<?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>, <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> <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#V2" title="SUPINF_2:attr.2">bounded_below</a>  &amp; <font color="Olive" title="b4">F2</font> is  <a href="supinf_2.html#V2" title="SUPINF_2:attr.2">bounded_below</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#V2" title="SUPINF_2:attr.2">bounded_below</a> </span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#V2" title="SUPINF_2:attr.2">bounded_below</a>  &amp; <font color="Olive" title="b2">F2</font> is  <a href="supinf_2.html#V2" title="SUPINF_2:attr.2">bounded_below</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#V2" title="SUPINF_2:attr.2">bounded_below</a> </span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#V2" title="SUPINF_2:attr.2">bounded_below</a>  &amp; <font color="Olive" title="b1">F2</font> is  <a href="supinf_2.html#V2" title="SUPINF_2:attr.2">bounded_below</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#V2" title="SUPINF_2:attr.2">bounded_below</a> </span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">F1</font> is  <a href="supinf_2.html#V2" title="SUPINF_2:attr.2">bounded_below</a>  &amp; <font color="Maroon" title="c5">F2</font> is  <a href="supinf_2.html#V2" title="SUPINF_2:attr.2">bounded_below</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#V2" title="SUPINF_2:attr.2">bounded_below</a>  )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:63"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
<font color="Maroon" title="c4">F1</font> is  <a href="supinf_2.html#V2" title="SUPINF_2:attr.2">bounded_below</a> 
 <span class="kw">and </span><br/><a NAME="E2:63"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
<font color="Maroon" title="c5">F2</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"> <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#V2" title="SUPINF_2:attr.2">bounded_below</a> </span><br/>

<a NAME="E3:63"/><span class="lab"><font color="Green" title="E30">A3</font></span>: 
not  <a href="supinf_2.html#K11" title="SUPINF_2:func.11">inf</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="supinf_2.html#D12" target="_self" title="SUPINF_2:def.12">Def12</a></span>;<br/>
<a NAME="E4:63"/><span class="lab"><font color="Green" title="E31">A4</font></span>: 
(  <a href="supinf_2.html#K11" title="SUPINF_2:func.11">inf</a> <font color="Maroon" title="c4">F1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp;  <a href="supinf_2.html#K11" title="SUPINF_2:func.11">inf</a> <font color="Maroon" title="c5">F2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  implies  <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> <span class="p1">(<span class="default"><a href="supinf_2.html#K11" title="SUPINF_2:func.11">inf</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#K11" title="SUPINF_2:func.11">inf</a> <font color="Maroon" title="c5">F2</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c6">a</font> =  <a href="supinf_2.html#K11" title="SUPINF_2:func.11">inf</a> <font color="Maroon" title="c4">F1</font>, <font color="Maroon" title="c7">b</font> =  <a href="supinf_2.html#K11" title="SUPINF_2:func.11">inf</a> <font color="Maroon" title="c5">F2</font> as   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> ;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:63_1"/><span class="lab"><font color="Green" title="E32">A5</font></span>: 
 <a href="supinf_2.html#K11" title="SUPINF_2:func.11">inf</a> <font color="Maroon" title="c4">F1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <span class="kw">and </span><br/><a NAME="E3:63_1"/><span class="lab"><font color="Green" title="E33">A6</font></span>: 
 <a href="supinf_2.html#K11" title="SUPINF_2:func.11">inf</a> <font color="Maroon" title="c5">F2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="Red">::  thesis: </font></span></a><span class="hide">  <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> <span class="p1">(<span class="default"><a href="supinf_2.html#K11" title="SUPINF_2:func.11">inf</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#K11" title="SUPINF_2:func.11">inf</a> <font color="Maroon" title="c5">F2</font></span>)</span></span><br/>

<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63_1"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="txt" href="#E3:63_1"><span class="lab"><font color="Green" title="E33">A6</font></span></a></span>;<br/>
<a NAME="E5:63_1"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K11" title="SUPINF_2:func.11">inf</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#K11" title="SUPINF_2:func.11">inf</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#D2" title="XXREAL_3:def.2">XXREAL_3:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:63_1"/>
 <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> <span class="p1">(<span class="default"><a href="supinf_2.html#K11" title="SUPINF_2:func.11">inf</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#K11" title="SUPINF_2:func.11">inf</a> <font color="Maroon" title="c5">F2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T12" title="XXREAL_0:th.12">XXREAL_0:12</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>
<a NAME="E5:63"/><span class="lab"><font color="Green" title="E32">A7</font></span>: 
(  <a href="supinf_2.html#K11" title="SUPINF_2:func.11">inf</a> <font color="Maroon" title="c4">F1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp;  <a href="supinf_2.html#K11" title="SUPINF_2:func.11">inf</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>  implies  <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> <span class="p1">(<span class="default"><a href="supinf_2.html#K11" title="SUPINF_2:func.11">inf</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#K11" title="SUPINF_2:func.11">inf</a> <font color="Maroon" title="c5">F2</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T7" title="XXREAL_0:th.7">XXREAL_0:7</a>, <a class="ref" href="xxreal_3.html#D2" title="XXREAL_3:def.2">XXREAL_3:def 2</a></span>;<br/>
<a NAME="E6:63"/><span class="lab"><font color="Green" title="E33">A8</font></span>: 
(  <a href="supinf_2.html#K11" title="SUPINF_2:func.11">inf</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;  <a href="supinf_2.html#K11" title="SUPINF_2:func.11">inf</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>  implies  <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> <span class="p1">(<span class="default"><a href="supinf_2.html#K11" title="SUPINF_2:func.11">inf</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#K11" title="SUPINF_2:func.11">inf</a> <font color="Maroon" title="c5">F2</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T7" title="XXREAL_0:th.7">XXREAL_0:7</a>, <a class="ref" href="xxreal_3.html#D2" title="XXREAL_3:def.2">XXREAL_3:def 2</a></span>;<br/>
<a NAME="E7:63"/><span class="lab"><font color="Green" title="E34">A9</font></span>: 
(  <a href="supinf_2.html#K11" title="SUPINF_2:func.11">inf</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;  <a href="supinf_2.html#K11" title="SUPINF_2:func.11">inf</a> <font color="Maroon" title="c5">F2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  implies  <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> <span class="p1">(<span class="default"><a href="supinf_2.html#K11" title="SUPINF_2:func.11">inf</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#K11" title="SUPINF_2:func.11">inf</a> <font color="Maroon" title="c5">F2</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T7" title="XXREAL_0:th.7">XXREAL_0:7</a>, <a class="ref" href="xxreal_3.html#D2" title="XXREAL_3:def.2">XXREAL_3:def 2</a></span>;<br/>
<a NAME="E8:63"/><span class="lab"><font color="Green" title="E35">A10</font></span>: 
not  <a href="supinf_2.html#K11" title="SUPINF_2:func.11">inf</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="ref" href="supinf_2.html#D12" target="_self" title="SUPINF_2:def.12">Def12</a></span>;<br/>
<a NAME="E9:63"/>
 <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> <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:63_2"/>
 <a href="supinf_2.html#K11" title="SUPINF_2:func.11">inf</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#K2" title="SUPINF_1:func.2">-infty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:63_2"/><span class="kw">then </span>
(  not  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K11" title="SUPINF_2:func.11">inf</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> or  <a href="supinf_2.html#K11" title="SUPINF_2:func.11">inf</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#K2" title="SUPINF_1:func.2">-infty</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:63_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:63"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="txt" href="#E3:63"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="txt" href="#E4:63"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="txt" href="#E5:63"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="txt" href="#E7:63"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="txt" href="#E6:63"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="ref" href="supinf_2.html#T36" target="_self" title="SUPINF_2:th.36">Th36</a>, <a class="ref" href="xxreal_0.html#T6" title="XXREAL_0:th.6">XXREAL_0:6</a>, <a class="ref" href="xxreal_3.html#D1" title="XXREAL_3:def.1">XXREAL_3:def 1</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">hence </span><a NAME="E10:63"/>
<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#V2" title="SUPINF_2:attr.2">bounded_below</a> 
 <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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