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

<b>let </b><font color="Maroon" title="c1">f</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c2">r1</font>, <font color="Maroon" title="c3">r2</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>



<b>assume </b><a NAME="E1:9"/><i><font color="Green" title="E5">A1</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">r1</font>,<font color="Maroon" title="c3">r2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 ;<br/>

<a NAME="E2:9"/><b>then </b><i><font color="Green" title="E6">A2</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r1</font> &amp; <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r2</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E3:9"/><b>then </b>
 <a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <font color="Maroon" title="c1">f</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="c1">f</font>
 
<b>by </b><i><a class="ref" href="rfinseq2.html#D2" target="_self" title="RFINSEQ2:def.2">Def2</a></i>;<br/>
<a NAME="E4:9"/><b>then </b><i><font color="Green" title="E7">A3</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <font color="Maroon" title="c1">f</font> &amp;  <a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <font color="Maroon" title="c1">f</font> <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="c1">f</font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E5:9"/><i><font color="Green" title="E8">A4</font></i>: 
( <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <font color="Maroon" title="c1">f</font></span>)</span> &amp; <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <font color="Maroon" title="c1">f</font> )
 
<b>by </b><i><a class="txt" href="rfinseq2.html#E2:9"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="rfinseq2.html#T2" target="_self" title="RFINSEQ2:th.2">Th2</a></i>;<br/>
<a NAME="E6:9"/><i><font color="Green" title="E9">A5</font></i>: 
( <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <font color="Maroon" title="c1">f</font></span>)</span> &amp; <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <font color="Maroon" title="c1">f</font> )
 
<b>by </b><i><a class="txt" href="rfinseq2.html#E2:9"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="rfinseq2.html#T2" target="_self" title="RFINSEQ2:th.2">Th2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_1"/>
( <font color="Maroon" title="c2">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">r2</font> or <font color="Maroon" title="c2">r1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">r2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1"><b>suppose </b></a><a NAME="E1:9_1_1"/>
<font color="Maroon" title="c2">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">r2</font>
 ;<br/><div class="add"><a NAME="E2:9_1_1"/><b>then </b><i><font color="Green" title="E10">A6</font></i>: 
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c2">r1</font>,<font color="Maroon" title="c3">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <font color="Maroon" title="c1">f</font>
 <b>by </b><i><a class="txt" href="rfinseq2.html#E2:9"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="rfinseq2.html#E5:9"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="xxreal_0.html#D8" title="XXREAL_0:def.8">XXREAL_0:def 8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_1_1_1_1"/>
(  <a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <font color="Maroon" title="c1">f</font> <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="c1">f</font> or  <a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_1_1_1"><b>case </b></a><a NAME="E1:9_1_1_1_1_1"/>
 <a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <font color="Maroon" title="c1">f</font> <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="c1">f</font>
 ;<br/><div class="add"><a NAME="E2:9_1_1_1_1_1"/><b>then </b>
 <a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="rfinseq2.html#E1:9"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/><a NAME="E3:9_1_1_1_1_1"/><b>then </b>
 <a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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/><a NAME="E4:9_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E11">A7</font></i>: 
 <a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="rfinseq2.html#E4:9"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E5:9_1_1_1_1_1"/><b>then </b>
 <a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c2">r1</font>,<font color="Maroon" title="c3">r2</font>
 <b>by </b><i><a class="txt" href="rfinseq2.html#E2:9"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/><a NAME="E6:9_1_1_1_1_1"/><b>then </b>
 <a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c2">r1</font>,<font color="Maroon" title="c3">r2</font>
 <b>by </b><i><a class="txt" href="rfinseq2.html#E2:9_1_1"><i><font color="Green" title="E10">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E7:9_1_1_1_1_1"/>
(  <a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c2">r1</font>,<font color="Maroon" title="c3">r2</font> &amp;  <a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <font color="Maroon" title="c2">r1</font>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c2">r1</font>,<font color="Maroon" title="c3">r2</font></span>)</span>,1,2 )
 <b>by </b><i><a class="txt" href="rfinseq2.html#E2:9"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="rfinseq2.html#E4:9_1_1_1_1_1"><i><font color="Green" title="E11">A7</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_1_1_2"><b>case </b></a><a NAME="E1:9_1_1_1_1_2"/>
 <a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font>
 ;<br/><div class="add"><a NAME="E2:9_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E11">A8</font></i>: 
 <a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <b>by </b><i><a class="txt" href="rfinseq2.html#E2:9"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="rfinseq2.html#E4:9"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E3:9_1_1_1_1_2"/>
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="c1">f</font>
 <b>by </b><i><a class="txt" href="rfinseq2.html#E2:9"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:9_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E12">A9</font></i>: 
<font color="Maroon" title="c2">r1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">r2</font>
 <b>by </b><i><a class="txt" href="rfinseq2.html#E2:9"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="rfinseq2.html#E2:9_1_1_1_1_2"><i><font color="Green" title="E11">A8</font></i></a>, <a class="ref" href="rfinseq2.html#D2" target="_self" title="RFINSEQ2:def.2">Def2</a></i>;<br/><a NAME="E5:9_1_1_1_1_2"/>
 <a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c2">r1</font>,<font color="Maroon" title="c3">r2</font>
 <b>by </b><i><a class="txt" href="rfinseq2.html#E2:9"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="rfinseq2.html#E2:9_1_1_1_1_2"><i><font color="Green" title="E11">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/><a NAME="E6:9_1_1_1_1_2"/><b>then </b>
 <a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c2">r1</font>,<font color="Maroon" title="c3">r2</font>
 <b>by </b><i><a class="txt" href="rfinseq2.html#E2:9_1_1"><i><font color="Green" title="E10">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E7:9_1_1_1_1_2"/>
(  <a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c2">r1</font>,<font color="Maroon" title="c3">r2</font> &amp;  <a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <font color="Maroon" title="c2">r1</font>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c2">r1</font>,<font color="Maroon" title="c3">r2</font></span>)</span>,1,2 )
 <b>by </b><i><a class="txt" href="rfinseq2.html#E2:9"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="rfinseq2.html#E2:9_1_1_1_1_2"><i><font color="Green" title="E11">A8</font></i></a>, <a class="txt" href="rfinseq2.html#E4:9_1_1_1_1_2"><i><font color="Green" title="E12">A9</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:9_1_1"/>
(  <a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c2">r1</font>,<font color="Maroon" title="c3">r2</font> &amp;  <a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <font color="Maroon" title="c2">r1</font>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c2">r1</font>,<font color="Maroon" title="c3">r2</font></span>)</span>,1,2 )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_2"><b>suppose </b></a><a NAME="E1:9_1_2"/>
<font color="Maroon" title="c2">r1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">r2</font>
 ;<br/><div class="add"><a NAME="E2:9_1_2"/><b>then </b><i><font color="Green" title="E10">A10</font></i>: 
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c2">r1</font>,<font color="Maroon" title="c3">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <font color="Maroon" title="c1">f</font>
 <b>by </b><i><a class="txt" href="rfinseq2.html#E2:9"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="rfinseq2.html#E6:9"><i><font color="Green" title="E9">A5</font></i></a>, <a class="ref" href="xxreal_0.html#D8" title="XXREAL_0:def.8">XXREAL_0:def 8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_1_2_1_1"/>
(  <a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <font color="Maroon" title="c1">f</font> <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="c1">f</font> or  <a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_2_1_1_1"><b>case </b></a><a NAME="E1:9_1_2_1_1_1"/>
 <a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <font color="Maroon" title="c1">f</font> <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="c1">f</font>
 ;<br/><div class="add"><a NAME="E2:9_1_2_1_1_1"/><b>then </b>
 <a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="rfinseq2.html#E1:9"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/><a NAME="E3:9_1_2_1_1_1"/><b>then </b>
 <a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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/><a NAME="E4:9_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E11">A11</font></i>: 
 <a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="rfinseq2.html#E4:9"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E5:9_1_2_1_1_1"/><b>then </b>
 <a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c2">r1</font>,<font color="Maroon" title="c3">r2</font>
 <b>by </b><i><a class="txt" href="rfinseq2.html#E2:9"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/><a NAME="E6:9_1_2_1_1_1"/><b>then </b>
 <a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c2">r1</font>,<font color="Maroon" title="c3">r2</font>
 <b>by </b><i><a class="txt" href="rfinseq2.html#E2:9_1_2"><i><font color="Green" title="E10">A10</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E7:9_1_2_1_1_1"/>
(  <a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c2">r1</font>,<font color="Maroon" title="c3">r2</font> &amp;  <a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <font color="Maroon" title="c2">r1</font>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c2">r1</font>,<font color="Maroon" title="c3">r2</font></span>)</span>,1,2 )
 <b>by </b><i><a class="txt" href="rfinseq2.html#E2:9"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="rfinseq2.html#E4:9_1_2_1_1_1"><i><font color="Green" title="E11">A11</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_2_1_1_2"><b>case </b></a><a NAME="E1:9_1_2_1_1_2"/>
 <a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font>
 ;<br/><div class="add"><a NAME="E2:9_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E11">A12</font></i>: 
 <a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <b>by </b><i><a class="txt" href="rfinseq2.html#E2:9"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="rfinseq2.html#E4:9"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E3:9_1_2_1_1_2"/>
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="c1">f</font>
 <b>by </b><i><a class="txt" href="rfinseq2.html#E2:9"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:9_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E12">A13</font></i>: 
<font color="Maroon" title="c2">r1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">r2</font>
 <b>by </b><i><a class="txt" href="rfinseq2.html#E2:9"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="rfinseq2.html#E2:9_1_2_1_1_2"><i><font color="Green" title="E11">A12</font></i></a>, <a class="ref" href="rfinseq2.html#D2" target="_self" title="RFINSEQ2:def.2">Def2</a></i>;<br/><a NAME="E5:9_1_2_1_1_2"/>
 <a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c2">r1</font>,<font color="Maroon" title="c3">r2</font>
 <b>by </b><i><a class="txt" href="rfinseq2.html#E2:9"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="rfinseq2.html#E2:9_1_2_1_1_2"><i><font color="Green" title="E11">A12</font></i></a>, <a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/><a NAME="E6:9_1_2_1_1_2"/><b>then </b>
 <a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c2">r1</font>,<font color="Maroon" title="c3">r2</font>
 <b>by </b><i><a class="txt" href="rfinseq2.html#E2:9_1_2"><i><font color="Green" title="E10">A10</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E7:9_1_2_1_1_2"/>
(  <a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c2">r1</font>,<font color="Maroon" title="c3">r2</font> &amp;  <a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <font color="Maroon" title="c2">r1</font>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c2">r1</font>,<font color="Maroon" title="c3">r2</font></span>)</span>,1,2 )
 <b>by </b><i><a class="txt" href="rfinseq2.html#E2:9"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="rfinseq2.html#E2:9_1_2_1_1_2"><i><font color="Green" title="E11">A12</font></i></a>, <a class="txt" href="rfinseq2.html#E4:9_1_2_1_1_2"><i><font color="Green" title="E12">A13</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:9_1_2"/>
(  <a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c2">r1</font>,<font color="Maroon" title="c3">r2</font> &amp;  <a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <font color="Maroon" title="c2">r1</font>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c2">r1</font>,<font color="Maroon" title="c3">r2</font></span>)</span>,1,2 )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
