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

<span class="kw">let </span><font color="Maroon" title="c1">f1</font>, <font color="Maroon" title="c2">f2</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c1">f1</font>,<font color="Olive" title="b1">r</font>)</span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c2">f2</font>,<font color="Olive" title="b1">r</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">f2</font></span>)</span>,<font color="Olive" title="b1">r</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">r</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c3">r</font>)</span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">r</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">f2</font></span>)</span>,<font color="Maroon" title="c3">r</font>)</span><br/>



<span class="kw">set </span><font color="Maroon" title="c4">I1</font> =  <a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c3">r</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c5">I2</font> =  <a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">r</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c6">f12</font> = <font color="Maroon" title="c1">f1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">f2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">I12</font> =  <a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">f2</font></span>)</span>,<font color="Maroon" title="c3">r</font>);<br/>
<a NAME="E1:2"/><span class="lab"><font color="Green" title="E3">A1</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">f2</font></span>)</span>,<font color="Maroon" title="c3">r</font>)</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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">f2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_9.html#D3" title="EUCLID_9:def.3">EUCLID_9:def 3</a></span>;<br/>
<a NAME="E2:2"/><span class="lab"><font color="Green" title="E4">A2</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f2</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c3">r</font>)</span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">r</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c3">r</font>)</span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">r</font>)</span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>;<br/>
<a NAME="E3:2"/><span class="lab"><font color="Green" title="E5">A3</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c3">r</font>)</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">f1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_9.html#D3" title="EUCLID_9:def.3">EUCLID_9:def 3</a></span>;<br/>
<a NAME="E4:2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E6">A4</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c3">r</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<a NAME="E5:2"/><span class="lab"><font color="Green" title="E7">A5</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">r</font>)</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="c2">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_9.html#D3" title="EUCLID_9:def.3">EUCLID_9:def 3</a></span>;<br/>
<a NAME="E6:2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">r</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<a NAME="E7:2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E8">A6</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c3">r</font>)</span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">r</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> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">f2</font></span>)</span>,<font color="Maroon" title="c3">r</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E4:2"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="txt" href="#E2:2"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">i</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c3">r</font>)</span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">r</font>)</span>)</span></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c3">r</font>)</span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">r</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">f2</font></span>)</span>,<font color="Maroon" title="c3">r</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">assume </span><a NAME="E1:2_1"/><span class="lab"><font color="Green" title="E9">A7</font></span>: 
<font color="Maroon" title="c8">i</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c3">r</font>)</span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">r</font>)</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c3">r</font>)</span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">r</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">f2</font></span>)</span>,<font color="Maroon" title="c3">r</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><a NAME="E2:2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A8</font></span>: 
<span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">f2</font></span>)</span>,<font color="Maroon" title="c3">r</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c3">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E7:2"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="ref" href="euclid_9.html#D3" title="EUCLID_9:def.3">EUCLID_9:def 3</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1"/>
( <font color="Maroon" title="c8">i</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> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c3">r</font>)</span>)</span> or  ex <font color="Olive" title="b1">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">r</font>)</span>)</span> &amp; <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c3">r</font>)</span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Olive" title="b1">j</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_1"/><span class="lab"><font color="Green" title="E11">A9</font></span>: 
<font color="Maroon" title="c8">i</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> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c3">r</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c3">r</font>)</span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">r</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">f2</font></span>)</span>,<font color="Maroon" title="c3">r</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:2_1_1_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c3">r</font>)</span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">r</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c3">r</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:2_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c3">r</font>)</span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">r</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">f2</font></span>)</span>,<font color="Maroon" title="c3">r</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="txt" href="#E1:2_1_1_1"><span class="lab"><font color="Green" title="E11">A9</font></span></a>, <a class="ref" href="euclid_9.html#D3" title="EUCLID_9:def.3">EUCLID_9:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2"/>
 ex <font color="Olive" title="b1">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">r</font>)</span>)</span> &amp; <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c3">r</font>)</span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Olive" title="b1">j</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c3">r</font>)</span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">r</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">f2</font></span>)</span>,<font color="Maroon" title="c3">r</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:2_1_1_2"/><span class="lab"><font color="Green" title="E11">A10</font></span>: 
<font color="Maroon" title="c9">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> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">r</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2"/><span class="lab"><font color="Green" title="E12">A11</font></span>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c3">r</font>)</span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c9">j</font>
 ;<br/><a NAME="E5:2_1_1_2"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c3">r</font>)</span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">r</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">r</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="txt" href="#E4:2"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="txt" href="#E3:2_1_1_2"><span class="lab"><font color="Green" title="E11">A10</font></span></a>, <a class="txt" href="#E4:2_1_1_2"><span class="lab"><font color="Green" title="E12">A11</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E6:2_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c3">r</font>)</span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">r</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">f2</font></span>)</span>,<font color="Maroon" title="c3">r</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="txt" href="#E3:2_1_1_2"><span class="lab"><font color="Green" title="E11">A10</font></span></a>, <a class="ref" href="euclid_9.html#D3" title="EUCLID_9:def.3">EUCLID_9:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E9:2"/>
<span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c3">r</font>)</span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">r</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">f2</font></span>)</span>,<font color="Maroon" title="c3">r</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="ref" href="finseq_1.html#T13" title="FINSEQ_1:th.13">FINSEQ_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
