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<b>let </b><font color="Maroon" title="c1">F</font>, <font color="Maroon" title="c2">F1</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ;<br/><b>let </b><font color="Maroon" title="c3">k</font>, <font color="Maroon" title="c4">ma</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:9"/><i><font color="Green" title="E9">A1</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">F</font>
 <b>and </b><br/><a NAME="E2:9"/><i><font color="Green" title="E10">A2</font></i>: 
<font color="Maroon" title="c4">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sfmastr3.html#K1" title="SFMASTR3:func.1">min_at</a> <font color="Maroon" title="c1">F</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">F</font></span>)</span>
 <b>and </b><br/><a NAME="E3:9"/><i><font color="Green" title="E11">A3</font></i>: 
<font color="Maroon" title="c1">F</font> <a href="sfmastr3.html#R2" title="SFMASTR3:pred.2">is_split_at</a> <font color="Maroon" title="c3">k</font>
 <b>and </b><br/><a NAME="E4:9"/><i><font color="Green" title="E12">A4</font></i>: 
<font color="Maroon" title="c1">F</font> <a href="sfmastr3.html#R1" title="SFMASTR3:pred.1">is_non_decreasing_on</a> 1,<font color="Maroon" title="c3">k</font>
 <b>and </b><br/><a NAME="E5:9"/><i><font color="Green" title="E13">A5</font></i>: 
<font color="Maroon" title="c2">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">ma</font></span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c4">ma</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon" title="c5">Fk</font> = <font color="Maroon" title="c1">F</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">ma</font></span>)</span>;<br/>
<a NAME="E6:9"/><i><font color="Green" title="E14">A6</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">ma</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sfmastr3.html#E5:9"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="funct_7.html#T32" title="FUNCT_7:th.32">FUNCT_7:32</a></i>;<br/>
<a NAME="E7:9"/><i><font color="Green" title="E15">A7</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">ma</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="funct_7.html#T32" title="FUNCT_7:th.32">FUNCT_7:32</a></i>;<br/>
<b>let </b><font color="Maroon" title="c6">i</font>, <font color="Maroon" title="c7">j</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="sfmastr3.html#D4" target="_self" title="SFMASTR3:def.4">SFMASTR3:def 4</a><br/>

<b>assume </b><a NAME="E8:9"/><i><font color="Green" title="E16">A8</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/>

<a NAME="E9:9"/><i><font color="Green" title="E17">A9</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E10:9"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c7">j</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="txt" href="sfmastr3.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="sfmastr3.html#E8:9"><i><font color="Green" title="E16">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:9"/><b>then </b><i><font color="Green" title="E18">A10</font></i>: 
<font color="Maroon" title="c7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F1</font>
 
<b>by </b><i><a class="txt" href="sfmastr3.html#E6:9"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="sfmastr3.html#E7:9"><i><font color="Green" title="E15">A7</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E12:9"/><i><font color="Green" title="E19">A11</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F1</font>
 
<b>by </b><i><a class="txt" href="sfmastr3.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="sfmastr3.html#E6:9"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="sfmastr3.html#E7:9"><i><font color="Green" title="E15">A7</font></i></a>, <a class="txt" href="sfmastr3.html#E9:9"><i><font color="Green" title="E17">A9</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</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="c7">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="sfmastr3.html#E8:9"><i><font color="Green" title="E16">A8</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<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"/><i><font color="Green" title="E20">A12</font></i>: 
<font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:9_1_1"/><b>then </b><i><font color="Green" title="E21">A13</font></i>: 
<font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:9_1_1"/>
<font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="sfmastr3.html#E8:9"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="sfmastr3.html#E1:9_1_1"><i><font color="Green" title="E20">A12</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E4:9_1_1"/><b>then </b><i><font color="Green" title="E22">A14</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">ma</font>
 <b>by </b><i><a class="txt" href="sfmastr3.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="sfmastr3.html#E2:9"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="sfmastr3.html#E9:9"><i><font color="Green" title="E17">A9</font></i></a>, <a class="ref" href="sfmastr3.html#T3" target="_self" title="SFMASTR3:th.3">Th3</a></i>;<br/><a NAME="E5:9_1_1"/><i><font color="Green" title="E23">A15</font></i>: 
<font color="Maroon" title="c7">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">ma</font>
 <b>by </b><i><a class="txt" href="sfmastr3.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="sfmastr3.html#E2:9"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="sfmastr3.html#E9:9"><i><font color="Green" title="E17">A9</font></i></a>, <a class="txt" href="sfmastr3.html#E1:9_1_1"><i><font color="Green" title="E20">A12</font></i></a>, <a class="ref" href="sfmastr3.html#T3" target="_self" title="SFMASTR3:th.3">Th3</a></i>;<br/><i><font color="Green" title="E24">A16</font></i>: <font color="Maroon" title="c2">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">ma</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
<b>by </b><i><a class="txt" href="sfmastr3.html#E5:9"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="sfmastr3.html#E4:9_1_1"><i><font color="Green" title="E22">A14</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>
<br/>.= 
<font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
<b>by </b><i><a class="txt" href="sfmastr3.html#E8:9"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="sfmastr3.html#E1:9_1_1"><i><font color="Green" title="E20">A12</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>
;<br/><font color="Maroon" title="c2">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">ma</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
<b>by </b><i><a class="txt" href="sfmastr3.html#E5:9"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="sfmastr3.html#E5:9_1_1"><i><font color="Green" title="E23">A15</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>
<br/>.= 
<font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
<b>by </b><i><a class="txt" href="sfmastr3.html#E1:9_1_1"><i><font color="Green" title="E20">A12</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>
;<br/><b>hence </b><a NAME="E8:9_1_1"/>
<font color="Maroon" title="c2">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
 <b>by </b><i><a class="txt" href="sfmastr3.html#E4:9"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="sfmastr3.html#E8:9"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="sfmastr3.html#E2:9_1_1"><i><font color="Green" title="E21">A13</font></i></a>, <a class="txt" href="sfmastr3.html#E6:9_1_1"><i><font color="Green" title="E24">A16</font></i></a>, <a class="ref" href="sfmastr3.html#D4" target="_self" title="SFMASTR3:def.4">Def4</a></i>;<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"/><i><font color="Green" title="E20">A17</font></i>: 
<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><b>thus </b><a NAME="E2:9_1_2"/>
<font color="Maroon" title="c2">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1"><b>proof </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"/>
( <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> or <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font> )
 </span><b>by </b><i><a class="txt" href="sfmastr3.html#E8:9"><i><font color="Green" title="E16">A8</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:9_1_2_1_1_1"/><i><font color="Green" title="E21">A18</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font>
 ;<br/><div class="add"><a NAME="E2:9_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">ma</font>
 <b>by </b><i><a class="txt" href="sfmastr3.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="sfmastr3.html#E2:9"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="sfmastr3.html#E9:9"><i><font color="Green" title="E17">A9</font></i></a>, <a class="txt" href="sfmastr3.html#E1:9_1_2"><i><font color="Green" title="E20">A17</font></i></a>, <a class="ref" href="sfmastr3.html#T3" target="_self" title="SFMASTR3:th.3">Th3</a></i>;<br/><b>then </b><i><font color="Green" title="E22">A19</font></i>: <font color="Maroon" title="c2">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">ma</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
<b>by </b><i><a class="txt" href="sfmastr3.html#E5:9"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>
<br/>.= 
<font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
<b>by </b><i><a class="txt" href="sfmastr3.html#E1:9_1_2"><i><font color="Green" title="E20">A17</font></i></a>, <a class="txt" href="sfmastr3.html#E1:9_1_2_1_1_1"><i><font color="Green" title="E21">A18</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>
;<br/><a NAME="E4:9_1_2_1_1_1"/><i><font color="Green" title="E23">A20</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="sfmastr3.html#E1:9_1_2"><i><font color="Green" title="E20">A17</font></i></a>, <a class="txt" href="sfmastr3.html#E1:9_1_2_1_1_1"><i><font color="Green" title="E21">A18</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E5:9_1_2_1_1_1"/><i><font color="Green" title="E24">A21</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font>
 <b>by </b><i><a class="txt" href="sfmastr3.html#E1:9_1_2"><i><font color="Green" title="E20">A17</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>thus </b><a NAME="E6:9_1_2_1_1_1"/>
<font color="Maroon" title="c2">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1_1_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_2_1_1_1_2_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_1_2_1"/>
( <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">ma</font> or <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">ma</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_2_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:9_1_2_1_1_1_2_1_1"/>
<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">ma</font>
 ;<br/><div class="add"><a NAME="E2:9_1_2_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c2">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="sfmastr3.html#E5:9"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="sfmastr3.html#E6:9"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="sfmastr3.html#E12:9"><i><font color="Green" title="E19">A11</font></i></a>, <a class="txt" href="sfmastr3.html#E1:9_1_2"><i><font color="Green" title="E20">A17</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><b>hence </b><a NAME="E3:9_1_2_1_1_1_2_1_1"/>
<font color="Maroon" title="c2">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
 <b>by </b><i><a class="txt" href="sfmastr3.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="sfmastr3.html#E3:9"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="sfmastr3.html#E8:9"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="sfmastr3.html#E1:9_1_2"><i><font color="Green" title="E20">A17</font></i></a>, <a class="txt" href="sfmastr3.html#E3:9_1_2_1_1_1"><i><font color="Green" title="E22">A19</font></i></a>, <a class="txt" href="sfmastr3.html#E4:9_1_2_1_1_1"><i><font color="Green" title="E23">A20</font></i></a>, <a class="txt" href="sfmastr3.html#E5:9_1_2_1_1_1"><i><font color="Green" title="E24">A21</font></i></a>, <a class="ref" href="sfmastr3.html#D5" target="_self" title="SFMASTR3:def.5">Def5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_2_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:9_1_2_1_1_1_2_1_2"/><i><font color="Green" title="E25">A22</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">ma</font>
 ;<br/><div class="add"><a NAME="E2:9_1_2_1_1_1_2_1_2"/>
<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">ma</font>
 <b>by </b><i><a class="txt" href="sfmastr3.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="sfmastr3.html#E2:9"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="sfmastr3.html#E9:9"><i><font color="Green" title="E17">A9</font></i></a>, <a class="ref" href="sfmastr3.html#T3" target="_self" title="SFMASTR3:th.3">Th3</a></i>;<br/><a NAME="E3:9_1_2_1_1_1_2_1_2"/><b>then </b><i><font color="Green" title="E26">A23</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">ma</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:9_1_2_1_1_1_2_1_2"/><i><font color="Green" title="E27">A24</font></i>: 
<font color="Maroon" title="c4">ma</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="txt" href="sfmastr3.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="sfmastr3.html#E2:9"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="sfmastr3.html#E9:9"><i><font color="Green" title="E17">A9</font></i></a>, <a class="ref" href="sfmastr3.html#T3" target="_self" title="SFMASTR3:th.3">Th3</a></i>;<br/><font color="Maroon" title="c2">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">ma</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
<b>by </b><i><a class="txt" href="sfmastr3.html#E5:9"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="sfmastr3.html#E1:9_1_2"><i><font color="Green" title="E20">A17</font></i></a>, <a class="txt" href="sfmastr3.html#E1:9_1_2_1_1_1_2_1_2"><i><font color="Green" title="E25">A22</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>
<br/>.= 
<font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">ma</font>
<b>by </b><i><a class="txt" href="sfmastr3.html#E6:9"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="sfmastr3.html#E7:9"><i><font color="Green" title="E15">A7</font></i></a>, <a class="txt" href="sfmastr3.html#E11:9"><i><font color="Green" title="E18">A10</font></i></a>, <a class="txt" href="sfmastr3.html#E1:9_1_2"><i><font color="Green" title="E20">A17</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>
;<br/><b>hence </b><a NAME="E6:9_1_2_1_1_1_2_1_2"/>
<font color="Maroon" title="c2">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
 <b>by </b><i><a class="txt" href="sfmastr3.html#E3:9"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="sfmastr3.html#E8:9"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="sfmastr3.html#E3:9_1_2_1_1_1"><i><font color="Green" title="E22">A19</font></i></a>, <a class="txt" href="sfmastr3.html#E4:9_1_2_1_1_1"><i><font color="Green" title="E23">A20</font></i></a>, <a class="txt" href="sfmastr3.html#E3:9_1_2_1_1_1_2_1_2"><i><font color="Green" title="E26">A23</font></i></a>, <a class="txt" href="sfmastr3.html#E4:9_1_2_1_1_1_2_1_2"><i><font color="Green" title="E27">A24</font></i></a>, <a class="ref" href="sfmastr3.html#D5" target="_self" title="SFMASTR3:def.5">Def5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:9_1_2_1_1_2"/>
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:9_1_2_1_1_2"/>
<font color="Maroon" title="c2">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div>
