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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#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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">M</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Olive" title="b1">S</font><br/>  for <font color="Olive" title="b3">f</font>, <font color="Olive" title="b4">g</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> holds <br/> ( <font color="Olive" title="b3">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Olive" title="b4">g</font>,<font color="Olive" title="b2">M</font> iff (  <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Olive" title="b3">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a>  <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Olive" title="b4">g</font>,<font color="Olive" title="b2">M</font> &amp;  <a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Olive" title="b3">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a>  <a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Olive" title="b4">g</font>,<font color="Olive" title="b2">M</font> ) )</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c2">S</font><br/>  for <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">g</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> holds <br/> ( <font color="Olive" title="b2">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Olive" title="b3">g</font>,<font color="Olive" title="b1">M</font> iff (  <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Olive" title="b2">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a>  <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Olive" title="b3">g</font>,<font color="Olive" title="b1">M</font> &amp;  <a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Olive" title="b2">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a>  <a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Olive" title="b3">g</font>,<font color="Olive" title="b1">M</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">M</font> be   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>, <font color="Olive" title="b2">g</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> holds <br/> ( <font color="Olive" title="b1">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Olive" title="b2">g</font>,<font color="Maroon" title="c3">M</font> iff (  <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Olive" title="b1">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a>  <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Olive" title="b2">g</font>,<font color="Maroon" title="c3">M</font> &amp;  <a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Olive" title="b1">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a>  <a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Olive" title="b2">g</font>,<font color="Maroon" title="c3">M</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">g</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">M</font> iff (  <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c4">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a>  <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">M</font> &amp;  <a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c4">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a>  <a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">M</font> ) )</span><br/>







<div><span class="kw">hereby </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c4">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a>  <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">M</font> &amp;  <a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c4">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a>  <a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c4">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">M</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:48_1"/>
<font color="Maroon" title="c4">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">M</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c4">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a>  <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">M</font> &amp;  <a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c4">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a>  <a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">M</font> )</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">E</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font><span class="kw"> such that </span><br/><a NAME="E3:48_1"/><span class="lab"><font color="Green" title="E42">A1</font></span>: 
( <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c6">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">E</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">E</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace1.html#D10" onmouseover="rs('lpspace1/D10')" onmouseout="rh()">LPSPACE1:def 10</a></span>;<br/><a NAME="E4:48_1"/>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">E</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">E</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_3.html#T44" onmouseover="rs('rfunct_3/T44')" onmouseout="rh()">RFUNCT_3:44</a></span>;<br/><a NAME="E5:48_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">E</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">E</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="ref" href="rfunct_3.html#T44" onmouseover="rs('rfunct_3/T44')" onmouseout="rh()">RFUNCT_3:44</a></span>;<br/><span class="kw">hence </span><a NAME="E6:48_1"/>
 <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c4">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a>  <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="ref" href="lpspace1.html#D10" onmouseover="rs('lpspace1/D10')" onmouseout="rh()">LPSPACE1:def 10</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c4">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a>  <a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">M</font></span><br/><a NAME="E7:48_1"/>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">E</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">E</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_3.html#T45" onmouseover="rs('rfunct_3/T45')" onmouseout="rh()">RFUNCT_3:45</a></span>;<br/><a NAME="E8:48_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">E</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">E</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="ref" href="rfunct_3.html#T45" onmouseover="rs('rfunct_3/T45')" onmouseout="rh()">RFUNCT_3:45</a></span>;<br/><span class="kw">hence </span><a NAME="E9:48_1"/>
 <a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c4">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a>  <a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="ref" href="lpspace1.html#D10" onmouseover="rs('lpspace1/D10')" onmouseout="rh()">LPSPACE1:def 10</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:48"/><span class="lab"><font color="Green" title="E42">A2</font></span>: 
 <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c4">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a>  <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">M</font>
 <span class="kw">and </span><br/><a NAME="E3:48"/><span class="lab"><font color="Green" title="E43">A3</font></span>: 
 <a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c4">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a>  <a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">M</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">M</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">E1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font><span class="kw"> such that </span><br/><a NAME="E5:48"/><span class="lab"><font color="Green" title="E44">A4</font></span>: 
( <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c6">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <span class="p1">(<span class="default"><a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="ref" href="lpspace1.html#D10" onmouseover="rs('lpspace1/D10')" onmouseout="rh()">LPSPACE1:def 10</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">E2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font><span class="kw"> such that </span><br/><a NAME="E7:48"/><span class="lab"><font color="Green" title="E45">A5</font></span>: 
( <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">E2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <span class="p1">(<span class="default"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">E2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">E2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="lpspace1.html#D10" onmouseover="rs('lpspace1/D10')" onmouseout="rh()">LPSPACE1:def 10</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">E</font> = <font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font>;<br/>
<a NAME="E8:48"/>
<font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c6">E1</font></span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">E2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure1.html#T10" onmouseover="rs('measure1/T10')" onmouseout="rh()">MEASURE1:10</a></span>;<br/>
<a NAME="E9:48"/><span class="kw">then </span>
<font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="txt" href="#E7:48"><span class="lab"><font color="Green" title="E45">A5</font></span></a></span>;<br/>
<a NAME="E10:48"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A6</font></span>: 
<font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T51" onmouseover="rs('supinf_2/T51')" onmouseout="rh()">SUPINF_2:51</a></span>;<br/>
<a NAME="E11:48"/>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="ref" href="subset_1.html#T21" onmouseover="rs('subset_1/T21')" onmouseout="rh()">SUBSET_1:21</a>, <a class="ref" href="relat_1.html#T74" onmouseover="rs('relat_1/T74')" onmouseout="rh()">RELAT_1:74</a></span>;<br/>
<a NAME="E12:48"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A7</font></span>: 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T21" onmouseover="rs('subset_1/T21')" onmouseout="rh()">SUBSET_1:21</a>, <a class="ref" href="relat_1.html#T74" onmouseover="rs('relat_1/T74')" onmouseout="rh()">RELAT_1:74</a></span>;<br/>
<a NAME="E13:48"/>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">E2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="ref" href="subset_1.html#T21" onmouseover="rs('subset_1/T21')" onmouseout="rh()">SUBSET_1:21</a>, <a class="ref" href="relat_1.html#T74" onmouseover="rs('relat_1/T74')" onmouseout="rh()">RELAT_1:74</a></span>;<br/>
<a NAME="E14:48"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A8</font></span>: 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T21" onmouseover="rs('subset_1/T21')" onmouseout="rh()">SUBSET_1:21</a>, <a class="ref" href="relat_1.html#T74" onmouseover="rs('relat_1/T74')" onmouseout="rh()">RELAT_1:74</a></span>;<br/>
<a NAME="E15:48"/><span class="lab"><font color="Green" title="E49">A9</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">g</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_3.html#T34" onmouseover="rs('rfunct_3/T34')" onmouseout="rh()">RFUNCT_3:34</a></span>;<br/>
<a NAME="E16:48"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_3.html#D10" onmouseover="rs('rfunct_3/D10')" onmouseout="rh()">RFUNCT_3:def 10</a></span>;<br/>
<a NAME="E17:48"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_3.html#T44" onmouseover="rs('rfunct_3/T44')" onmouseout="rh()">RFUNCT_3:44</a></span>;<br/>
<a NAME="E18:48"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:48"><span class="lab"><font color="Green" title="E47">A7</font></span></a>, <a class="ref" href="rfunct_3.html#T44" onmouseover="rs('rfunct_3/T44')" onmouseout="rh()">RFUNCT_3:44</a></span>;<br/>
<a NAME="E19:48"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A10</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_3.html#D10" onmouseover="rs('rfunct_3/D10')" onmouseout="rh()">RFUNCT_3:def 10</a></span>;<br/>
<a NAME="E20:48"/>
 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">X</font>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</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="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:48_2"/><span class="lab"><font color="Green" title="E51">A11</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">x</font></span><br/>

<a NAME="E2:48_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A12</font></span>: 
( <font color="Maroon" title="c9">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c9">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T57" onmouseover="rs('relat_1/T57')" onmouseout="rh()">RELAT_1:57</a></span>;<br/>
<a NAME="E3:48_2"/><span class="lab"><font color="Green" title="E53">A13</font></span>: 
( <font color="Maroon" title="c9">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">g</font> &amp; <font color="Maroon" title="c9">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:48"><span class="lab"><font color="Green" title="E50">A10</font></span></a>, <a class="txt" href="#E1:48_2"><span class="lab"><font color="Green" title="E51">A11</font></span></a>, <a class="ref" href="relat_1.html#T57" onmouseover="rs('relat_1/T57')" onmouseout="rh()">RELAT_1:57</a></span>;<br/>
<a NAME="E4:48_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_2"><span class="lab"><font color="Green" title="E51">A11</font></span></a>, <a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</a></span>;<br/>
<a NAME="E5:48_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_3.html#T34" onmouseover="rs('rfunct_3/T34')" onmouseout="rh()">RFUNCT_3:34</a></span>;<br/>
<a NAME="E6:48_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_2"><span class="lab"><font color="Green" title="E52">A12</font></span></a>, <a class="txt" href="#E15:48"><span class="lab"><font color="Green" title="E49">A9</font></span></a>, <a class="ref" href="valued_1.html#T13" onmouseover="rs('valued_1/T13')" onmouseout="rh()">VALUED_1:13</a></span>;<br/>
<a NAME="E7:48_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_2"><span class="lab"><font color="Green" title="E52">A12</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<a NAME="E8:48_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_2"><span class="lab"><font color="Green" title="E52">A12</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<a NAME="E9:48_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_2"><span class="lab"><font color="Green" title="E52">A12</font></span></a>, <a class="txt" href="#E12:48"><span class="lab"><font color="Green" title="E47">A7</font></span></a>, <a class="txt" href="#E14:48"><span class="lab"><font color="Green" title="E48">A8</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<a NAME="E10:48_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_2"><span class="lab"><font color="Green" title="E52">A12</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<a NAME="E11:48_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">max+</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K19" title="RFUNCT_3:func.19">max-</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_2"><span class="lab"><font color="Green" title="E53">A13</font></span></a>, <a class="txt" href="#E15:48"><span class="lab"><font color="Green" title="E49">A9</font></span></a>, <a class="ref" href="valued_1.html#T13" onmouseover="rs('valued_1/T13')" onmouseout="rh()">VALUED_1:13</a></span>;<br/>
<a NAME="E12:48_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_3.html#T34" onmouseover="rs('rfunct_3/T34')" onmouseout="rh()">RFUNCT_3:34</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:48_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">E1</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c7">E2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_2"><span class="lab"><font color="Green" title="E53">A13</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>
<span class="kw">hence </span><a NAME="E21:48"/>
<font color="Maroon" title="c4">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:48"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="txt" href="#E19:48"><span class="lab"><font color="Green" title="E50">A10</font></span></a>, <a class="ref" href="partfun1.html#T5" onmouseover="rs('partfun1/T5')" onmouseout="rh()">PARTFUN1:5</a>, <a class="ref" href="lpspace1.html#D10" onmouseover="rs('lpspace1/D10')" onmouseout="rh()">LPSPACE1:def 10</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
