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

<b>let </b><font color="Maroon" title="c1">X</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Y</font>, <font color="Olive" title="b2">Z</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>   st <font color="Olive" title="b1">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  &amp; <font color="Olive" title="b2">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  holds <br/>for <font color="Olive" title="b3">F1</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">Y</font><br/> for <font color="Olive" title="b4">F2</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b2">Z</font>  st <font color="Olive" title="b3">F1</font> is <a href="supinf_2.html#V1" title="SUPINF_2:attr.1">bounded_above</a> &amp; <font color="Olive" title="b4">F2</font> is <a href="supinf_2.html#V1" title="SUPINF_2:attr.1">bounded_above</a> holds <br/><font color="Olive" title="b3">F1</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">+</a> <font color="Olive" title="b4">F2</font> is <a href="supinf_2.html#V1" title="SUPINF_2:attr.1">bounded_above</a></span><br/><b>let </b><font color="Maroon" title="c2">Y</font>, <font color="Maroon" title="c3">Z</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  &amp; <font color="Maroon" title="c3">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  implies for <font color="Olive" title="b1">F1</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font><br/> for <font color="Olive" title="b2">F2</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">Z</font>  st <font color="Olive" title="b1">F1</font> is <a href="supinf_2.html#V1" title="SUPINF_2:attr.1">bounded_above</a> &amp; <font color="Olive" title="b2">F2</font> is <a href="supinf_2.html#V1" title="SUPINF_2:attr.1">bounded_above</a> holds <br/><font color="Olive" title="b1">F1</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">+</a> <font color="Olive" title="b2">F2</font> is <a href="supinf_2.html#V1" title="SUPINF_2:attr.1">bounded_above</a> )</span><br/>



<b>assume </b><a NAME="E1:67"/><i><font color="Green" title="E51">A1</font></i>: 
( <font color="Maroon" title="c2">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  &amp; <font color="Maroon" title="c3">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F1</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font><br/> for <font color="Olive" title="b2">F2</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">Z</font>  st <font color="Olive" title="b1">F1</font> is <a href="supinf_2.html#V1" title="SUPINF_2:attr.1">bounded_above</a> &amp; <font color="Olive" title="b2">F2</font> is <a href="supinf_2.html#V1" title="SUPINF_2:attr.1">bounded_above</a> holds <br/><font color="Olive" title="b1">F1</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">+</a> <font color="Olive" title="b2">F2</font> is <a href="supinf_2.html#V1" title="SUPINF_2:attr.1">bounded_above</a></span><br/>

<b>let </b><font color="Maroon" title="c4">F1</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F2</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">Z</font>  st <font color="Maroon" title="c4">F1</font> is <a href="supinf_2.html#V1" title="SUPINF_2:attr.1">bounded_above</a> &amp; <font color="Olive" title="b1">F2</font> is <a href="supinf_2.html#V1" title="SUPINF_2:attr.1">bounded_above</a> holds <br/><font color="Maroon" title="c4">F1</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">+</a> <font color="Olive" title="b1">F2</font> is <a href="supinf_2.html#V1" title="SUPINF_2:attr.1">bounded_above</a></span><br/><b>let </b><font color="Maroon" title="c5">F2</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">Z</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">F1</font> is <a href="supinf_2.html#V1" title="SUPINF_2:attr.1">bounded_above</a> &amp; <font color="Maroon" title="c5">F2</font> is <a href="supinf_2.html#V1" title="SUPINF_2:attr.1">bounded_above</a> implies <font color="Maroon" title="c4">F1</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">+</a> <font color="Maroon" title="c5">F2</font> is <a href="supinf_2.html#V1" title="SUPINF_2:attr.1">bounded_above</a> )</span><br/>



<b>assume </b><a NAME="E2:67"/><i><font color="Green" title="E52">A2</font></i>: 
( <font color="Maroon" title="c4">F1</font> is <a href="supinf_2.html#V1" title="SUPINF_2:attr.1">bounded_above</a> &amp; <font color="Maroon" title="c5">F2</font> is <a href="supinf_2.html#V1" title="SUPINF_2:attr.1">bounded_above</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">F1</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">+</a> <font color="Maroon" title="c5">F2</font> is <a href="supinf_2.html#V1" title="SUPINF_2:attr.1">bounded_above</a></span><br/>

<a NAME="E3:67"/><i><font color="Green" title="E53">A3</font></i>: 
( not <!--HUMANRECFAILED-->(  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> &amp;  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z</font> ) &amp; not <!--HUMANRECFAILED-->(  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> &amp;  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z</font> ) )
 
<b>by </b><i><a class="txt" href="supinf_2.html#E1:67"><i><font color="Green" title="E51">A1</font></i></a>, <a class="ref" href="supinf_1.html#T1" title="SUPINF_1:th.1">SUPINF_1:1</a></i>;<br/>
<a NAME="E4:67"/><i><font color="Green" title="E54">A4</font></i>: 
( not  <a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <font color="Maroon" title="c4">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; not  <a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <font color="Maroon" title="c5">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 
<b>by </b><i><a class="txt" href="supinf_2.html#E2:67"><i><font color="Green" title="E52">A2</font></i></a>, <a class="ref" href="supinf_2.html#D11" target="_self" title="SUPINF_2:def.11">Def11</a></i>;<br/>
<a NAME="E5:67"/><i><font color="Green" title="E55">A5</font></i>: 
(  <a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <font color="Maroon" title="c4">F1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  &amp;  <a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <font color="Maroon" title="c5">F2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  implies <span class="p1">(<span class="default"><a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1"/><i><font color="Green" title="E56">A6</font></i>: 
(  <a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <font color="Maroon" title="c4">F1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  &amp;  <a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <font color="Maroon" title="c5">F2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><br/>

<b>reconsider </b><font color="Maroon" title="c6">a</font> =  <a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <font color="Maroon" title="c4">F1</font>, <font color="Maroon" title="c7">b</font> =  <a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <font color="Maroon" title="c5">F2</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c8">a</font> = <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c9">b</font> = <font color="Maroon" title="c7">b</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="supinf_2.html#E1:67_1"><i><font color="Green" title="E56">A6</font></i></a></i>;<br/>
<a NAME="E4:67_1"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c9">b</font>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T1" target="_self" title="SUPINF_2:th.1">Th1</a></i>;<br/>
<b>hence </b><a NAME="E5:67_1"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="ref" href="xxreal_0.html#T9" title="XXREAL_0:th.9">XXREAL_0:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E6:67"/><i><font color="Green" title="E56">A7</font></i>: 
(  <a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <font color="Maroon" title="c4">F1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  &amp;  <a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <font color="Maroon" title="c5">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  implies <span class="p1">(<span class="default"><a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 
<b>by </b><i><a class="ref" href="supinf_2.html#D2" target="_self" title="SUPINF_2:def.2">Def2</a>, <a class="ref" href="supinf_1.html#T1" title="SUPINF_1:th.1">SUPINF_1:1</a>, <a class="ref" href="xxreal_0.html#T7" title="XXREAL_0:th.7">XXREAL_0:7</a></i>;<br/>
<a NAME="E7:67"/><i><font color="Green" title="E57">A8</font></i>: 
(  <a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <font color="Maroon" title="c4">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp;  <a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <font color="Maroon" title="c5">F2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  implies <span class="p1">(<span class="default"><a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 
<b>by </b><i><a class="ref" href="supinf_2.html#D2" target="_self" title="SUPINF_2:def.2">Def2</a>, <a class="ref" href="supinf_1.html#T1" title="SUPINF_1:th.1">SUPINF_1:1</a>, <a class="ref" href="xxreal_0.html#T7" title="XXREAL_0:th.7">XXREAL_0:7</a></i>;<br/>
<a NAME="E8:67"/><i><font color="Green" title="E58">A9</font></i>: 
(  <a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <font color="Maroon" title="c4">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp;  <a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <font color="Maroon" title="c5">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  implies <span class="p1">(<span class="default"><a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 
<b>by </b><i><a class="ref" href="supinf_2.html#D2" target="_self" title="SUPINF_2:def.2">Def2</a>, <a class="ref" href="xxreal_0.html#T7" title="XXREAL_0:th.7">XXREAL_0:7</a></i>;<br/>
<a NAME="E9:67"/>
 <a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">+</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_2"/>
not  <a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">+</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:67_2"/><b>then </b>
(  not  <a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">+</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  or  <a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">+</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<b>hence </b><a NAME="E3:67_2"/>
contradiction
 <b>by </b><i><a class="txt" href="supinf_2.html#E3:67"><i><font color="Green" title="E53">A3</font></i></a>, <a class="txt" href="supinf_2.html#E4:67"><i><font color="Green" title="E54">A4</font></i></a>, <a class="txt" href="supinf_2.html#E5:67"><i><font color="Green" title="E55">A5</font></i></a>, <a class="txt" href="supinf_2.html#E6:67"><i><font color="Green" title="E56">A7</font></i></a>, <a class="txt" href="supinf_2.html#E7:67"><i><font color="Green" title="E57">A8</font></i></a>, <a class="txt" href="supinf_2.html#E8:67"><i><font color="Green" title="E58">A9</font></i></a>, <a class="ref" href="supinf_2.html#T35" target="_self" title="SUPINF_2:th.35">Th35</a>, <a class="ref" href="supinf_1.html#D7" title="SUPINF_1:def.7">SUPINF_1:def 7</a>, <a class="ref" href="xxreal_0.html#T4" title="XXREAL_0:th.4">XXREAL_0:4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:67"/>
<font color="Maroon" title="c4">F1</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">+</a> <font color="Maroon" title="c5">F2</font> is <a href="supinf_2.html#V1" title="SUPINF_2:attr.1">bounded_above</a>
 <b>by </b><i><a class="ref" href="supinf_2.html#D11" target="_self" title="SUPINF_2:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
