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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> ;<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">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/>







<b>assume </b><a NAME="E1:60"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="supinf_1.html#K3">REAL</a> 
 ;<br/>

<a NAME="E2:60"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> &amp;  <a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E3:60"/><b>then </b>
<font color="Maroon">F</font> <a href="supinf_2.html#K10" target="_self">.</a> <font color="Maroon">x</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_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:60"/><b>then </b>
<font color="Maroon">F</font> <a href="supinf_2.html#K10" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:60"/><b>then </b><i><font color="Green">E29</font></i>: 
( not <font color="Maroon">F</font> <a href="supinf_2.html#K10" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; not <font color="Maroon">F</font> <a href="supinf_2.html#K10" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 
<b>by </b><i>, <a class="ref" href="supinf_1.html#T1">SUPINF_1:1</a>, <a class="ref" href="supinf_1.html#T6">SUPINF_1:6</a></i>;<br/>
<a NAME="E6:60"/>
(  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">F</font> <a href="supinf_2.html#K10" target="_self">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">F</font> <a href="supinf_2.html#K10" target="_self">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 
<b>by </b><i><a class="ref" href="supinf_1.html#T20">SUPINF_1:20</a>, <a class="ref" href="supinf_1.html#T21">SUPINF_1:21</a></i>;<br/>
<b>hence </b><a NAME="E7:60"/>
(  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">F</font> <a href="supinf_2.html#K10" target="_self">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">F</font> <a href="supinf_2.html#K10" target="_self">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i>, <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/>


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