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

<span class="kw">let </span><font color="Maroon" title="c2">z1</font>, <font color="Maroon" title="c3">z2</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">m</font> &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">m</font> holds <br/><font color="Maroon" title="c2">z1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">IFEQ</a> (<font color="Olive" title="b1">i</font>,2,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="funcop_1.html#K16" title="FUNCOP_1:func.16">IFEQ</a> (<font color="Olive" title="b1">i</font>,3,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>)</span>)</span>) ) &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">m</font> &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">m</font> holds <br/><font color="Maroon" title="c3">z2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">IFEQ</a> (<font color="Olive" title="b1">i</font>,2,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="funcop_1.html#K16" title="FUNCOP_1:func.16">IFEQ</a> (<font color="Olive" title="b1">i</font>,3,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>)</span>)</span>) ) implies <font color="Maroon" title="c2">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:35_1_2"/><span class="lab"><font color="Green" title="E29">A14</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">m</font>
 <span class="kw">and </span><br/><a NAME="E2:35_1_2"/><span class="lab"><font color="Green" title="E30">A15</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">m</font> holds <br/><font color="Maroon" title="c2">z1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">IFEQ</a> (<font color="Olive" title="b1">i</font>,2,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="funcop_1.html#K16" title="FUNCOP_1:func.16">IFEQ</a> (<font color="Olive" title="b1">i</font>,3,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>)</span>)</span>)
 <span class="kw">and </span><br/><a NAME="E3:35_1_2"/><span class="lab"><font color="Green" title="E31">A16</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">m</font>
 <span class="kw">and </span><br/><a NAME="E4:35_1_2"/><span class="lab"><font color="Green" title="E32">A17</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">m</font> holds <br/><font color="Maroon" title="c3">z2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">IFEQ</a> (<font color="Olive" title="b1">i</font>,2,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="funcop_1.html#K16" title="FUNCOP_1:func.16">IFEQ</a> (<font color="Olive" title="b1">i</font>,3,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>)</span>)</span>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z2</font></span><br/>

<span class="lab"><font color="Green" title="E33">A18</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">m</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">z2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_2"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">z2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>
;<br/>
<div><span class="lab"><font color="Green" title="E34">A19</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="35_1_2_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">m</font> holds <br/><font color="Maroon" title="c2">z1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">j</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">j</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">m</font> implies <font color="Maroon" title="c2">z1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">j</font> )</span><br/><span class="kw">assume </span><a NAME="E1:35_1_2_2"/><span class="lab"><font color="Green" title="E35">A20</font></span>: 
<font color="Maroon" title="c4">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">m</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">z1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">j</font></span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="35_1_2_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">z1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">j</font></span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_2_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_2_2_1_1"/>
( <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or ( <font color="Maroon" title="c4">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2 &amp; <font color="Maroon" title="c4">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 3 ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_2_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_2_2_1_1_1"/>
<font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">z1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">j</font></span><br/><div class="add"><a NAME="E2:35_1_2_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A21</font></span>: 
 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">IFEQ</a> (<font color="Maroon" title="c4">j</font>,2,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="funcop_1.html#K16" title="FUNCOP_1:func.16">IFEQ</a> (<font color="Maroon" title="c4">j</font>,3,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">j</font></span>)</span>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#D8" onmouseover="rs('funcop_1/D8')" onmouseout="rh()">FUNCOP_1:def 8</a></span>;<br/><a NAME="E3:35_1_2_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">z2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_2"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="txt" href="#E1:35_1_2_2"><span class="lab"><font color="Green" title="E35">A20</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:35_1_2_2_1_1_1"/>
<font color="Maroon" title="c2">z1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_2"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="txt" href="#E1:35_1_2_2"><span class="lab"><font color="Green" title="E35">A20</font></span></a>, <a class="txt" href="#E2:35_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E36">A21</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="35_1_2_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_2_2_1_1_2"/><span class="lab"><font color="Green" title="E36">A22</font></span>: 
<font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">z1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">j</font></span><br/><div class="add"><a NAME="E2:35_1_2_2_1_1_2"/><span class="lab"><font color="Green" title="E37">A23</font></span>: 
( <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 implies  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">IFEQ</a> (<font color="Maroon" title="c4">j</font>,2,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="funcop_1.html#K16" title="FUNCOP_1:func.16">IFEQ</a> (<font color="Maroon" title="c4">j</font>,3,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">j</font></span>)</span>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2 )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="35_1_2_2_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:35_1_2_2_1_1_2_1"/><span class="lab"><font color="Green" title="E38">A24</font></span>: 
<font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">IFEQ</a> (<font color="Maroon" title="c4">j</font>,2,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="funcop_1.html#K16" title="FUNCOP_1:func.16">IFEQ</a> (<font color="Maroon" title="c4">j</font>,3,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">j</font></span>)</span>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span><br/>

<span class="kw">then </span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">IFEQ</a> (<font color="Maroon" title="c4">j</font>,2,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="funcop_1.html#K16" title="FUNCOP_1:func.16">IFEQ</a> (<font color="Maroon" title="c4">j</font>,3,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">j</font></span>)</span>)</span>)</span>) = 
 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">IFEQ</a> (<font color="Maroon" title="c4">j</font>,3,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">j</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#D8" onmouseover="rs('funcop_1/D8')" onmouseout="rh()">FUNCOP_1:def 8</a></span>
<br/>.= 
<font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_2_2_1_1_2_1"><span class="lab"><font color="Green" title="E38">A24</font></span></a>, <a class="ref" href="funcop_1.html#D8" onmouseover="rs('funcop_1/D8')" onmouseout="rh()">FUNCOP_1:def 8</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E3:35_1_2_2_1_1_2_1"/>
 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">IFEQ</a> (<font color="Maroon" title="c4">j</font>,2,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="funcop_1.html#K16" title="FUNCOP_1:func.16">IFEQ</a> (<font color="Maroon" title="c4">j</font>,3,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">j</font></span>)</span>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><a NAME="E3:35_1_2_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">z2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_2"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="txt" href="#E1:35_1_2_2"><span class="lab"><font color="Green" title="E35">A20</font></span></a>, <a class="txt" href="#E1:35_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E36">A22</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:35_1_2_2_1_1_2"/>
<font color="Maroon" title="c2">z1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_2"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="txt" href="#E1:35_1_2_2"><span class="lab"><font color="Green" title="E35">A20</font></span></a>, <a class="txt" href="#E1:35_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E36">A22</font></span></a>, <a class="txt" href="#E2:35_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E37">A23</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="35_1_2_2_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:35_1_2_2_1_1_3"/><span class="lab"><font color="Green" title="E36">A25</font></span>: 
( <font color="Maroon" title="c4">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2 &amp; <font color="Maroon" title="c4">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 3 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">z1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">j</font></span><br/><div class="add"><a NAME="E2:35_1_2_2_1_1_3"/><span class="lab"><font color="Green" title="E37">A26</font></span>: 
( <font color="Maroon" title="c4">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2 &amp; <font color="Maroon" title="c4">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 3 implies  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">IFEQ</a> (<font color="Maroon" title="c4">j</font>,2,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="funcop_1.html#K16" title="FUNCOP_1:func.16">IFEQ</a> (<font color="Maroon" title="c4">j</font>,3,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">j</font></span>)</span>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">j</font> )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="35_1_2_2_1_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:35_1_2_2_1_1_3_1"/><span class="lab"><font color="Green" title="E38">A27</font></span>: 
<font color="Maroon" title="c4">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2
 <span class="kw">and </span><br/><a NAME="E2:35_1_2_2_1_1_3_1"/><span class="lab"><font color="Green" title="E39">A28</font></span>: 
<font color="Maroon" title="c4">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 3
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">IFEQ</a> (<font color="Maroon" title="c4">j</font>,2,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="funcop_1.html#K16" title="FUNCOP_1:func.16">IFEQ</a> (<font color="Maroon" title="c4">j</font>,3,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">j</font></span>)</span>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">j</font></span><br/>

 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">IFEQ</a> (<font color="Maroon" title="c4">j</font>,2,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="funcop_1.html#K16" title="FUNCOP_1:func.16">IFEQ</a> (<font color="Maroon" title="c4">j</font>,3,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">j</font></span>)</span>)</span>)</span>) = 
 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">IFEQ</a> (<font color="Maroon" title="c4">j</font>,3,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">j</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_2_2_1_1_3_1"><span class="lab"><font color="Green" title="E38">A27</font></span></a>, <a class="ref" href="funcop_1.html#D8" onmouseover="rs('funcop_1/D8')" onmouseout="rh()">FUNCOP_1:def 8</a></span>
<br/>.= 
<font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_2_2_1_1_3_1"><span class="lab"><font color="Green" title="E39">A28</font></span></a>, <a class="ref" href="funcop_1.html#D8" onmouseover="rs('funcop_1/D8')" onmouseout="rh()">FUNCOP_1:def 8</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:35_1_2_2_1_1_3_1"/>
 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">IFEQ</a> (<font color="Maroon" title="c4">j</font>,2,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="funcop_1.html#K16" title="FUNCOP_1:func.16">IFEQ</a> (<font color="Maroon" title="c4">j</font>,3,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">j</font></span>)</span>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">j</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><a NAME="E3:35_1_2_2_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c3">z2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_2"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="txt" href="#E1:35_1_2_2"><span class="lab"><font color="Green" title="E35">A20</font></span></a>, <a class="txt" href="#E1:35_1_2_2_1_1_3"><span class="lab"><font color="Green" title="E36">A25</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:35_1_2_2_1_1_3"/>
<font color="Maroon" title="c2">z1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_2"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="txt" href="#E1:35_1_2_2"><span class="lab"><font color="Green" title="E35">A20</font></span></a>, <a class="txt" href="#E1:35_1_2_2_1_1_3"><span class="lab"><font color="Green" title="E36">A25</font></span></a>, <a class="txt" href="#E2:35_1_2_2_1_1_3"><span class="lab"><font color="Green" title="E37">A26</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_2_2"/>
<font color="Maroon" title="c2">z1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">j</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">m</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">z1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_2"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">z1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E8:35_1_2"/>
<font color="Maroon" title="c2">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_2"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="txt" href="#E6:35_1_2"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="ref" href="finseq_1.html#T13" onmouseover="rs('finseq_1/T13')" onmouseout="rh()">FINSEQ_1:13</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
