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<b>thus </b><a NAME="E1:2_1_1"/>
( <font color="Maroon" title="c2">f</font> is <a href="seq_1.html#V1" title="SEQ_1:attr.1">finite</a> implies for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> holds <br/><span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:2_1_1_1"/><i><font color="Green" title="E1">Z</font></i>: 
<font color="Maroon" title="c2">f</font> is <a href="seq_1.html#V1" title="SEQ_1:attr.1">finite</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>

<b>assume </b><a NAME="E2:2_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font>
 ;<br/>

<a NAME="E3:2_1_1_1"/><b>then </b><i><font color="Green" title="E2">A</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E4:2_1_1_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> 
 
<b>by </b><i><a class="txt" href="mesfunc2.html#E1:2_1_1_1"><i><font color="Green" title="E1">Z</font></i></a>, <a class="ref" href="seq_1.html#D1" title="SEQ_1:def.1">SEQ_1:def 1</a></i>;<br/>
<a NAME="E5:2_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> 
 
<b>by </b><i><a class="txt" href="mesfunc2.html#E3:2_1_1_1"><i><font color="Green" title="E2">A</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:2_1_1_1"/>
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="ref" href="extreal2.html#T102" title="EXTREAL2:th.102">EXTREAL2:102</a></i>;<br/>


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

<b>assume </b><a NAME="E2:2_1_1"/><i><font color="Green" title="E1">Z</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> holds <br/><span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 ;<br/>

<b>let </b><font color="Maroon" title="c3">e</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="seq_1.html#D1" title="SEQ_1:def.1">SEQ_1:def 1</a><br/>

<b>assume </b><a NAME="E3:2_1_1"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:2_1_1"/><i><font color="Green" title="E2">W1</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font>
 <b>and </b><br/><a NAME="E6:2_1_1"/><i><font color="Green" title="E3">W2</font></i>: 
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c4">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/>
<b>reconsider </b><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">x</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="mesfunc2.html#E5:2_1_1"><i><font color="Green" title="E2">W1</font></i></a></i>;<br/>
<a NAME="E8:2_1_1"/>
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c5">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<b>by </b><i><a class="txt" href="mesfunc2.html#E5:2_1_1"><i><font color="Green" title="E2">W1</font></i></a>, <a class="txt" href="mesfunc2.html#E2:2_1_1"><i><font color="Green" title="E1">Z</font></i></a></i>;<br/>
<a NAME="E9:2_1_1"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> 
 
<b>by </b><i><a class="ref" href="extreal2.html#T102" title="EXTREAL2:th.102">EXTREAL2:102</a></i>;<br/>
<b>hence </b><a NAME="E10:2_1_1"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> 
 <b>by </b><i><a class="txt" href="mesfunc2.html#E6:2_1_1"><i><font color="Green" title="E3">W2</font></i></a></i>;<br/>


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