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<b>thus </b><a NAME="E1:72_2_1"/>
( <font color="Maroon" title="c1">f</font> is <a href="valued_0.html#V2" title="VALUED_0:attr.2">ext-real-valued</a> implies 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>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f</font> holds <br/><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> is <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:72_2_1_1"/><i><font color="Green" title="E13">A6</font></i>: 
<font color="Maroon" title="c1">f</font> is <a href="valued_0.html#V2" title="VALUED_0:attr.2">ext-real-valued</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E2:72_2_1_1"/><i><font color="Green" title="E14">A7</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c3">f</font> = <font color="Maroon" title="c1">f</font> as  <a href="valued_0.html#V2" title="VALUED_0:attr.2">ext-real-valued</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <b>by </b><i><a class="txt" href="valued_0.html#E1:72_2_1_1"><i><font color="Green" title="E13">A6</font></i></a></i>;<br/>
<a NAME="E4:72_2_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="txt" href="valued_0.html#E2:72_2_1_1"><i><font color="Green" title="E14">A7</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<b>hence </b><a NAME="E5:72_2_1_1"/>
<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> is <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>
 <b>by </b><i><a class="ref" href="membered.html#D2" title="MEMBERED:def.2">MEMBERED:def 2</a></i>;<br/>


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

<b>assume </b><a NAME="E2:72_2_1"/><i><font color="Green" title="E13">A8</font></i>: 
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>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f</font> holds <br/><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> is <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c2">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>,<a class="ref" href="valued_0.html#D2" target="_self" title="VALUED_0:def.2">VALUED_0:def 2</a><br/>

<b>assume </b><a NAME="E3:72_2_1"/>
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:72_2_1"/><i><font color="Green" title="E14">A9</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f</font>
 <b>and </b><br/><a NAME="E6:72_2_1"/><i><font color="Green" title="E15">A10</font></i>: 
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E7:72_2_1"/>
<font color="Maroon" title="c2">y</font> is <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>
 
<b>by </b><i><a class="txt" href="valued_0.html#E2:72_2_1"><i><font color="Green" title="E13">A8</font></i></a>, <a class="txt" href="valued_0.html#E5:72_2_1"><i><font color="Green" title="E14">A9</font></i></a>, <a class="txt" href="valued_0.html#E6:72_2_1"><i><font color="Green" title="E15">A10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:72_2_1"/>
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> 
 <b>by </b><i><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></i>;<br/>


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