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<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  st <font color="Olive" title="b1">f</font> is  <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a>  holds <br/> <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">f</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> is  <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a>  implies  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> )</span><br/>



<span class="kw">assume </span><a NAME="E1:38"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
<font color="Maroon" title="c2">f</font> is  <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span><br/>

<a NAME="E2:38"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> or <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:38_1"/><span class="lab"><font color="Green" title="E22">A2</font></span>: 
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">y</font> = <font color="Maroon" title="c3">y</font> as   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:38_1"/><span class="lab"><font color="Green" title="E23">A3</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">and </span><br/><a NAME="E5:38_1"/><span class="lab"><font color="Green" title="E24">A4</font></span>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_1"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c5">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38_1"><span class="lab"><font color="Green" title="E23">A3</font></span></a></span>;<br/>
<a NAME="E7:38_1"/><span class="lab"><font color="Green" title="E25">A5</font></span>: 
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="txt" href="#E4:38_1"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="ref" href="mesfunc2.html#D1" target="_self" title="MESFUNC2:def.1">Def1</a></span>;<br/>
<a NAME="E8:38_1"/><span class="kw">then </span>
 <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:38_1"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="ref" href="extreal2.html#T10" title="EXTREAL2:th.10">EXTREAL2:10</a></span>;<br/>
<a NAME="E9:38_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A6</font></span>: 
 <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#D3" title="XXREAL_3:def.3">XXREAL_3:def 3</a></span>;<br/>
<a NAME="E10:38_1"/>
<font color="Maroon" title="c4">y</font> <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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:38_1"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="txt" href="#E7:38_1"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="ref" href="extreal2.html#T10" title="EXTREAL2:th.10">EXTREAL2:10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:38_1"/>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:38_1"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:38"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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