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<div class="add">

<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> ;<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#K6" title="NUMBERS:func.6">ExtREAL</a> ;<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>;<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>;<br/>







<a NAME="E1:48"/>
<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> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<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="p1">(<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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>proof </b></a><div class="add">

<a NAME="E1:48_1"/><i><font color="Green" title="E38">A1</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:48_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<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="p1">(<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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:48_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c7">x</font> = <font color="Maroon" title="c6">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c8">a</font> = <font color="Maroon" title="c4">F</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c9">b</font> = <font color="Maroon" title="c5">G</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> ;<br/>
<a NAME="E4:48_1_1"/><i><font color="Green" title="E39">A2</font></i>: 
<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="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c9">b</font>
 
<b>by </b><i><a class="ref" href="supinf_2.html#D9" target="_self" title="SUPINF_2:def.9">Def9</a></i>;<br/>
<a NAME="E5:48_1_1"/><i><font color="Green" title="E40">A3</font></i>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K9" title="SUPINF_2:func.9">rng</a> <font color="Maroon" title="c4">F</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></i>;<br/>
<a NAME="E6:48_1_1"/>
<font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K9" title="SUPINF_2:func.9">rng</a> <font color="Maroon" title="c5">G</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></i>;<br/>
<b>hence </b><a NAME="E7:48_1_1"/>
<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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<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="p1">(<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>
 <b>by </b><i><a class="txt" href="supinf_2.html#E4:48_1_1"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="supinf_2.html#E5:48_1_1"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="supinf_2.html#D5" target="_self" title="SUPINF_2:def.5">Def5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:48_1"/>
<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> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<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="p1">(<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>
 <b>by </b><i><a class="txt" href="supinf_2.html#E1:48_1"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:48"/>
 <a href="supinf_2.html#K9" title="SUPINF_2:func.9">rng</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="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<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="p1">(<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>
 <b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>


</div>
