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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="Red">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>



<span class="lab"><font color="Green" title="E15">A1</font></span>:  <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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc1.html#D5" target="_self" title="MESFUNC1:def.5">Def5</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc1.html#D5" target="_self" title="MESFUNC1:def.5">Def5</a></span>
;<br/>
<a NAME="E2:20"/><span class="lab"><font color="Green" title="E16">A2</font></span>: 
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="20_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E1:20_2"/><span class="lab"><font color="Green" title="E17">A3</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E2:20_2"/><span class="lab"><font color="Green" title="E18">A4</font></span>: 
<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#K1" title="EXTREAL1:func.1">*</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_2"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="ref" href="mesfunc1.html#D5" target="_self" title="MESFUNC1:def.5">Def5</a></span>;<br/>
<span class="kw">thus </span><a NAME="E3:20_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E1:20_2"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E2:20_2"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="ref" href="mesfunc1.html#D5" target="_self" title="MESFUNC1:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E3:20"/>
<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
