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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</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  <font color="Olive" title="b1">f1</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <font color="Olive" title="b2">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f2</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <font color="Olive" title="b1">f1</font></span><br/><b>let </b><font color="Maroon" title="c2">f1</font>, <font color="Maroon" title="c3">f2</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">f1</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <font color="Maroon" title="c3">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f2</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <font color="Maroon" title="c2">f1</font></span><br/>



<i><font color="Green" title="E15">A1</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f2</font></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc1.html#D5" target="_self" title="MESFUNC1:def.5">Def5</a></i>
<br/>.= 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <font color="Maroon" title="c2">f1</font></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc1.html#D5" target="_self" title="MESFUNC1:def.5">Def5</a></i>
;<br/>
<a NAME="E2:19"/>
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">C</font>  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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">x</font> )</span><br/>

<b>assume </b><a NAME="E1:19_2"/><i><font color="Green" title="E16">A2</font></i>: 
<font color="Maroon" title="c4">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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">x</font></span><br/>

<a NAME="E2:19_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D5" target="_self" title="MESFUNC1:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E3:19_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="mesfunc1.html#E1:19"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="mesfunc1.html#E1:19_2"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="mesfunc1.html#D5" target="_self" title="MESFUNC1:def.5">Def5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:19"/>
<font color="Maroon" title="c2">f1</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <font color="Maroon" title="c3">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f2</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <font color="Maroon" title="c2">f1</font>
 <b>by </b><i><a class="txt" href="mesfunc1.html#E1:19"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
