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<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">Y</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K6">ExtREAL</a> , <font color="Maroon">Z</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">F</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">G</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Z</font>;<br/>







<a NAME="E1:47"/>
<font color="Maroon">F</font> <a href="supinf_2.html#K11" target="_self">+</a> <font color="Maroon">G</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K5" target="_self">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">G</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<a NAME="E1:47_1"/><i><font color="Green">E24</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="supinf_2.html#K11" target="_self">+</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:47_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="supinf_2.html#K11" target="_self">+</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K5" target="_self">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">G</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:47_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> ;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">F</font> <a href="supinf_2.html#K10" target="_self">.</a> <font color="Maroon">x</font>, <font color="Maroon">b</font> = <font color="Maroon">G</font> <a href="supinf_2.html#K10" target="_self">.</a> <font color="Maroon">x</font> as   <a href="supinf_1.html#NM3">R_eal</a> ;<br/>
<a NAME="E4:47_1_1"/><i><font color="Green">E28</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="supinf_2.html#K11" target="_self">+</a> <font color="Maroon">G</font></span>)</span> <a href="supinf_2.html#K10" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="supinf_2.html#K2" target="_self">+</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E5:47_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T6">FUNCT_2:6</a></i>;<br/>
<a NAME="E6:47_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T6">FUNCT_2:6</a></i>;<br/>
<b>hence </b><a NAME="E7:47_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="supinf_2.html#K11" target="_self">+</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K5" target="_self">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="supinf_2.html#T9" target="_self">Th9</a>, <a class="ref" href="supinf_2.html#T2" target="_self">Th2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:47_1"/>
<font color="Maroon">F</font> <a href="supinf_2.html#K11" target="_self">+</a> <font color="Maroon">G</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K5" target="_self">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_2.html#T8" target="_self">Th8</a>, <a class="ref" href="funct_2.html#T5">FUNCT_2:5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:47"/>
 <a href="supinf_2.html#K7" target="_self">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="supinf_2.html#K11" target="_self">+</a> <font color="Maroon">G</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K5" target="_self">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>


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