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

<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>



<b>set </b><font color="Maroon" title="c4">ab</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> ) </span>}</span> ;<br/>
<a NAME="E1:75"/><i><font color="Green" title="E50">A1</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/>
<b>thus </b><a NAME="E2:75"/>
( <font color="Maroon" title="c3">f</font> <a href="rfunct_3.html#R2" title="RFUNCT_3:pred.2">is_convex_on</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> implies ( <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> &amp; ( for <font color="Olive" title="b1">p</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 holds <br/>for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Olive" title="b3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> holds <br/><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Olive" title="b3">y</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">x</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b3">y</font></span>)</span></span>)</span> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:75_1"/><i><font color="Green" title="E51">A2</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="rfunct_3.html#R2" title="RFUNCT_3:pred.2">is_convex_on</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 ;<br/>

<b>hence </b><a NAME="E2:75_1"/>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="ref" href="rfunct_3.html#D13" target="_self" title="RFUNCT_3:def.13">Def13</a></i>;<br/>

<b>let </b><font color="Maroon" title="c5">p</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E3:75_1"/><i><font color="Green" title="E52">A3</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">p</font> &amp; <font color="Maroon" title="c5">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ;<br/>

<a NAME="E4:75_1"/><b>then </b><i><font color="Green" title="E53">A4</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">p</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/>
<b>let </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E5:75_1"/><i><font color="Green" title="E54">A5</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )
 ;<br/>

<a NAME="E6:75_1"/><b>then </b><i><font color="Green" title="E55">A6</font></i>: 
(  ex <font color="Olive" title="b1">r1</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r1</font> &amp; <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> ) &amp;  ex <font color="Olive" title="b1">r2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font> &amp; <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r2</font> &amp; <font color="Olive" title="b1">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> ) )
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E1:75"><i><font color="Green" title="E50">A1</font></i></a></i>;<br/>
<a NAME="E7:75_1"/><b>then </b><i><font color="Green" title="E56">A7</font></i>: 
( <font color="Maroon" title="c5">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c5">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c6">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c2">b</font> )
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E3:75_1"><i><font color="Green" title="E52">A3</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>
<a NAME="E8:75_1"/><i><font color="Green" title="E57">A8</font></i>: 
( <span class="p1">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c7">y</font> &amp; <span class="p1">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c7">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c2">b</font> )
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E4:75_1"><i><font color="Green" title="E53">A4</font></i></a>, <a class="txt" href="rfunct_3.html#E6:75_1"><i><font color="Green" title="E55">A6</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>
<a NAME="E9:75_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font>
 
;<br/>
<a NAME="E10:75_1"/><b>then </b><i><font color="Green" title="E58">A9</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c7">y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E7:75_1"><i><font color="Green" title="E56">A7</font></i></a>, <a class="txt" href="rfunct_3.html#E8:75_1"><i><font color="Green" title="E57">A8</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E11:75_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font>
 
;<br/>
<a NAME="E12:75_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font>
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E7:75_1"><i><font color="Green" title="E56">A7</font></i></a>, <a class="txt" href="rfunct_3.html#E8:75_1"><i><font color="Green" title="E57">A8</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E13:75_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E10:75_1"><i><font color="Green" title="E58">A9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E14:75_1"/>
<font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c7">y</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">y</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E1:75"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="rfunct_3.html#E1:75_1"><i><font color="Green" title="E51">A2</font></i></a>, <a class="txt" href="rfunct_3.html#E3:75_1"><i><font color="Green" title="E52">A3</font></i></a>, <a class="txt" href="rfunct_3.html#E5:75_1"><i><font color="Green" title="E54">A5</font></i></a>, <a class="ref" href="rfunct_3.html#D13" target="_self" title="RFUNCT_3:def.13">Def13</a></i>;<br/>


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

<b>assume </b><a NAME="E3:75"/><i><font color="Green" title="E51">A10</font></i>: 
( <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> &amp; ( for <font color="Olive" title="b1">p</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 holds <br/>for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Olive" title="b3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> holds <br/><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Olive" title="b3">y</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">x</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b3">y</font></span>)</span></span>)</span> ) )
 ;<br/>

<b>hence </b><a NAME="E4:75"/>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="rfunct_3.html#D13" target="_self" title="RFUNCT_3:def.13">RFUNCT_3:def 13</a><br/>

<b>let </b><font color="Maroon" title="c5">p</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E5:75"/><i><font color="Green" title="E52">A11</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">p</font> &amp; <font color="Maroon" title="c5">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ;<br/>

<b>let </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E6:75"/>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )
 ;<br/>

<b>hence </b><a NAME="E7:75"/>
<font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c7">y</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">y</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E3:75"><i><font color="Green" title="E51">A10</font></i></a>, <a class="txt" href="rfunct_3.html#E5:75"><i><font color="Green" title="E52">A11</font></i></a></i>;<br/>


</div>
