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<span class="kw">let </span><font color="Maroon" title="c1">C</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">C</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> holds <br/> (  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">1 <a href="mesfunc1.html#K10" title="MESFUNC1:func.10">/</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> &amp; (  for <font color="Olive" title="b2">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b2">c</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> <span class="p1">(<span class="default">1 <a href="mesfunc1.html#K10" title="MESFUNC1:func.10">/</a> <font color="Olive" title="b1">f</font></span>)</span> holds <br/><span class="p1">(<span class="default">1 <a href="mesfunc1.html#K10" title="MESFUNC1:func.10">/</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="mesfunc1.html#K9" title="MESFUNC1:func.9">1.</a> <a href="extreal1.html#K2" title="EXTREAL1:func.2">/</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">c</font></span>)</span> ) )</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">C</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"> (  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">1 <a href="mesfunc1.html#K10" title="MESFUNC1:func.10">/</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> &amp; (  for <font color="Olive" title="b1">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b1">c</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> <span class="p1">(<span class="default">1 <a href="mesfunc1.html#K10" title="MESFUNC1:func.10">/</a> <font color="Maroon" title="c2">f</font></span>)</span> holds <br/><span class="p1">(<span class="default">1 <a href="mesfunc1.html#K10" title="MESFUNC1:func.10">/</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="mesfunc1.html#K9" title="MESFUNC1:func.9">1.</a> <a href="extreal1.html#K2" title="EXTREAL1:func.2">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">c</font></span>)</span> ) )</span><br/>



<span class="kw">thus </span><a NAME="E1:16"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">1 <a href="mesfunc1.html#K10" title="MESFUNC1:func.10">/</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc1.html#D9" target="_self" title="MESFUNC1:def.9">Def9</a></span>; <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">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b1">c</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> <span class="p1">(<span class="default">1 <a href="mesfunc1.html#K10" title="MESFUNC1:func.10">/</a> <font color="Maroon" title="c2">f</font></span>)</span> holds <br/><span class="p1">(<span class="default">1 <a href="mesfunc1.html#K10" title="MESFUNC1:func.10">/</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="mesfunc1.html#K9" title="MESFUNC1:func.9">1.</a> <a href="extreal1.html#K2" title="EXTREAL1:func.2">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">c</font></span>)</span></span><br/>

<span class="kw">thus </span><a NAME="E2:16"/>
 for <font color="Olive" title="b1">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b1">c</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> <span class="p1">(<span class="default">1 <a href="mesfunc1.html#K10" title="MESFUNC1:func.10">/</a> <font color="Maroon" title="c2">f</font></span>)</span> holds <br/><span class="p1">(<span class="default">1 <a href="mesfunc1.html#K10" title="MESFUNC1:func.10">/</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="mesfunc1.html#K9" title="MESFUNC1:func.9">1.</a> <a href="extreal1.html#K2" title="EXTREAL1:func.2">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">c</font></span>)</span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C</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">c</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> <span class="p1">(<span class="default">1 <a href="mesfunc1.html#K10" title="MESFUNC1:func.10">/</a> <font color="Maroon" title="c2">f</font></span>)</span> implies <span class="p1">(<span class="default">1 <a href="mesfunc1.html#K10" title="MESFUNC1:func.10">/</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="mesfunc1.html#K9" title="MESFUNC1:func.9">1.</a> <a href="extreal1.html#K2" title="EXTREAL1:func.2">/</a> <span class="p1">(<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="c3">c</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:16_1"/>
<font color="Maroon" title="c3">c</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> <span class="p1">(<span class="default">1 <a href="mesfunc1.html#K10" title="MESFUNC1:func.10">/</a> <font color="Maroon" title="c2">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default">1 <a href="mesfunc1.html#K10" title="MESFUNC1:func.10">/</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="mesfunc1.html#K9" title="MESFUNC1:func.9">1.</a> <a href="extreal1.html#K2" title="EXTREAL1:func.2">/</a> <span class="p1">(<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="c3">c</font></span>)</span></span><br/>

<a NAME="E2:16_1"/><span class="kw">then </span>
<span class="p1">(<span class="default">1 <a href="mesfunc1.html#K10" title="MESFUNC1:func.10">/</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> 1</span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">/</a> <span class="p1">(<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="c3">c</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc1.html#D9" target="_self" title="MESFUNC1:def.9">Def9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:16_1"/>
<span class="p1">(<span class="default">1 <a href="mesfunc1.html#K10" title="MESFUNC1:func.10">/</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="mesfunc1.html#K9" title="MESFUNC1:func.9">1.</a> <a href="extreal1.html#K2" title="EXTREAL1:func.2">/</a> <span class="p1">(<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="c3">c</font></span>)</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>


</div>
