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<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#V3" title="SUPINF_2:attr.3">bounded</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#V3" title="SUPINF_2:attr.3">bounded</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#V3" title="SUPINF_2:attr.3">bounded</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#V3" title="SUPINF_2:attr.3">bounded</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#V3" title="SUPINF_2:attr.3">bounded</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#V3" title="SUPINF_2:attr.3">bounded</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#V3" title="SUPINF_2:attr.3">bounded</a>  implies <font color="Maroon" title="c3">F</font> is  <a href="supinf_2.html#V3" title="SUPINF_2:attr.3">bounded</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:56_1"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
<font color="Maroon" title="c3">F</font> is  <a href="supinf_2.html#V3" title="SUPINF_2:attr.3">bounded</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#V3" title="SUPINF_2:attr.3">bounded</a> </span><br/><a NAME="E2:56_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A2</font></span>: 
 <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#V2" title="SUPINF_2:attr.2">bounded_below</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T40" target="_self" title="SUPINF_2:th.40">Th40</a></span>;<br/><a NAME="E3:56_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="txt" href="#E1:56_1"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="supinf_2.html#T41" target="_self" title="SUPINF_2:th.41">Th41</a></span>;<br/><span class="kw">hence </span><a NAME="E4:56_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#V3" title="SUPINF_2:attr.3">bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_1"><span class="lab"><font color="Green" title="E24">A2</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">assume </span><a NAME="E2:56"/><span class="lab"><font color="Green" title="E23">A3</font></span>: 
 <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#V3" title="SUPINF_2:attr.3">bounded</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#V3" title="SUPINF_2:attr.3">bounded</a> </span><br/>

<a NAME="E3:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A4</font></span>: 
<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="ref" href="supinf_2.html#T41" target="_self" title="SUPINF_2:th.41">Th41</a></span>;<br/>
<a NAME="E4:56"/>
<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="txt" href="#E2:56"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="ref" href="supinf_2.html#T40" target="_self" title="SUPINF_2:th.40">Th40</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:56"/>
<font color="Maroon" title="c3">F</font> is  <a href="supinf_2.html#V3" title="SUPINF_2:attr.3">bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56"><span class="lab"><font color="Green" title="E24">A4</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/>


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