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<span class="kw">let </span><font color="Maroon" title="c1">f</font> be   <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</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">  for <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <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="Olive" title="b1">r1</font>,<font color="Olive" title="b2">r2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> holds <br/>(  <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="Olive" title="b1">r1</font>,<font color="Olive" title="b2">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#K16" title="FUNCOP_1:func.16">IFEQ</a> (<font color="Olive" title="b1">r1</font>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Olive" title="b1">r1</font>,<font color="Olive" title="b2">r2</font>)</span>)</span>,1,2) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">r1</font>, <font color="Maroon" title="c3">r2</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</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="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> implies (  <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#K16" title="FUNCOP_1:func.16">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) ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:9"/><span class="lab"><font color="Green" title="E5">A1</font></span>: 
<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>
 ; <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="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#K16" title="FUNCOP_1:func.16">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) )</span><br/>

<a NAME="E2:9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E6">A2</font></span>: 
 <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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" onmouseover="rs('finseq_1/T44')" onmouseout="rh()">FINSEQ_1:44</a></span>;<br/>
<a NAME="E3:9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A3</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfinseq2.html#T2" target="_self" onmouseover="rs('rfinseq2/T2')" onmouseout="rh()">Th2</a></span>;<br/>
<a NAME="E4:9"/><span class="lab"><font color="Green" title="E8">A4</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E5">A1</font></span></a></span>;<br/>
<a NAME="E5:9"/><span class="lab"><font color="Green" title="E9">A5</font></span>: 
 <a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <font color="Maroon" title="c1">f</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">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="rfinseq2.html#D2" target="_self" onmouseover="rs('rfinseq2/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<a NAME="E6:9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A6</font></span>: 
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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E7:9"/><span class="lab"><font color="Green" title="E11">A7</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="rfinseq2.html#T2" target="_self" onmouseover="rs('rfinseq2/T2')" onmouseout="rh()">Th2</a></span>;<br/>
<a NAME="E8:9"/><span class="lab"><font color="Green" title="E12">A8</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E5">A1</font></span></a></span>;<br/>
<a NAME="E9:9"/><span class="lab"><font color="Green" title="E13">A9</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_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="E1:9_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"><span class="kw">suppose </span></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>
 ; <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="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#K16" title="FUNCOP_1:func.16">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) )</span><br/><div class="add"><a NAME="E2:9_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A10</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:9"><span class="lab"><font color="Green" title="E12">A8</font></span></a>, <a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="ref" href="xxreal_0.html#D9" onmouseover="rs('xxreal_0/D9')" onmouseout="rh()">XXREAL_0:def 9</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="9_1_1_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"> ( (  <a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> &amp;  <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#K16" title="FUNCOP_1:func.16">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) ) 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> &amp;  <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#K16" title="FUNCOP_1:func.16">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) ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_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="E1:9_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#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="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"><span class="kw">case </span></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#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</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"> (  <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#K16" title="FUNCOP_1:func.16">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) )</span><br/><div class="add"><a NAME="E2:9_1_1_1_1_1"/><span class="kw">then </span>
 <a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="finseq_1.html#T44" onmouseover="rs('finseq_1/T44')" onmouseout="rh()">FINSEQ_1:44</a></span>;<br/><a NAME="E3:9_1_1_1_1_1"/><span class="kw">then </span>
 <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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E4:9_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A11</font></span>: 
 <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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:9"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E5:9_1_1_1_1_1"/><span class="kw">then </span>
 <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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:9"><span class="lab"><font color="Green" title="E12">A8</font></span></a>, <a class="ref" href="xxreal_0.html#T17" onmouseover="rs('xxreal_0/T17')" onmouseout="rh()">XXREAL_0:17</a></span>;<br/><a NAME="E6:9_1_1_1_1_1"/><span class="kw">then </span>
 <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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_1_1"><span class="lab"><font color="Green" title="E14">A10</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><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#K16" title="FUNCOP_1:func.16">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) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:9"><span class="lab"><font color="Green" title="E12">A8</font></span></a>, <a class="txt" href="#E4:9_1_1_1_1_1"><span class="lab"><font color="Green" title="E15">A11</font></span></a>, <a class="ref" href="funcop_1.html#D8" onmouseover="rs('funcop_1/D8')" onmouseout="rh()">FUNCOP_1:def 8</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="9_1_1_1_1_2"><span class="kw">case </span></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>
 ; <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="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#K16" title="FUNCOP_1:func.16">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) )</span><br/><div class="add"><a NAME="E2:9_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A12</font></span>: 
 <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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="txt" href="#E9:9"><span class="lab"><font color="Green" title="E13">A9</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E3:9_1_1_1_1_2"/><span class="kw">then </span>
 <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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="ref" href="xxreal_0.html#T17" onmouseover="rs('xxreal_0/T17')" onmouseout="rh()">XXREAL_0:17</a></span>;<br/><a NAME="E4:9_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A13</font></span>: 
 <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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_1_1"><span class="lab"><font color="Green" title="E14">A10</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E5:9_1_1_1_1_2"/>
1 <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">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E6:9_1_1_1_1_2"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="txt" href="#E8:9"><span class="lab"><font color="Green" title="E12">A8</font></span></a>, <a class="txt" href="#E4:9"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="txt" href="#E2:9_1_1_1_1_2"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="ref" href="rfinseq2.html#D2" target="_self" onmouseover="rs('rfinseq2/D2')" onmouseout="rh()">Def2</a></span>;<br/><span class="kw">hence </span><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#K16" title="FUNCOP_1:func.16">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) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="txt" href="#E2:9_1_1_1_1_2"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="txt" href="#E4:9_1_1_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="ref" href="funcop_1.html#D8" onmouseover="rs('funcop_1/D8')" onmouseout="rh()">FUNCOP_1:def 8</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="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#K16" title="FUNCOP_1:func.16">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) )
 ; <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="9_1_2"><span class="kw">suppose </span></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>
 ; <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="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#K16" title="FUNCOP_1:func.16">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) )</span><br/><div class="add"><a NAME="E2:9_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A14</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="txt" href="#E7:9"><span class="lab"><font color="Green" title="E11">A7</font></span></a>, <a class="ref" href="xxreal_0.html#D9" onmouseover="rs('xxreal_0/D9')" onmouseout="rh()">XXREAL_0:def 9</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="9_1_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"> ( (  <a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> &amp;  <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#K16" title="FUNCOP_1:func.16">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) ) 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> &amp;  <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#K16" title="FUNCOP_1:func.16">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) ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_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="E1:9_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#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="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"><span class="kw">case </span></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#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</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"> (  <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#K16" title="FUNCOP_1:func.16">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) )</span><br/><div class="add"><a NAME="E2:9_1_2_1_1_1"/><span class="kw">then </span>
 <a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="finseq_1.html#T44" onmouseover="rs('finseq_1/T44')" onmouseout="rh()">FINSEQ_1:44</a></span>;<br/><a NAME="E3:9_1_2_1_1_1"/><span class="kw">then </span>
 <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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E4:9_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A15</font></span>: 
 <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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:9"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E5:9_1_2_1_1_1"/><span class="kw">then </span>
 <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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:9"><span class="lab"><font color="Green" title="E12">A8</font></span></a>, <a class="ref" href="xxreal_0.html#T17" onmouseover="rs('xxreal_0/T17')" onmouseout="rh()">XXREAL_0:17</a></span>;<br/><a NAME="E6:9_1_2_1_1_1"/><span class="kw">then </span>
 <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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_1_2"><span class="lab"><font color="Green" title="E14">A14</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><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#K16" title="FUNCOP_1:func.16">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) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:9"><span class="lab"><font color="Green" title="E12">A8</font></span></a>, <a class="txt" href="#E4:9_1_2_1_1_1"><span class="lab"><font color="Green" title="E15">A15</font></span></a>, <a class="ref" href="funcop_1.html#D8" onmouseover="rs('funcop_1/D8')" onmouseout="rh()">FUNCOP_1:def 8</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="9_1_2_1_1_2"><span class="kw">case </span></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>
 ; <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="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#K16" title="FUNCOP_1:func.16">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) )</span><br/><div class="add"><a NAME="E2:9_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A16</font></span>: 
 <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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="txt" href="#E9:9"><span class="lab"><font color="Green" title="E13">A9</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E3:9_1_2_1_1_2"/><span class="kw">then </span>
 <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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="ref" href="xxreal_0.html#T17" onmouseover="rs('xxreal_0/T17')" onmouseout="rh()">XXREAL_0:17</a></span>;<br/><a NAME="E4:9_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A17</font></span>: 
 <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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_1_2"><span class="lab"><font color="Green" title="E14">A14</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E5:9_1_2_1_1_2"/>
1 <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">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E6:9_1_2_1_1_2"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="txt" href="#E8:9"><span class="lab"><font color="Green" title="E12">A8</font></span></a>, <a class="txt" href="#E4:9"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="txt" href="#E2:9_1_2_1_1_2"><span class="lab"><font color="Green" title="E15">A16</font></span></a>, <a class="ref" href="rfinseq2.html#D2" target="_self" onmouseover="rs('rfinseq2/D2')" onmouseout="rh()">Def2</a></span>;<br/><span class="kw">hence </span><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#K16" title="FUNCOP_1:func.16">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) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="txt" href="#E2:9_1_2_1_1_2"><span class="lab"><font color="Green" title="E15">A16</font></span></a>, <a class="txt" href="#E4:9_1_2_1_1_2"><span class="lab"><font color="Green" title="E16">A17</font></span></a>, <a class="ref" href="funcop_1.html#D8" onmouseover="rs('funcop_1/D8')" onmouseout="rh()">FUNCOP_1:def 8</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="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#K16" title="FUNCOP_1:func.16">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) )
 ; <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>
