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<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> <br/> for <font color="Olive" title="b3">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><br/> for <font color="Olive" title="b4">G</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> holds   <a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">+</a> <font color="Olive" title="b4">G</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <font color="Olive" title="b3">F</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="Olive" title="b4">G</font></span>)</span></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"> 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><br/> for <font color="Olive" title="b2">G</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> holds   <a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">+</a> <font color="Olive" title="b2">G</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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> <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="Olive" title="b2">G</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">G</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> holds   <a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">+</a> <font color="Olive" title="b1">G</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <font color="Maroon" title="c4">F</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="Olive" title="b1">G</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c5">G</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">  <a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">+</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <font color="Maroon" title="c4">F</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">G</font></span>)</span></span><br/>







<a NAME="E1:45"/><i><font color="Green" title="E34">A3</font></i>: 
 <a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">+</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K9" title="SUPINF_2:func.9">rng</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K9" title="SUPINF_2:func.9">rng</a> <font color="Maroon" title="c5">G</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T34" target="_self" title="SUPINF_2:th.34">Th34</a>, <a class="ref" href="xxreal_2.html#T59" title="XXREAL_2:th.59">XXREAL_2:59</a></i>;<br/>
<a NAME="E2:45"/>
 <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K9" title="SUPINF_2:func.9">rng</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K9" title="SUPINF_2:func.9">rng</a> <font color="Maroon" title="c5">G</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K9" title="SUPINF_2:func.9">rng</a> <font color="Maroon" title="c4">F</font></span>)</span></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#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K9" title="SUPINF_2:func.9">rng</a> <font color="Maroon" title="c5">G</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T26" target="_self" title="SUPINF_2:th.26">Th26</a></i>;<br/>
<b>hence </b><a NAME="E3:45"/>
 <a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">+</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <font color="Maroon" title="c4">F</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">G</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:45"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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