<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be    <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><br/>  for <font color="Olive" title="b2">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/>  for <font color="Olive" title="b3">a</font> being   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> holds  <font color="Olive" title="b2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K20" title="MESFUNC1:func.20">eq_dom</a> (<font color="Olive" title="b1">f</font>,<font color="Olive" title="b3">a</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K19" title="MESFUNC1:func.19">great_eq_dom</a> (<font color="Olive" title="b1">f</font>,<font color="Olive" title="b3">a</font>)</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">less_eq_dom</a> (<font color="Olive" title="b1">f</font>,<font color="Olive" title="b3">a</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">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">  for <font color="Olive" title="b1">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/>  for <font color="Olive" title="b2">a</font> being   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> holds  <font color="Olive" title="b1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K20" title="MESFUNC1:func.20">eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">a</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K19" title="MESFUNC1:func.19">great_eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">a</font>)</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">less_eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">a</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">A</font> be    <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">a</font> being   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> holds  <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K20" title="MESFUNC1:func.20">eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">a</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K19" title="MESFUNC1:func.19">great_eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">a</font>)</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">less_eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">a</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">a</font> be   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</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="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K20" title="MESFUNC1:func.20">eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">a</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K19" title="MESFUNC1:func.19">great_eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">a</font>)</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">less_eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">a</font>)</span>)</span></span><br/>







<a NAME="E1:31"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K20" title="MESFUNC1:func.20">eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">a</font>)</span>)</span> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K19" title="MESFUNC1:func.19">great_eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">a</font>)</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">less_eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">a</font>)</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <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"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K20" title="MESFUNC1:func.20">eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">a</font>)</span>)</span> implies <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K19" title="MESFUNC1:func.19">great_eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">a</font>)</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">less_eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">a</font>)</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_1"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K20" title="MESFUNC1:func.20">eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">a</font>)</span>)</span>
 ; <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="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K19" title="MESFUNC1:func.19">great_eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">a</font>)</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">less_eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">a</font>)</span>)</span></span><br/>

<a NAME="E2:31_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A2</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E3:31_1"/><span class="lab"><font color="Green" title="E30">A3</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K20" title="MESFUNC1:func.20">eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">a</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E4:31_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A4</font></span>: 
<font color="Maroon" title="c4">a</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="ref" href="mesfunc1.html#D15" target="_self" title="MESFUNC1:def.15">Def16</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="#E1:31_1"><span class="lab"><font color="Green" title="E28">A1</font></span></a></span>;<br/>
<a NAME="E6:31_1"/><span class="lab"><font color="Green" title="E32">A5</font></span>: 
<font color="Maroon" title="c6">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> <font color="Maroon" title="c2">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31_1"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="ref" href="mesfunc1.html#D15" target="_self" title="MESFUNC1:def.15">Def16</a></span>;<br/>
<a NAME="E7:31_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K19" title="MESFUNC1:func.19">great_eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">a</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:31_1"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="mesfunc1.html#D14" target="_self" title="MESFUNC1:def.14">Def15</a></span>;<br/>
<a NAME="E8:31_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A6</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K19" title="MESFUNC1:func.19">great_eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">a</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31_1"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E9:31_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K17" title="MESFUNC1:func.17">less_eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">a</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:31_1"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="txt" href="#E6:31_1"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="ref" href="mesfunc1.html#D12" target="_self" title="MESFUNC1:def.12">Def13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:31_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K19" title="MESFUNC1:func.19">great_eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">a</font>)</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">less_eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">a</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:31_1"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</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>
<a NAME="E2:31"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A7</font></span>: 
<font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K20" title="MESFUNC1:func.20">eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">a</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K19" title="MESFUNC1:func.19">great_eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">a</font>)</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">less_eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">a</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E3:31"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K19" title="MESFUNC1:func.19">great_eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">a</font>)</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">less_eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">a</font>)</span>)</span> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K20" title="MESFUNC1:func.20">eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">a</font>)</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <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"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K19" title="MESFUNC1:func.19">great_eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">a</font>)</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">less_eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">a</font>)</span>)</span> implies <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K20" title="MESFUNC1:func.20">eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">a</font>)</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_2"/><span class="lab"><font color="Green" title="E29">A8</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K19" title="MESFUNC1:func.19">great_eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">a</font>)</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">less_eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">a</font>)</span>)</span>
 ; <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="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K20" title="MESFUNC1:func.20">eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">a</font>)</span>)</span></span><br/>

<a NAME="E2:31_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A9</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K19" title="MESFUNC1:func.19">great_eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">a</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E3:31_2"/><span class="lab"><font color="Green" title="E31">A10</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K17" title="MESFUNC1:func.17">less_eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">a</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_2"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E4:31_2"/><span class="lab"><font color="Green" title="E32">A11</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31_2"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E5:31_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K19" title="MESFUNC1:func.19">great_eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">a</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31_2"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E6:31_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A12</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="ref" href="mesfunc1.html#D14" target="_self" title="MESFUNC1:def.14">Def15</a></span>;<br/>
<a NAME="E7:31_2"/><span class="lab"><font color="Green" title="E34">A13</font></span>: 
<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> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31_2"><span class="lab"><font color="Green" title="E31">A10</font></span></a>, <a class="ref" href="mesfunc1.html#D12" target="_self" title="MESFUNC1:def.12">Def13</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="#E1:31_2"><span class="lab"><font color="Green" title="E29">A8</font></span></a></span>;<br/>
<a NAME="E9:31_2"/><span class="lab"><font color="Green" title="E35">A14</font></span>: 
<font color="Maroon" title="c6">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> <font color="Maroon" title="c2">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31_2"><span class="lab"><font color="Green" title="E31">A10</font></span></a>, <a class="ref" href="mesfunc1.html#D12" target="_self" title="MESFUNC1:def.12">Def13</a></span>;<br/>
<a NAME="E10:31_2"/>
<font color="Maroon" title="c4">a</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="c6">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:31_2"><span class="lab"><font color="Green" title="E33">A12</font></span></a>, <a class="txt" href="#E7:31_2"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E11:31_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K20" title="MESFUNC1:func.20">eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">a</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:31_2"><span class="lab"><font color="Green" title="E35">A14</font></span></a>, <a class="ref" href="mesfunc1.html#D15" target="_self" title="MESFUNC1:def.15">Def16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:31_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K20" title="MESFUNC1:func.20">eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">a</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:31_2"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</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>
<a NAME="E4:31"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K19" title="MESFUNC1:func.19">great_eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">a</font>)</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">less_eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">a</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K20" title="MESFUNC1:func.20">eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">a</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:31"/>
<font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K20" title="MESFUNC1:func.20">eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">a</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K19" title="MESFUNC1:func.19">great_eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">a</font>)</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">less_eq_dom</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">a</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</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>
