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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/> (  <a href="supinf_2.html#K11" title="SUPINF_2:func.11">inf</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K14" title="SUPINF_2:func.14">-</a> <font color="Olive" title="b2">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <font color="Olive" title="b2">F</font></span>)</span> &amp;  <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="Olive" title="b2">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="Olive" title="b2">F</font></span>)</span> )</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/> (  <a href="supinf_2.html#K11" title="SUPINF_2:func.11">inf</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K14" title="SUPINF_2:func.14">-</a> <font color="Olive" title="b1">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <font color="Olive" title="b1">F</font></span>)</span> &amp;  <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="Olive" title="b1">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="Olive" title="b1">F</font></span>)</span> )</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"> (  <a href="supinf_2.html#K11" title="SUPINF_2:func.11">inf</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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <font color="Maroon" title="c3">F</font></span>)</span> &amp;  <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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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> )</span><br/>





<span class="kw">thus </span> <a href="supinf_2.html#K11" title="SUPINF_2:func.11">inf</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="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K6" title="SUPINF_2:func.6">-</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K9" title="SUPINF_2:func.9">rng</a> <font color="Maroon" title="c3">F</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T37" target="_self" title="SUPINF_2:th.37">Th37</a></span>
<br/>.= 
 <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <font color="Maroon" title="c3">F</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T32" target="_self" title="SUPINF_2:th.32">Th32</a></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#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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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></span><br/>

<span class="kw">thus </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="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K6" title="SUPINF_2:func.6">-</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K9" title="SUPINF_2:func.9">rng</a> <font color="Maroon" title="c3">F</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T37" target="_self" title="SUPINF_2:th.37">Th37</a></span>
<br/>.= 
 <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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T33" target="_self" title="SUPINF_2:th.33">Th33</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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