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

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">Y</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> <br/> for <font color="Olive" title="b2">F</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> holds <br/> ( <font color="Olive" title="b2">F</font> is  <a href="supinf_2.html#V2" title="SUPINF_2:attr.2">bounded_below</a>  iff  <a href="supinf_2.html#K14" title="SUPINF_2:func.14">-</a> <font color="Olive" title="b2">F</font> is  <a href="supinf_2.html#V1" title="SUPINF_2:attr.1">bounded_above</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">Y</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F</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> holds <br/> ( <font color="Olive" title="b1">F</font> is  <a href="supinf_2.html#V2" title="SUPINF_2:attr.2">bounded_below</a>  iff  <a href="supinf_2.html#K14" title="SUPINF_2:func.14">-</a> <font color="Olive" title="b1">F</font> is  <a href="supinf_2.html#V1" title="SUPINF_2:attr.1">bounded_above</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">F</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">F</font> is  <a href="supinf_2.html#V2" title="SUPINF_2:attr.2">bounded_below</a>  iff  <a href="supinf_2.html#K14" title="SUPINF_2:func.14">-</a> <font color="Maroon" title="c3">F</font> is  <a href="supinf_2.html#V1" title="SUPINF_2:attr.1">bounded_above</a>  )</span><br/>





<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="supinf_2.html#K14" title="SUPINF_2:func.14">-</a> <font color="Maroon" title="c3">F</font> is  <a href="supinf_2.html#V1" title="SUPINF_2:attr.1">bounded_above</a>  implies <font color="Maroon" title="c3">F</font> is  <a href="supinf_2.html#V2" title="SUPINF_2:attr.2">bounded_below</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:55_1"/>
<font color="Maroon" title="c3">F</font> is  <a href="supinf_2.html#V2" title="SUPINF_2:attr.2">bounded_below</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="supinf_2.html#K14" title="SUPINF_2:func.14">-</a> <font color="Maroon" title="c3">F</font> is  <a href="supinf_2.html#V1" title="SUPINF_2:attr.1">bounded_above</a> </span><br/><a NAME="E2:55_1"/><span class="kw">then </span>
 <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K11" title="SUPINF_2:func.11">inf</a> <font color="Maroon" title="c3">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#D12" target="_self" title="SUPINF_2:def.12">Def12</a></span>;<br/><a NAME="E3:55_1"/><span class="kw">then </span>
 <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K11" title="SUPINF_2:func.11">inf</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T40" title="XXREAL_3:th.40">XXREAL_3:40</a></span>;<br/><a NAME="E4:55_1"/><span class="kw">then </span>
 <a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K14" title="SUPINF_2:func.14">-</a> <font color="Maroon" title="c3">F</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 class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T38" target="_self" title="SUPINF_2:th.38">Th38</a>, <a class="ref" href="xxreal_3.html#T5" title="XXREAL_3:th.5">XXREAL_3:5</a></span>;<br/><span class="kw">hence </span><a NAME="E5:55_1"/>
 <a href="supinf_2.html#K14" title="SUPINF_2:func.14">-</a> <font color="Maroon" title="c3">F</font> is  <a href="supinf_2.html#V1" title="SUPINF_2:attr.1">bounded_above</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#D11" target="_self" title="SUPINF_2:def.11">Def11</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">assume </span><a NAME="E2:55"/>
 <a href="supinf_2.html#K14" title="SUPINF_2:func.14">-</a> <font color="Maroon" title="c3">F</font> is  <a href="supinf_2.html#V1" title="SUPINF_2:attr.1">bounded_above</a> 
 ; <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">F</font> is  <a href="supinf_2.html#V2" title="SUPINF_2:attr.2">bounded_below</a> </span><br/>

<a NAME="E3:55"/><span class="kw">then </span>
 <a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K14" title="SUPINF_2:func.14">-</a> <font color="Maroon" title="c3">F</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 class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#D11" target="_self" title="SUPINF_2:def.11">Def11</a></span>;<br/>
<a NAME="E4:55"/><span class="kw">then </span>
 <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K11" title="SUPINF_2:func.11">inf</a> <font color="Maroon" title="c3">F</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 class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T38" target="_self" title="SUPINF_2:th.38">Th38</a></span>;<br/>
<a NAME="E5:55"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A1</font></span>: 
 <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</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"><a href="supinf_2.html#K11" title="SUPINF_2:func.11">inf</a> <font color="Maroon" title="c3">F</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T40" title="XXREAL_3:th.40">XXREAL_3:40</a></span>;<br/>
<a NAME="E6:55"/>
 <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#D3" title="XXREAL_3:def.3">XXREAL_3:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:55"/>
<font color="Maroon" title="c3">F</font> is  <a href="supinf_2.html#V2" title="SUPINF_2:attr.2">bounded_below</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:55"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="supinf_2.html#D12" target="_self" title="SUPINF_2:def.12">Def12</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>
