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

<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="rlvect_1.html#L2">RLSStruct</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a> ;<br/>



<b>assume </b><a NAME="E1:41"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">x</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>  holds <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/>

<b>thus </b><a NAME="E2:41"/>
( <font color="Maroon">f</font> is <a href="convfun1.html#V1" target="_self">convex</a> implies for <font color="Olive">x1</font>, <font color="Olive">x2</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font><br/> for <font color="Olive">p</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;</a> 1 holds <br/><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">x1</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Olive">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Olive">x2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Olive">p</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Olive">x1</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Olive">p</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Olive">x2</font></span>)</span></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1"/>
<font color="Maroon">f</font> is <a href="convfun1.html#V1" target="_self">convex</a>
 ;<br/>

<a NAME="E2:41_1"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="convfun1.html#K6" target="_self">epigraph</a> <font color="Maroon">f</font> is <a href="convex1.html#V1">convex</a>
 
<b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">x1</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>, <font color="Maroon">x2</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E3:41_1"/><i><font color="Green">E41</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;</a> 1 )
 ;<br/>

<b>set </b><font color="Maroon">p1</font> =  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font>;<br/>
<a NAME="E4:41_1"/><i><font color="Green">E45</font></i>: 
 <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="measure6.html#D1">MEASURE6:def 1</a></i>;<br/>
<a NAME="E5:41_1"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="extreal1.html#T18">EXTREAL1:18</a></i>;<br/>
<a NAME="E6:41_1"/><i><font color="Green">E47</font></i>: 
1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/>
<b>set </b><font color="Maroon">p2</font> =  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span>;<br/>
<a NAME="E7:41_1"/><i><font color="Green">E48</font></i>: 
 <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="measure6.html#D1">MEASURE6:def 1</a></i>;<br/>
<a NAME="E8:41_1"/><b>then </b><i><font color="Green">E70</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="convfun1.html#E2:41_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="extreal1.html#T18">EXTREAL1:18</a></i>;<br/>
<a NAME="E9:41_1"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  )
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_1_1"/>
( ( <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  ) or ( <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  ) or ( <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) or ( <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) )
 </span><b>by </b><i><a class="txt" href="convfun1.html#E3:41_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="supinf_2.html#T2">SUPINF_2:2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1"><b>suppose </b></a><a NAME="E1:41_1_1_1"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">w1</font> = <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x1</font> as   <a href="real_1.html#NM1">Real</a> ;<br/><b>reconsider </b><font color="Maroon">w2</font> = <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x2</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i/>;<br/><a NAME="E4:41_1_1_1"/><i><font color="Green">E73</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">w1</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="convfun1.html#K6" target="_self">epigraph</a> <font color="Maroon">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:41_1_1_1_1"/>
<font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">w1</font>
 
<b>by </b><i><a class="ref" href="measure6.html#D1">MEASURE6:def 1</a></i>;<br/>
<a NAME="E2:41_1_1_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">w1</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Olive">y</font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E3:41_1_1_1_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">w1</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="convfun1.html#K6" target="_self">epigraph</a> <font color="Maroon">f</font>
 ;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon">u1</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">w1</font></span><a href="domain_1.html#K1">]</a></span> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="convfun1.html#K3" target="_self">Prod_of_RLS</a> <font color="Maroon">X</font>,<a href="convfun1.html#K4" target="_self">RLS_Real</a> </span>)</span> ;<br/><a NAME="E6:41_1_1_1"/><i><font color="Green">E74</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x2</font>,<font color="Maroon">w2</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="convfun1.html#K6" target="_self">epigraph</a> <font color="Maroon">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:41_1_1_1_2"/>
<font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">w2</font>
 
<b>by </b><i><a class="ref" href="measure6.html#D1">MEASURE6:def 1</a></i>;<br/>
<a NAME="E2:41_1_1_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x2</font>,<font color="Maroon">w2</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Olive">y</font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E3:41_1_1_1_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x2</font>,<font color="Maroon">w2</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="convfun1.html#K6" target="_self">epigraph</a> <font color="Maroon">f</font>
 ;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon">u2</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x2</font>,<font color="Maroon">w2</font></span><a href="domain_1.html#K1">]</a></span> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="convfun1.html#K3" target="_self">Prod_of_RLS</a> <font color="Maroon">X</font>,<a href="convfun1.html#K4" target="_self">RLS_Real</a> </span>)</span> ;<br/><a NAME="E8:41_1_1_1"/><i><font color="Green">E75</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="convfun1.html#K6" target="_self">epigraph</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="convfun1.html#E1:41"><i><font color="Green">E34</font></i></a>, , , , <a class="ref" href="convex1.html#D2">CONVEX1:def 2</a></i>;<br/><a NAME="E9:41_1_1_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">w1</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i/>;<br/><a NAME="E10:41_1_1_1"/>
<span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">w2</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i/>;<br/><a NAME="E11:41_1_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">w1</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">w2</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="convfun1.html#K6" target="_self">epigraph</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="convfun1.html#E4:41_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="convfun1.html#E5:41_1"><i><font color="Green">E46</font></i></a>, </i>;<br/><a NAME="E12:41_1_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">w1</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">w2</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Olive">y</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">x0</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>, <font color="Maroon">y0</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a> <b> such that </b><br/><a NAME="E14:41_1_1_1"/><i><font color="Green">E77</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">w1</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">w2</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x0</font>,<font color="Maroon">y0</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">y0</font> )
 ;<br/><a NAME="E15:41_1_1_1"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">x0</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x2</font></span>)</span> &amp; <font color="Maroon">y0</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">w1</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">w2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="convfun1.html#E6:41_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E16:41_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">w1</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">w2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x2</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E17:41_1_1_1"/>
<font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="convfun1.html#E6:41_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="convfun1.html#E7:41_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="measure6.html#D1">MEASURE6:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_2"><b>suppose </b></a><a NAME="E1:41_1_1_2"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 ;<br/><div class="add"><a NAME="E2:41_1_1_2"/><b>then </b><i><font color="Green">E102</font></i>: 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i>, <a class="ref" href="extreal1.html#D1">EXTREAL1:def 1</a></i>;<br/><a NAME="E3:41_1_1_2"/>
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x2</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_2_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">w2</font> = <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x2</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i/>;<br/>
<a NAME="E2:41_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">w2</font>
 
<b>by </b><i>, <a class="ref" href="extreal1.html#T13">EXTREAL1:13</a></i>;<br/>
<b>hence </b><a NAME="E3:41_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x2</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 ;<br/>


</div><b>end;</b></div><a NAME="E4:41_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i>, <a class="ref" href="supinf_1.html#T6">SUPINF_1:6</a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><b>hence </b><a NAME="E5:41_1_1_2"/>
<font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_1.html#T20">SUPINF_1:20</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_3"><b>suppose </b></a><a NAME="E1:41_1_1_3"/><i><font color="Green">E103</font></i>: 
( <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 ;<br/><div class="add"><a NAME="E2:41_1_1_3"/><b>then </b><i><font color="Green">E105</font></i>: 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i>, <a class="ref" href="extreal1.html#D1">EXTREAL1:def 1</a></i>;<br/><a NAME="E3:41_1_1_3"/>
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x1</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_3_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">w1</font> = <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x1</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i/>;<br/>
<a NAME="E2:41_1_1_3_1"/>
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">w1</font>
 
<b>by </b><i>, <a class="ref" href="extreal1.html#T13">EXTREAL1:13</a></i>;<br/>
<b>hence </b><a NAME="E3:41_1_1_3_1"/>
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x1</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 ;<br/>


</div><b>end;</b></div><a NAME="E4:41_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i>, <a class="ref" href="supinf_1.html#T6">SUPINF_1:6</a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><b>hence </b><a NAME="E5:41_1_1_3"/>
<font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_1.html#T20">SUPINF_1:20</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_4"><b>suppose </b></a><a NAME="E1:41_1_1_4"/>
( <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 ;<br/><div class="add"><a NAME="E2:41_1_1_4"/><b>then </b>
( <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i>, , <a class="ref" href="extreal1.html#D1">EXTREAL1:def 1</a></i>;<br/><a NAME="E3:41_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><b>hence </b><a NAME="E4:41_1_1_4"/>
<font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_1.html#T20">SUPINF_1:20</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E3:41"/>
( ( for <font color="Olive">x1</font>, <font color="Olive">x2</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font><br/> for <font color="Olive">p</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;</a> 1 holds <br/><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">x1</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Olive">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Olive">x2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Olive">p</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Olive">x1</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Olive">p</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Olive">x2</font></span>)</span></span>)</span> ) implies <font color="Maroon">f</font> is <a href="convfun1.html#V1" target="_self">convex</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_2"/><i><font color="Green">E106</font></i>: 
for <font color="Olive">x1</font>, <font color="Olive">x2</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font><br/> for <font color="Olive">p</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;</a> 1 holds <br/><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">x1</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Olive">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Olive">x2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Olive">p</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Olive">x1</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Olive">p</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Olive">x2</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:41_2"/>
for <font color="Olive">u1</font>, <font color="Olive">u2</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="convfun1.html#K3" target="_self">Prod_of_RLS</a> <font color="Maroon">X</font>,<a href="convfun1.html#K4" target="_self">RLS_Real</a> </span>)</span><br/> for <font color="Olive">p</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;</a> 1 &amp; <font color="Olive">u1</font> <a href="hidden.html#R2">in</a>  <a href="convfun1.html#K6" target="_self">epigraph</a> <font color="Maroon">f</font> &amp; <font color="Olive">u2</font> <a href="hidden.html#R2">in</a>  <a href="convfun1.html#K6" target="_self">epigraph</a> <font color="Maroon">f</font> holds <br/><span class="p1">(<span class="default"><font color="Olive">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">u1</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Olive">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Olive">u2</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="convfun1.html#K6" target="_self">epigraph</a> <font color="Maroon">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u1</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="convfun1.html#K3" target="_self">Prod_of_RLS</a> <font color="Maroon">X</font>,<a href="convfun1.html#K4" target="_self">RLS_Real</a> </span>)</span>, <font color="Maroon">u2</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="convfun1.html#K3" target="_self">Prod_of_RLS</a> <font color="Maroon">X</font>,<a href="convfun1.html#K4" target="_self">RLS_Real</a> </span>)</span>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E1:41_2_1"/><i><font color="Green">E107</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;</a> 1 &amp; <font color="Maroon">u1</font> <a href="hidden.html#R2">in</a>  <a href="convfun1.html#K6" target="_self">epigraph</a> <font color="Maroon">f</font> &amp; <font color="Maroon">u2</font> <a href="hidden.html#R2">in</a>  <a href="convfun1.html#K6" target="_self">epigraph</a> <font color="Maroon">f</font> )
 ;<br/>

<b>thus </b><a NAME="E2:41_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="convfun1.html#K6" target="_self">epigraph</a> <font color="Maroon">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:41_2_1_1"/>
<font color="Maroon">u1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Olive">y</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">x1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>, <font color="Maroon">y1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a> <b> such that </b><br/><a NAME="E3:41_2_1_1"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">y1</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">y1</font> )
 ;<br/>
<a NAME="E4:41_2_1_1"/>
<font color="Maroon">u2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Olive">y</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">x2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>, <font color="Maroon">y2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a> <b> such that </b><br/><a NAME="E6:41_2_1_1"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">u2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x2</font>,<font color="Maroon">y2</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">y2</font> )
 ;<br/>
<a NAME="E7:41_2_1_1"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E8:41_2_1_1"/>
 <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font>
 
<b>by </b><i><a class="ref" href="measure6.html#D1">MEASURE6:def 1</a></i>;<br/>
<a NAME="E9:41_2_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i>, <a class="ref" href="supinf_1.html#T18">SUPINF_1:18</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">w1</font> = <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x1</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, <a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/>
<a NAME="E11:41_2_1_1"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E12:41_2_1_1"/>
 <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font>
 
<b>by </b><i><a class="ref" href="measure6.html#D1">MEASURE6:def 1</a></i>;<br/>
<a NAME="E13:41_2_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i>, <a class="ref" href="supinf_1.html#T18">SUPINF_1:18</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">w2</font> = <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x2</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, <a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/>
<a NAME="E15:41_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">w1</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">w2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x2</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E16:41_2_1_1"/><b>then </b>
 <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">w1</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">w2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="measure6.html#D1">MEASURE6:def 1</a></i>;<br/>
<a NAME="E17:41_2_1_1"/><b>then </b><i><font color="Green">E127</font></i>: 
<font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">w1</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">w2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="convfun1.html#E1:41"><i><font color="Green">E34</font></i></a>, </i>;<br/>
<a NAME="E18:41_2_1_1"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E19:41_2_1_1"/>
 <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">w1</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">w2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">w1</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">w2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="measure6.html#D1">MEASURE6:def 1</a></i>;<br/>
<a NAME="E20:41_2_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i>, <a class="ref" href="supinf_1.html#T18">SUPINF_1:18</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">w</font> = <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x2</font></span>)</span></span>)</span> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, <a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/>
<a NAME="E22:41_2_1_1"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">w</font>
 
<b>by </b><i><a class="ref" href="measure6.html#D1">MEASURE6:def 1</a></i>;<br/>
<a NAME="E23:41_2_1_1"/><b>then </b><i><font color="Green">E132</font></i>: 
<font color="Maroon">w</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">w1</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">w2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="measure6.html#T13">MEASURE6:13</a></i>;<br/>
<a NAME="E24:41_2_1_1"/>
<font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">w1</font>
 
<b>by </b><i><a class="ref" href="measure6.html#D1">MEASURE6:def 1</a></i>;<br/>
<a NAME="E25:41_2_1_1"/><b>then </b>
<font color="Maroon">w1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y1</font>
 
<b>by </b><i>, <a class="ref" href="measure6.html#T13">MEASURE6:13</a></i>;<br/>
<a NAME="E26:41_2_1_1"/><b>then </b><i><font color="Green">E133</font></i>: 
<font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">w1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">y1</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E27:41_2_1_1"/>
<font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">w2</font>
 
<b>by </b><i><a class="ref" href="measure6.html#D1">MEASURE6:def 1</a></i>;<br/>
<a NAME="E28:41_2_1_1"/><b>then </b><i><font color="Green">E134</font></i>: 
<font color="Maroon">w2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y2</font>
 
<b>by </b><i>, <a class="ref" href="measure6.html#T13">MEASURE6:13</a></i>;<br/>
<a NAME="E29:41_2_1_1"/>
1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/>
<a NAME="E30:41_2_1_1"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">w2</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">y2</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E31:41_2_1_1"/><b>then </b><i><font color="Green">E135</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">w1</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">w2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">w1</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">y2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E32:41_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">w1</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">y2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">y1</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">y2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E33:41_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">w1</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">w2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">y1</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">y2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E34:41_2_1_1"/><b>then </b>
<font color="Maroon">w</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">y1</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">y2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E35:41_2_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">y1</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">y2</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="measure6.html#T13">MEASURE6:13</a></i>;<br/>
<a NAME="E36:41_2_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">y1</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">y2</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Olive">y</font> </span>}</span> 
 
;<br/>
<a NAME="E37:41_2_1_1"/><b>then </b><i><font color="Green">E136</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">y1</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">y2</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="convfun1.html#K6" target="_self">epigraph</a> <font color="Maroon">f</font>
 
;<br/>
<a NAME="E38:41_2_1_1"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">y1</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E39:41_2_1_1"/>
<span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">y2</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E40:41_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="convfun1.html#K6" target="_self">epigraph</a> <font color="Maroon">f</font>
 <b>by </b><i>, , </i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E3:41_2"/><b>then </b>
 <a href="convfun1.html#K6" target="_self">epigraph</a> <font color="Maroon">f</font> is <a href="convex1.html#V1">convex</a>
 
<b>by </b><i><a class="ref" href="convex1.html#D2">CONVEX1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:41_2"/>
<font color="Maroon">f</font> is <a href="convfun1.html#V1" target="_self">convex</a>
 <b>by </b><i/>;<br/>


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


</div>
