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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> ;<br/><b>let </b><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> ;<br/><b>let </b><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>;<br/>





<b>thus </b><a NAME="E1:53"/>
 <a href="supinf_2.html#K9" title="SUPINF_2:func.9">rng</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="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="supinf_2.html#K6" title="SUPINF_2:func.6">-</a> <span class="p1">(<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>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:53_1"/><i><font color="Green" title="E41">A1</font></i>: 
<font color="Maroon" title="c4">y</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> <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>
 ;<br/>

<a NAME="E2:53_1"/><i><font color="Green" title="E42">A2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">F</font> <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="E3:53_1"/><i><font color="Green" title="E43">A3</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</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> <font color="Maroon" title="c1">X</font> &amp;  <a href="supinf_2.html#K9" title="SUPINF_2:func.9">rng</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="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="supinf_2.html#K6" title="SUPINF_2:func.6">-</a> <font color="Maroon" title="c2">Y</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">y</font> = <font color="Maroon" title="c4">y</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> <b>by </b><i><a class="txt" href="supinf_2.html#E1:53_1"><i><font color="Green" title="E41">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:53_1"/><i><font color="Green" title="E44">A4</font></i>: 
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">a</font> )
 <b>by </b><i><a class="txt" href="supinf_2.html#E1:53_1"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="supinf_2.html#E3:53_1"><i><font color="Green" title="E43">A3</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">a</font> = <font color="Maroon" title="c6">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="supinf_2.html#E6:53_1"><i><font color="Green" title="E44">A4</font></i></a></i>;<br/>
<a NAME="E8:53_1"/>
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c5">y</font> <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"><font color="Maroon" title="c3">F</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">a</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="supinf_2.html#E6:53_1"><i><font color="Green" title="E44">A4</font></i></a>, <a class="ref" href="supinf_2.html#D10" target="_self" title="SUPINF_2:def.10">Def10</a></i>;<br/>
<a NAME="E9:53_1"/><b>then </b>
 <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <font color="Maroon" title="c5">y</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="c3">F</font>
 
<b>by </b><i><a class="txt" href="supinf_2.html#E2:53_1"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E10:53_1"/><b>then </b>
 <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K6" title="SUPINF_2:func.6">-</a> <span class="p1">(<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>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T23" target="_self" title="SUPINF_2:th.23">Th23</a></i>;<br/>
<b>hence </b><a NAME="E11:53_1"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K6" title="SUPINF_2:func.6">-</a> <span class="p1">(<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>
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:53"/>
 <a href="supinf_2.html#K6" title="SUPINF_2:func.6">-</a> <span class="p1">(<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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="supinf_2.html#K9" title="SUPINF_2:func.9">rng</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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:53_2"/><i><font color="Green" title="E41">A5</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K6" title="SUPINF_2:func.6">-</a> <span class="p1">(<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>
 ;<br/>

<a NAME="E2:53_2"/><i><font color="Green" title="E42">A6</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</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> <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="E3:53_2"/><i><font color="Green" title="E43">A7</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="supinf_2.html#K9" title="SUPINF_2:func.9">rng</a> <font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">y</font> = <font color="Maroon" title="c4">y</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> <b>by </b><i><a class="txt" href="supinf_2.html#E1:53_2"><i><font color="Green" title="E41">A5</font></i></a></i>;<br/>
<a NAME="E5:53_2"/>
 <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K6" title="SUPINF_2:func.6">-</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>
 
<b>by </b><i><a class="txt" href="supinf_2.html#E1:53_2"><i><font color="Green" title="E41">A5</font></i></a>, <a class="ref" href="supinf_2.html#T23" target="_self" title="SUPINF_2:th.23">Th23</a></i>;<br/>
<a NAME="E6:53_2"/><b>then </b>
 <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <font color="Maroon" title="c5">y</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="c3">F</font>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T22" target="_self" title="SUPINF_2:th.22">Th22</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:53_2"/><i><font color="Green" title="E44">A8</font></i>: 
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">a</font> )
 <b>by </b><i><a class="txt" href="supinf_2.html#E3:53_2"><i><font color="Green" title="E43">A7</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">a</font> = <font color="Maroon" title="c6">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="supinf_2.html#E8:53_2"><i><font color="Green" title="E44">A8</font></i></a></i>;<br/>
<a NAME="E10:53_2"/>
<font color="Maroon" title="c5">y</font> <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"><font color="Maroon" title="c3">F</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="supinf_2.html#E8:53_2"><i><font color="Green" title="E44">A8</font></i></a></i>;<br/>
<a NAME="E11:53_2"/><b>then </b>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">a</font>
 
<b>by </b><i><a class="ref" href="supinf_2.html#D10" target="_self" title="SUPINF_2:def.10">Def10</a></i>;<br/>
<b>hence </b><a NAME="E12:53_2"/>
<font color="Maroon" title="c4">y</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> <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>
 <b>by </b><i><a class="txt" href="supinf_2.html#E2:53_2"><i><font color="Green" title="E42">A6</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>


</div>
