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

<span class="kw">let </span><font color="Maroon" title="c1">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><br/>  for <font color="Olive" title="b2">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">f</font> <a href="rfunct_4.html#R1" title="RFUNCT_4:pred.1">is_strictly_convex_on</a> <font color="Olive" title="b2">X</font> iff <font color="Olive" title="b1">f</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c1">r</font> <a href="rfunct_4.html#R1" title="RFUNCT_4:pred.1">is_strictly_convex_on</a> <font color="Olive" title="b2">X</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Maroon" title="c2">f</font> <a href="rfunct_4.html#R1" title="RFUNCT_4:pred.1">is_strictly_convex_on</a> <font color="Olive" title="b1">X</font> iff <font color="Maroon" title="c2">f</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c1">r</font> <a href="rfunct_4.html#R1" title="RFUNCT_4:pred.1">is_strictly_convex_on</a> <font color="Olive" title="b1">X</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> <a href="rfunct_4.html#R1" title="RFUNCT_4:pred.1">is_strictly_convex_on</a> <font color="Maroon" title="c3">X</font> iff <font color="Maroon" title="c2">f</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c1">r</font> <a href="rfunct_4.html#R1" title="RFUNCT_4:pred.1">is_strictly_convex_on</a> <font color="Maroon" title="c3">X</font> )</span><br/>





<a NAME="E1:13"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T3" title="VALUED_1:th.3">VALUED_1:3</a></span>;<br/>
<a NAME="E2:13"/><span class="lab"><font color="Green" title="E9">A2</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c1">r</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c1">r</font></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:13_1"/><span class="lab"><font color="Green" title="E10">A3</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c1">r</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span><br/>

<span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c1">r</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T3" title="VALUED_1:th.3">VALUED_1:3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c1">r</font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c1">r</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="txt" href="#E1:13_1"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="ref" href="valued_1.html#T3" title="VALUED_1:th.3">VALUED_1:3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c1">r</font></span>)</span>

;<br/>
<span class="kw">hence </span><a NAME="E3:13_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:13"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c1">r</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T3" title="VALUED_1:th.3">VALUED_1:3</a></span>;<br/>
<a NAME="E4:13"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="txt" href="#E2:13"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="ref" href="partfun1.html#T5" title="PARTFUN1:th.5">PARTFUN1:5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:13"/>
( <font color="Maroon" title="c2">f</font> <a href="rfunct_4.html#R1" title="RFUNCT_4:pred.1">is_strictly_convex_on</a> <font color="Maroon" title="c3">X</font> iff <font color="Maroon" title="c2">f</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c1">r</font> <a href="rfunct_4.html#R1" title="RFUNCT_4:pred.1">is_strictly_convex_on</a> <font color="Maroon" title="c3">X</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="rfunct_4.html#E12"><span class="lab"><font color="Green" title="E7">Lm1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
