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

<span class="kw">let </span><font color="Maroon" title="c1">C</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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 <font color="Maroon" title="c1">C</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  holds  <span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Olive" title="b1">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Olive" title="b1">f</font></span>)</span></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 <font color="Maroon" title="c1">C</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c2">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c2">f</font></span>)</span></span><br/>



<a NAME="E1:28"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc2.html#T19" target="_self" title="MESFUNC2:th.19">Th19</a></span>;<br/>
<a NAME="E2:28"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c2">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a></span>;<br/>
<a NAME="E3:28"/><span class="lab"><font color="Green" title="E21">A3</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 <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> holds <br/><span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c2">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> implies <span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c2">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:28_1"/><span class="lab"><font color="Green" title="E22">A4</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c2">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font></span><br/>

<div><span class="lab"><font color="Green" title="E23">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:28_1_1_1"/>
( <span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font> or <span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc2.html#T20" target="_self" title="MESFUNC2:th.20">Th20</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:28_1_1_1_1"/><span class="lab"><font color="Green" title="E24">A6</font></span>: 
<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c2">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font></span><br/><div class="add"><span class="lab"><font color="Green" title="E25">A7</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font></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="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_1_1_1_1"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="ref" href="mesfunc2.html#T21" target="_self" title="MESFUNC2:th.21">Th21</a></span>
<br/>.= 
<font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T4" title="XXREAL_3:th.4">XXREAL_3:4</a></span>
;<br/><a NAME="E3:28_1_1_1_1"/><span class="lab"><font color="Green" title="E26">A8</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_1"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="ref" href="mesfunc2.html#D2" target="_self" title="MESFUNC2:def.2">Def2</a></span>;<br/><a NAME="E4:28_1_1_1_1"/><span class="lab"><font color="Green" title="E27">A9</font></span>: 
<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="extreal2.html#K1" title="EXTREAL2:func.1">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font></span>)</span>,<a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28_1_1_1_1"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="ref" href="mesfunc2.html#D2" target="_self" title="MESFUNC2:def.2">Def2</a></span>;<br/><a NAME="E5:28_1_1_1_1"/><span class="lab"><font color="Green" title="E28">A10</font></span>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_1_1_1_1"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="txt" href="#E4:28_1_1_1_1"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="ref" href="xxreal_0.html#D10" title="XXREAL_0:def.10">XXREAL_0:def 10</a></span>;<br/><span class="lab"><font color="Green" title="E29">A11</font></span>: <font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font> = 
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:28_1_1_1_1"><span class="lab"><font color="Green" title="E28">A10</font></span></a>, <a class="ref" href="extreal1.html#D3" title="EXTREAL1:def.3">EXTREAL1:def 3</a></span>
<br/>.= 
<span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c2">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E1:28_1"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a></span>
;<br/><span class="kw">thus </span><a NAME="E7:28_1_1_1_1"/>
<span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c2">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E1:28_1"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E2:28_1_1_1_1"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="txt" href="#E6:28_1_1_1_1"><span class="lab"><font color="Green" title="E29">A11</font></span></a>, <a class="ref" href="mesfunc1.html#D3" title="MESFUNC1:def.3">MESFUNC1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:28_1_1_1_2"/><span class="lab"><font color="Green" title="E24">A12</font></span>: 
<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c2">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font></span><br/><div class="add"><span class="lab"><font color="Green" title="E25">A13</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font></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="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font></span>)</span> = 
<a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_1"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E1:28_1_1_1_2"><span class="lab"><font color="Green" title="E24">A12</font></span></a>, <a class="ref" href="mesfunc2.html#T22" target="_self" title="MESFUNC2:th.22">Th22</a></span>
<br/>.= 
 <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T4" title="XXREAL_3:th.4">XXREAL_3:4</a></span>
;<br/><a NAME="E3:28_1_1_1_2"/><span class="lab"><font color="Green" title="E26">A14</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_1"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="ref" href="mesfunc2.html#D2" target="_self" title="MESFUNC2:def.2">Def2</a></span>;<br/><a NAME="E4:28_1_1_1_2"/><span class="lab"><font color="Green" title="E27">A15</font></span>: 
<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="extreal2.html#K1" title="EXTREAL2:func.1">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font></span>)</span>,<a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28_1_1_1_2"><span class="lab"><font color="Green" title="E26">A14</font></span></a>, <a class="ref" href="mesfunc2.html#D2" target="_self" title="MESFUNC2:def.2">Def2</a></span>;<br/><a NAME="E5:28_1_1_1_2"/><span class="lab"><font color="Green" title="E28">A16</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_1_1_1_2"><span class="lab"><font color="Green" title="E24">A12</font></span></a>, <a class="txt" href="#E4:28_1_1_1_2"><span class="lab"><font color="Green" title="E27">A15</font></span></a>, <a class="ref" href="xxreal_0.html#D10" title="XXREAL_0:def.10">XXREAL_0:def 10</a></span>;<br/><span class="lab"><font color="Green" title="E29">A17</font></span>:  <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font></span>)</span> = 
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:28_1_1_1_2"><span class="lab"><font color="Green" title="E28">A16</font></span></a>, <a class="ref" href="extreal2.html#T55" title="EXTREAL2:th.55">EXTREAL2:55</a></span>
<br/>.= 
<span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c2">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E1:28_1"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a></span>
;<br/><span class="kw">thus </span><a NAME="E7:28_1_1_1_2"/>
<span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c2">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E1:28_1"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E2:28_1_1_1_2"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="txt" href="#E6:28_1_1_1_2"><span class="lab"><font color="Green" title="E29">A17</font></span></a>, <a class="ref" href="mesfunc1.html#D3" title="MESFUNC1:def.3">MESFUNC1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E3:28_1"/>
<span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c2">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28_1"><span class="lab"><font color="Green" title="E23">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E4:28"/>
<span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c2">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
