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

<b>let </b><font color="Maroon" title="c1">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> , <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c2">g</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a> , <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon" title="c3">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a> ;<br/>





<b>assume </b><a NAME="E1:43"/><i><font color="Green" title="E32">A1</font></i>: 
( <font color="Maroon" title="c3">X</font> <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="c1">f</font> &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> ( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> implies <font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font> ) &amp; ( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> implies <font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) ) ) )
 ;<br/>

<a NAME="E2:43"/><i><font color="Green" title="E33">A2</font></i>: 
for <font color="Olive" title="b1">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>   holds <font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a> ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">v</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_1_1"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> or  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1"><b>suppose </b></a><a NAME="E1:43_1_1_1"/><i><font color="Green" title="E34">A3</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ;<br/><div class="add"><a NAME="E2:43_1_1_1"/>
<font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> 
 ;<br/><b>hence </b><a NAME="E3:43_1_1_1"/>
<font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c4">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 <b>by </b><i><a class="txt" href="convfun1.html#E1:43"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="convfun1.html#E1:43_1_1_1"><i><font color="Green" title="E34">A3</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_2"><b>suppose </b></a><a NAME="E1:43_1_1_2"/>
not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:43_1_1_2"/>
<font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c4">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 <b>by </b><i><a class="txt" href="convfun1.html#E1:43"><i><font color="Green" title="E32">A1</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:43"/><i><font color="Green" title="E34">A4</font></i>: 
for <font color="Olive" title="b1">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>   st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> holds <br/><font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a> ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">v</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<b>assume </b><a NAME="E2:43_2"/>
<font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ;<br/>

<a NAME="E3:43_2"/><b>then </b>
( <font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font> )
 
<b>by </b><i><a class="txt" href="convfun1.html#E1:43"><i><font color="Green" title="E32">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:43_2"/>
<font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> 
 ;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E4:43"/>
( <font color="Maroon" title="c2">g</font> is <a href="convfun1.html#V1" title="CONVFUN1:attr.1">convex</a> implies ( <font color="Maroon" title="c1">f</font> <a href="rfunct_3.html#R2" title="RFUNCT_3:pred.2">is_convex_on</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c3">X</font> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_3"/><i><font color="Green" title="E35">A5</font></i>: 
<font color="Maroon" title="c2">g</font> is <a href="convfun1.html#V1" title="CONVFUN1:attr.1">convex</a>
 ;<br/>

<b>thus </b><a NAME="E2:43_3"/>
<font color="Maroon" title="c1">f</font> <a href="rfunct_3.html#R2" title="RFUNCT_3:pred.2">is_convex_on</a> <font color="Maroon" title="c3">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:43_3_1"/>
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">x1</font>, <font color="Olive" title="b3">x2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Olive" title="b3">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Olive" title="b2">x1</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Olive" title="b3">x2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> holds <br/><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Olive" title="b2">x1</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Olive" title="b3">x2</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="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">x1</font></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b3">x2</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">p</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:43_3_1_1"/><i><font color="Green" title="E36">A6</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ;<br/>

<b>let </b><font color="Maroon" title="c5">x1</font>, <font color="Maroon" title="c6">x2</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E2:43_3_1_1"/>
( <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c6">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c5">x1</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c6">x2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> )
 ;<br/>

<a NAME="E3:43_3_1_1"/><b>then </b><i><font color="Green" title="E37">A7</font></i>: 
( <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c5">x1</font> &amp; <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c6">x2</font> &amp; <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c5">x1</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c6">x2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c5">x1</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c6">x2</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="convfun1.html#E1:43"><i><font color="Green" title="E32">A1</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_3_1_1_1"/>
( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or ( <font color="Maroon" title="c4">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_3_1_1_1_1"><b>suppose </b></a><a NAME="E1:43_3_1_1_1_1"/>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:43_3_1_1_1_1"/>
<font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c5">x1</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c6">x2</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="c4">p</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x1</font></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x2</font></span>)</span></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_3_1_1_1_2"><b>suppose </b></a><a NAME="E1:43_3_1_1_1_2"/>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:43_3_1_1_1_2"/>
<font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c5">x1</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c6">x2</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="c4">p</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x1</font></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x2</font></span>)</span></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_3_1_1_1_3"><b>suppose </b></a><a NAME="E1:43_3_1_1_1_3"/>
( <font color="Maroon" title="c4">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 ;<br/><div class="add"><a NAME="E2:43_3_1_1_1_3"/><b>then </b><i><font color="Green" title="E38">A8</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 )
 <b>by </b><i><a class="txt" href="convfun1.html#E1:43_3_1_1"><i><font color="Green" title="E36">A6</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">v1</font> = <font color="Maroon" title="c5">x1</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>  ;<br/><b>reconsider </b><font color="Maroon" title="c8">v2</font> = <font color="Maroon" title="c6">x2</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>  ;<br/><a NAME="E5:43_3_1_1_1_3"/><i><font color="Green" title="E39">A9</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">v1</font></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">v2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c7">v1</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c8">v2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="convfun1.html#E3:43_3_1_1"><i><font color="Green" title="E37">A7</font></i></a>, <a class="txt" href="convfun1.html#E33"><i><font color="Green" title="E30">Lm18</font></i></a></i>;<br/><a NAME="E6:43_3_1_1_1_3"/>
( <font color="Maroon" title="c4">p</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c7">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c5">x1</font> &amp; <span class="p1">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c8">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c6">x2</font> )
 <b>by </b><i><a class="txt" href="convfun1.html#E8"><i><font color="Green" title="E6">Lm1</font></i></a></i>;<br/><a NAME="E7:43_3_1_1_1_3"/><b>then </b>
<font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c7">v1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c8">v2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c5">x1</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c6">x2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="convfun1.html#E3:43_3_1_1"><i><font color="Green" title="E37">A7</font></i></a>, <a class="txt" href="convfun1.html#E9"><i><font color="Green" title="E7">Lm2</font></i></a></i>;<br/><b>hence </b><a NAME="E8:43_3_1_1_1_3"/>
<font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c5">x1</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c6">x2</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="c4">p</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x1</font></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="convfun1.html#E2:43"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="convfun1.html#E1:43_3"><i><font color="Green" title="E35">A5</font></i></a>, <a class="txt" href="convfun1.html#E2:43_3_1_1_1_3"><i><font color="Green" title="E38">A8</font></i></a>, <a class="txt" href="convfun1.html#E5:43_3_1_1_1_3"><i><font color="Green" title="E39">A9</font></i></a>, <a class="ref" href="convfun1.html#T9" target="_self" title="CONVFUN1:th.9">Th9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E2:43_3_1"/>
<font color="Maroon" title="c1">f</font> <a href="rfunct_3.html#R2" title="RFUNCT_3:pred.2">is_convex_on</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="convfun1.html#E1:43"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="rfunct_3.html#D13" title="RFUNCT_3:def.13">RFUNCT_3:def 13</a></i>;<br/>


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

<b>thus </b><a NAME="E3:43_3"/>
<font color="Maroon" title="c3">X</font> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_3_2"><b>proof </b></a><div class="add">

<a NAME="E1:43_3_2"/>
for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a> <br/> for <font color="Olive" title="b3">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="b3">p</font> &amp; <font color="Olive" title="b3">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 &amp; <font color="Olive" title="b1">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Olive" title="b2">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b3">p</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Olive" title="b1">v1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Olive" title="b3">p</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Olive" title="b2">v2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_3_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">v1</font>, <font color="Maroon" title="c5">v2</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a> ;<br/><b>let </b><font color="Maroon" title="c6">p</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>



<b>assume </b><a NAME="E1:43_3_2_1"/><i><font color="Green" title="E36">A10</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Maroon" title="c6">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 &amp; <font color="Maroon" title="c4">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c5">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> )
 ;<br/>

<a NAME="E2:43_3_2_1"/><b>then </b><i><font color="Green" title="E37">A11</font></i>: 
<font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c4">v1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c5">v2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c4">v1</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c6">p</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c5">v2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="convfun1.html#E2:43"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="convfun1.html#E1:43_3"><i><font color="Green" title="E35">A5</font></i></a>, <a class="ref" href="convfun1.html#T9" target="_self" title="CONVFUN1:th.9">Th9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">w1</font> = <font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c4">v1</font>, <font color="Maroon" title="c8">w2</font> = <font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c5">v2</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="convfun1.html#E3:43"><i><font color="Green" title="E34">A4</font></i></a>, <a class="txt" href="convfun1.html#E1:43_3_2_1"><i><font color="Green" title="E36">A10</font></i></a></i>;<br/>
<a NAME="E4:43_3_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c7">w1</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c8">w2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c4">v1</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c6">p</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c5">v2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="convfun1.html#E33"><i><font color="Green" title="E30">Lm18</font></i></a></i>;<br/>
<a NAME="E5:43_3_2_1"/><b>then </b><i><font color="Green" title="E38">A12</font></i>: 
<font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c4">v1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c5">v2</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<b>by </b><i><a class="txt" href="convfun1.html#E2:43_3_2_1"><i><font color="Green" title="E37">A11</font></i></a>, <a class="ref" href="xxreal_0.html#T9" title="XXREAL_0:th.9">XXREAL_0:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">x</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c4">v1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c5">v2</font></span>)</span> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<b>thus </b><a NAME="E7:43_3_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c4">v1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c5">v2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="convfun1.html#E1:43"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="convfun1.html#E5:43_3_2_1"><i><font color="Green" title="E38">A12</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:43_3_2"/>
<font color="Maroon" title="c3">X</font> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a>
 <b>by </b><i><a class="ref" href="convex1.html#D2" title="CONVEX1:def.2">CONVEX1:def 2</a></i>;<br/>


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


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

<b>thus </b><a NAME="E5:43"/>
( <font color="Maroon" title="c1">f</font> <a href="rfunct_3.html#R2" title="RFUNCT_3:pred.2">is_convex_on</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c3">X</font> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a> implies <font color="Maroon" title="c2">g</font> is <a href="convfun1.html#V1" title="CONVFUN1:attr.1">convex</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_4"/><i><font color="Green" title="E35">A13</font></i>: 
( <font color="Maroon" title="c1">f</font> <a href="rfunct_3.html#R2" title="RFUNCT_3:pred.2">is_convex_on</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c3">X</font> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a> )
 ;<br/>

<a NAME="E2:43_4"/>
for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a> <br/> for <font color="Olive" title="b3">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="b3">p</font> &amp; <font color="Olive" title="b3">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 holds <br/><font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">p</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Olive" title="b1">v1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Olive" title="b3">p</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Olive" title="b2">v2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Olive" title="b3">p</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b1">v1</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Olive" title="b3">p</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b2">v2</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">v1</font>, <font color="Maroon" title="c5">v2</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a> ;<br/><b>let </b><font color="Maroon" title="c6">p</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>



<b>assume </b><a NAME="E1:43_4_1"/><i><font color="Green" title="E36">A14</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Maroon" title="c6">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 )
 ;<br/>

<a NAME="E2:43_4_1"/><i><font color="Green" title="E37">A15</font></i>: 
<font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">p</font>
 
<b>by </b><i><a class="ref" href="measure6.html#D1" title="MEASURE6:def.1">MEASURE6:def 1</a></i>;<br/>
<a NAME="E3:43_4_1"/><i><font color="Green" title="E38">A16</font></i>: 
1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c6">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="convfun1.html#E1:43_4_1"><i><font color="Green" title="E36">A14</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/>
<a NAME="E4:43_4_1"/><i><font color="Green" title="E39">A17</font></i>: 
1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c6">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="measure6.html#D1" title="MEASURE6:def.1">MEASURE6:def 1</a></i>;<br/>
<a NAME="E5:43_4_1"/><i><font color="Green" title="E40">A18</font></i>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c6">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="convfun1.html#E3:43_4_1"><i><font color="Green" title="E38">A16</font></i></a>, <a class="ref" href="measure6.html#D1" title="MEASURE6:def.1">MEASURE6:def 1</a>, <a class="ref" href="supinf_2.html#D1" title="SUPINF_2:def.1">SUPINF_2:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">x1</font> = <font color="Maroon" title="c4">v1</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c8">x2</font> = <font color="Maroon" title="c5">v2</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_4_1_1"/>
( ( <font color="Maroon" title="c4">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c5">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> ) or ( <font color="Maroon" title="c4">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; not <font color="Maroon" title="c5">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> ) or ( not <font color="Maroon" title="c4">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c5">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> ) or ( not <font color="Maroon" title="c4">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; not <font color="Maroon" title="c5">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_4_1_1_1"><b>suppose </b></a><a NAME="E1:43_4_1_1_1"/><i><font color="Green" title="E41">A19</font></i>: 
( <font color="Maroon" title="c4">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c5">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> )
 ;<br/><div class="add"><a NAME="E2:43_4_1_1_1"/><b>then </b><i><font color="Green" title="E42">A20</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c4">v1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c5">v2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="convfun1.html#E1:43_4"><i><font color="Green" title="E35">A13</font></i></a>, <a class="txt" href="convfun1.html#E1:43_4_1"><i><font color="Green" title="E36">A14</font></i></a>, <a class="ref" href="convex1.html#D2" title="CONVEX1:def.2">CONVEX1:def 2</a></i>;<br/><a NAME="E3:43_4_1_1_1"/>
( <font color="Maroon" title="c6">p</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c4">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c7">x1</font> &amp; <span class="p1">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c5">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c8">x2</font> )
 <b>by </b><i><a class="txt" href="convfun1.html#E8"><i><font color="Green" title="E6">Lm1</font></i></a></i>;<br/><a NAME="E4:43_4_1_1_1"/><b>then </b><i><font color="Green" title="E43">A21</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c4">v1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c5">v2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c7">x1</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c8">x2</font></span>)</span>
 <b>by </b><i><a class="txt" href="convfun1.html#E9"><i><font color="Green" title="E7">Lm2</font></i></a></i>;<br/><a NAME="E5:43_4_1_1_1"/><b>then </b><i><font color="Green" title="E44">A22</font></i>: 
( <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c4">v1</font> &amp; <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c5">v2</font> &amp; <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c7">x1</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c8">x2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c4">v1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c5">v2</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="convfun1.html#E1:43"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="convfun1.html#E1:43_4_1_1_1"><i><font color="Green" title="E41">A19</font></i></a>, <a class="txt" href="convfun1.html#E2:43_4_1_1_1"><i><font color="Green" title="E42">A20</font></i></a></i>;<br/><a NAME="E6:43_4_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">x1</font></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">x2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c4">v1</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c6">p</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c5">v2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="convfun1.html#E33"><i><font color="Green" title="E30">Lm18</font></i></a></i>;<br/><b>hence </b><a NAME="E7:43_4_1_1_1"/>
<font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c4">v1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c5">v2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c4">v1</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c6">p</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c5">v2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="convfun1.html#E1:43_4"><i><font color="Green" title="E35">A13</font></i></a>, <a class="txt" href="convfun1.html#E1:43_4_1"><i><font color="Green" title="E36">A14</font></i></a>, <a class="txt" href="convfun1.html#E1:43_4_1_1_1"><i><font color="Green" title="E41">A19</font></i></a>, <a class="txt" href="convfun1.html#E2:43_4_1_1_1"><i><font color="Green" title="E42">A20</font></i></a>, <a class="txt" href="convfun1.html#E4:43_4_1_1_1"><i><font color="Green" title="E43">A21</font></i></a>, <a class="txt" href="convfun1.html#E5:43_4_1_1_1"><i><font color="Green" title="E44">A22</font></i></a>, <a class="ref" href="rfunct_3.html#D13" title="RFUNCT_3:def.13">RFUNCT_3:def 13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_4_1_1_2"><b>suppose </b></a><a NAME="E1:43_4_1_1_2"/><i><font color="Green" title="E41">A23</font></i>: 
( <font color="Maroon" title="c4">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; not <font color="Maroon" title="c5">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> )
 ;<br/><div class="add"><a NAME="E2:43_4_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c8">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="convfun1.html#E1:43"><i><font color="Green" title="E32">A1</font></i></a></i>;<br/><a NAME="E3:43_4_1_1_2"/><b>then </b><i><font color="Green" title="E42">A24</font></i>: 
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c6">p</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c5">v2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="convfun1.html#E3:43_4_1"><i><font color="Green" title="E38">A16</font></i></a>, <a class="txt" href="convfun1.html#E4:43_4_1"><i><font color="Green" title="E39">A17</font></i></a>, <a class="ref" href="extreal1.html#D1" title="EXTREAL1:def.1">EXTREAL1:def 1</a>, <a class="ref" href="supinf_2.html#D1" title="SUPINF_2:def.1">SUPINF_2:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">w1</font> = <font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c7">x1</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="convfun1.html#E3:43"><i><font color="Green" title="E34">A4</font></i></a>, <a class="txt" href="convfun1.html#E1:43_4_1_1_2"><i><font color="Green" title="E41">A23</font></i></a></i>;<br/><a NAME="E5:43_4_1_1_2"/>
<font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">p</font>
 <b>by </b><i><a class="ref" href="measure6.html#D1" title="MEASURE6:def.1">MEASURE6:def 1</a></i>;<br/><a NAME="E6:43_4_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">p</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c9">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c4">v1</font></span>)</span>
 <b>by </b><i><a class="ref" href="extreal1.html#T13" title="EXTREAL1:th.13">EXTREAL1:13</a></i>;<br/><a NAME="E7:43_4_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c4">v1</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c6">p</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c5">v2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="convfun1.html#E3:43_4_1_1_2"><i><font color="Green" title="E42">A24</font></i></a>, <a class="ref" href="supinf_2.html#D2" title="SUPINF_2:def.2">SUPINF_2:def 2</a></i>;<br/><b>hence </b><a NAME="E8:43_4_1_1_2"/>
<font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c4">v1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c5">v2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c4">v1</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c6">p</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c5">v2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T4" title="XXREAL_0:th.4">XXREAL_0:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_4_1_1_3"><b>suppose </b></a><a NAME="E1:43_4_1_1_3"/><i><font color="Green" title="E41">A25</font></i>: 
( not <font color="Maroon" title="c4">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c5">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> )
 ;<br/><div class="add"><a NAME="E2:43_4_1_1_3"/><b>then </b>
<font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="convfun1.html#E1:43"><i><font color="Green" title="E32">A1</font></i></a></i>;<br/><a NAME="E3:43_4_1_1_3"/><b>then </b><i><font color="Green" title="E42">A26</font></i>: 
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c4">v1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="convfun1.html#E1:43_4_1"><i><font color="Green" title="E36">A14</font></i></a>, <a class="txt" href="convfun1.html#E2:43_4_1"><i><font color="Green" title="E37">A15</font></i></a>, <a class="ref" href="extreal1.html#D1" title="EXTREAL1:def.1">EXTREAL1:def 1</a>, <a class="ref" href="supinf_2.html#D1" title="SUPINF_2:def.1">SUPINF_2:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">w2</font> = <font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c8">x2</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="convfun1.html#E3:43"><i><font color="Green" title="E34">A4</font></i></a>, <a class="txt" href="convfun1.html#E1:43_4_1_1_3"><i><font color="Green" title="E41">A25</font></i></a></i>;<br/><a NAME="E5:43_4_1_1_3"/>
1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c6">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="measure6.html#D1" title="MEASURE6:def.1">MEASURE6:def 1</a></i>;<br/><a NAME="E6:43_4_1_1_3"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c9">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c6">p</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c5">v2</font></span>)</span>
 <b>by </b><i><a class="ref" href="extreal1.html#T13" title="EXTREAL1:th.13">EXTREAL1:13</a></i>;<br/><a NAME="E7:43_4_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c4">v1</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c6">p</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c5">v2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="convfun1.html#E3:43_4_1_1_3"><i><font color="Green" title="E42">A26</font></i></a>, <a class="ref" href="supinf_2.html#D2" title="SUPINF_2:def.2">SUPINF_2:def 2</a></i>;<br/><b>hence </b><a NAME="E8:43_4_1_1_3"/>
<font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c4">v1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c5">v2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c4">v1</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c6">p</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c5">v2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T4" title="XXREAL_0:th.4">XXREAL_0:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_4_1_1_4"><b>suppose </b></a><a NAME="E1:43_4_1_1_4"/>
( not <font color="Maroon" title="c4">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; not <font color="Maroon" title="c5">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> )
 ;<br/><div class="add"><a NAME="E2:43_4_1_1_4"/><b>then </b>
( <font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c8">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 <b>by </b><i><a class="txt" href="convfun1.html#E1:43"><i><font color="Green" title="E32">A1</font></i></a></i>;<br/><a NAME="E3:43_4_1_1_4"/><b>then </b>
( <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c4">v1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c6">p</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c5">v2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 <b>by </b><i><a class="txt" href="convfun1.html#E1:43_4_1"><i><font color="Green" title="E36">A14</font></i></a>, <a class="txt" href="convfun1.html#E2:43_4_1"><i><font color="Green" title="E37">A15</font></i></a>, <a class="txt" href="convfun1.html#E5:43_4_1"><i><font color="Green" title="E40">A18</font></i></a>, <a class="ref" href="extreal1.html#D1" title="EXTREAL1:def.1">EXTREAL1:def 1</a>, <a class="ref" href="supinf_2.html#D1" title="SUPINF_2:def.1">SUPINF_2:def 1</a></i>;<br/><a NAME="E4:43_4_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c4">v1</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c6">p</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c5">v2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="ref" href="supinf_2.html#D2" title="SUPINF_2:def.2">SUPINF_2:def 2</a></i>;<br/><b>hence </b><a NAME="E5:43_4_1_1_4"/>
<font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c4">v1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c5">v2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c4">v1</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c6">p</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c5">v2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T4" title="XXREAL_0:th.4">XXREAL_0:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E3:43_4"/>
<font color="Maroon" title="c2">g</font> is <a href="convfun1.html#V1" title="CONVFUN1:attr.1">convex</a>
 <b>by </b><i><a class="txt" href="convfun1.html#E2:43"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="convfun1.html#T9" target="_self" title="CONVFUN1:th.9">Th9</a></i>;<br/>


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


</div>
