:5_1_1"/><span class="lab"><font color="Green" title="E8">A9</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font>
 <span class="kw">and </span><br/><a NAME="E14:5_1_1"/><span class="lab"><font color="Green" title="E9">A10</font></span>: 
<font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</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">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i</font></span>)</span></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i</font></span>)</span></span>)</span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c8">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="E16:5_1_1"/><span class="lab"><font color="Green" title="E10">A11</font></span>: 
<font color="Maroon" title="c8">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:5_1_1"><span class="lab"><font color="Green" title="E8">A9</font></span></a>, <a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></span>;<br/>
<a NAME="E17:5_1_1"/>
<font color="Maroon" title="c8">j</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> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:5_1_1"><span class="lab"><font color="Green" title="E8">A9</font></span></a>, <a class="txt" href="#E14:5_1_1"><span class="lab"><font color="Green" title="E9">A10</font></span></a>, <a class="txt" href="#E16:5_1_1"><span class="lab"><font color="Green" title="E10">A11</font></span></a></span>;<br/>
<a NAME="E18:5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A12</font></span>: 
<font color="Maroon" title="c8">j</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:5_1_1"><span class="lab"><font color="Green" title="E5">A3</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">then </span><span class="lab"><font color="Green" title="E12">A13</font></span>: <font color="Maroon" title="c6">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<font color="Maroon" title="c6">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">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="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:5_1_1"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="txt" href="#E18:5_1_1"><span class="lab"><font color="Green" title="E11">A12</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">j</font>

;<br/>
<a NAME="E20:5_1_1"/><span class="lab"><font color="Green" title="E13">A14</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i</font>
 
;<br/>
<a NAME="E21:5_1_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c7">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> <font color="Maroon" title="c1">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:5_1_1"><span class="lab"><font color="Green" title="E8">A9</font></span></a>, <a class="txt" href="#E14:5_1_1"><span class="lab"><font color="Green" title="E9">A10</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E22:5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A15</font></span>: 
<font color="Maroon" title="c7">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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:5_1_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">hence </span><font color="Maroon" title="c6">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<font color="Maroon" title="c6">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</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="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">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="#E6:5_1_1"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="txt" href="#E16:5_1_1"><span class="lab"><font color="Green" title="E10">A11</font></span></a>, <a class="txt" href="#E20:5_1_1"><span class="lab"><font color="Green" title="E13">A14</font></span></a>, <a class="txt" href="#E22:5_1_1"><span class="lab"><font color="Green" title="E14">A15</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i</font></span>)</span></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5_1_1"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E16:5_1_1"><span class="lab"><font color="Green" title="E10">A11</font></span></a>, <a class="txt" href="#E19:5_1_1"><span class="lab"><font color="Green" title="E12">A13</font></span></a></span>
;<br/> <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>
