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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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">S</font> being  <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">A</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">S</font><br/> for <font color="Olive" title="b3">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#K1" title="NUMBERS:func.1">REAL</a> <br/> for <font color="Olive" title="b4">a</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds  <font color="Olive" title="b2">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">eq_dom</a> <font color="Olive" title="b3">f</font>,<font color="Olive" title="b4">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#K8" title="SUBSET_1:func.8">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_eq_dom</a> <font color="Olive" title="b3">f</font>,<font color="Olive" title="b4">a</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_eq_dom</a> <font color="Olive" title="b3">f</font>,<font color="Olive" title="b4">a</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font>; <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">A</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font><br/> for <font color="Olive" title="b2">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#K1" title="NUMBERS:func.1">REAL</a> <br/> for <font color="Olive" title="b3">a</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds  <font color="Olive" title="b1">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">eq_dom</a> <font color="Olive" title="b2">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="b1">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_eq_dom</a> <font color="Olive" title="b2">f</font>,<font color="Olive" title="b3">a</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_eq_dom</a> <font color="Olive" title="b2">f</font>,<font color="Olive" title="b3">a</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">A</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font>; <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">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#K1" title="NUMBERS:func.1">REAL</a> <br/> for <font color="Olive" title="b2">a</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds  <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">eq_dom</a> <font color="Olive" title="b1">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="Maroon" title="c3">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_eq_dom</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">a</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_eq_dom</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">a</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">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#K1" title="NUMBERS:func.1">REAL</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">a</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds  <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">eq_dom</a> <font color="Maroon" title="c4">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#K8" title="SUBSET_1:func.8">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_eq_dom</a> <font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">a</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_eq_dom</a> <font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">a</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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="c3">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">eq_dom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">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#K8" title="SUBSET_1:func.8">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_eq_dom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_eq_dom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">a</font></span>)</span></span><br/>









<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">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#K8" title="SUBSET_1:func.8">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_eq_dom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_eq_dom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">a</font></span>)</span> implies <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#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">eq_dom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">a</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:59_1"/><span class="lab"><font color="Green" title="E36">A1</font></span>: 
<font color="Maroon" title="c6">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#K8" title="SUBSET_1:func.8">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_eq_dom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_eq_dom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">a</font></span>)</span>
 ; <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="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#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">eq_dom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">a</font></span>)</span></span><br/><a NAME="E2:59_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A2</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_eq_dom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">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="E3:59_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A3</font></span>: 
<font color="Maroon" title="c6">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="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc6.html#T4" title="MESFUNC6:th.4">MESFUNC6:4</a></span>;<br/><a NAME="E4:59_1"/><span class="lab"><font color="Green" title="E39">A4</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#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_eq_dom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_1"><span class="lab"><font color="Green" title="E36">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="E5:59_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#K17" title="MESFUNC1:func.17">great_eq_dom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">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="E6:59_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A5</font></span>: 
 ex <font color="Olive" title="b1">y1</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">y1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc6.html#T6" title="MESFUNC6:th.6">MESFUNC6:6</a></span>;<br/><a NAME="E7:59_1"/>
 ex <font color="Olive" title="b1">y2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Olive" title="b1">y2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_1"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="ref" href="mesfunc6.html#T4" title="MESFUNC6:th.4">MESFUNC6:4</a></span>;<br/><a NAME="E8:59_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:59_1"><span class="lab"><font color="Green" title="E40">A5</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E9:59_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A6</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#K12" title="RELAT_1:func.12">eq_dom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_1"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="ref" href="mesfunc6.html#T7" title="MESFUNC6:th.7">MESFUNC6:7</a></span>;<br/><a NAME="E10:59_1"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59_1"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E11:59_1"/>
<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#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">eq_dom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:59_1"><span class="lab"><font color="Green" title="E41">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="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E2:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A7</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_eq_dom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_eq_dom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">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#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">eq_dom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">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="Red">::  thesis: </font></span></a><span class="hide"> ( <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#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">eq_dom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">a</font></span>)</span> implies <font color="Maroon" title="c6">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#K8" title="SUBSET_1:func.8">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_eq_dom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_eq_dom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">a</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:59_2"/><span class="lab"><font color="Green" title="E37">A8</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#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">eq_dom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">a</font></span>)</span>
 ; <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="c6">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#K8" title="SUBSET_1:func.8">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_eq_dom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_eq_dom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">a</font></span>)</span></span><br/><a NAME="E2:59_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A9</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>
 <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="E3:59_2"/><span class="lab"><font color="Green" title="E39">A10</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#K12" title="RELAT_1:func.12">eq_dom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_2"><span class="lab"><font color="Green" title="E37">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="E4:59_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A11</font></span>: 
 ex <font color="Olive" title="b1">y</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc6.html#T7" title="MESFUNC6:th.7">MESFUNC6:7</a></span>;<br/><a NAME="E5:59_2"/><span class="lab"><font color="Green" title="E41">A12</font></span>: 
<font color="Maroon" title="c6">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="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_2"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="ref" href="mesfunc6.html#T7" title="MESFUNC6:th.7">MESFUNC6:7</a></span>;<br/><a NAME="E6:59_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#K17" title="MESFUNC1:func.17">great_eq_dom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59_2"><span class="lab"><font color="Green" title="E40">A11</font></span></a>, <a class="ref" href="mesfunc6.html#T6" title="MESFUNC6:th.6">MESFUNC6:6</a></span>;<br/><a NAME="E7:59_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A13</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#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_eq_dom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_2"><span class="lab"><font color="Green" title="E38">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="E8:59_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_eq_dom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59_2"><span class="lab"><font color="Green" title="E40">A11</font></span></a>, <a class="txt" href="#E5:59_2"><span class="lab"><font color="Green" title="E41">A12</font></span></a>, <a class="ref" href="mesfunc6.html#T4" title="MESFUNC6:th.4">MESFUNC6:4</a></span>;<br/><span class="kw">hence </span><a NAME="E9:59_2"/>
<font color="Maroon" title="c6">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#K8" title="SUBSET_1:func.8">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_eq_dom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_eq_dom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:59_2"><span class="lab"><font color="Green" title="E42">A13</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="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E4:59"/><span class="kw">then </span>
<font color="Maroon" title="c3">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">eq_dom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">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#K8" title="SUBSET_1:func.8">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_eq_dom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_eq_dom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">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="E5:59"/>
<font color="Maroon" title="c3">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">eq_dom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">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#K8" title="SUBSET_1:func.8">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_eq_dom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_eq_dom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E36">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="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
