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<span class="kw">let </span><font color="Maroon" title="c2">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="c2">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> being   <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a>  <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a><br/>  for <font color="Olive" title="b4">g</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a><br/>  for <font color="Olive" title="b5">E</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">S</font>  st  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">E</font> &amp; (  for <font color="Olive" title="b6">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Olive" title="b3">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b6">n</font> is <font color="Olive" title="b5">E</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) &amp; <font color="Olive" title="b2">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b5">E</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  &amp; (  for <font color="Olive" title="b6">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  ex <font color="Olive" title="b7">L</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">S</font> st <br/>( <font color="Olive" title="b7">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b5">E</font> &amp; <font color="Olive" title="b2">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Olive" title="b7">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; (  for <font color="Olive" title="b8">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b7">L</font> holds <br/><span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b3">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b6">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b8">x</font></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  ) ) ) &amp;  ex <font color="Olive" title="b6">G</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">S</font> st <br/>( <font color="Olive" title="b6">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b5">E</font> &amp; <font color="Olive" title="b2">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Olive" title="b6">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; (  for <font color="Olive" title="b7">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b7">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b5">E</font> holds <br/><font color="Olive" title="b3">f</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b7">x</font> is  <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</a>  ) &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">E</font> &amp; (  for <font color="Olive" title="b7">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b7">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b6">G</font> holds <br/><font color="Olive" title="b4">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">f</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b7">x</font></span>)</span> ) ) holds <br/> for <font color="Olive" title="b6">e</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b6">e</font> holds <br/> ex <font color="Olive" title="b7">F</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">S</font> st <br/>( <font color="Olive" title="b7">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b5">E</font> &amp; <font color="Olive" title="b2">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Olive" title="b7">F</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b6">e</font> &amp; (  for <font color="Olive" title="b8">p</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b8">p</font> holds <br/> ex <font color="Olive" title="b9">N</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b10">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b9">N</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b10">n</font> holds <br/> for <font color="Olive" title="b11">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b11">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b7">F</font> holds <br/><span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b3">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b10">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b11">x</font></span>)</span> <a href="xxreal_3.html#K10" title="XXREAL_3:func.10">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b4">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b11">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b8">p</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c2">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="c3">S</font><br/>  for <font color="Olive" title="b2">f</font> being   <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a>  <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a><br/>  for <font color="Olive" title="b3">g</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a><br/>  for <font color="Olive" title="b4">E</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font>  st  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">E</font> &amp; (  for <font color="Olive" title="b5">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Olive" title="b2">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b5">n</font> is <font color="Olive" title="b4">E</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) &amp; <font color="Olive" title="b1">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b4">E</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  &amp; (  for <font color="Olive" title="b5">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  ex <font color="Olive" title="b6">L</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font> st <br/>( <font color="Olive" title="b6">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">E</font> &amp; <font color="Olive" title="b1">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Olive" title="b6">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; (  for <font color="Olive" title="b7">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b7">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b6">L</font> holds <br/><span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b2">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b5">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b7">x</font></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  ) ) ) &amp;  ex <font color="Olive" title="b5">G</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font> st <br/>( <font color="Olive" title="b5">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">E</font> &amp; <font color="Olive" title="b1">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Olive" title="b5">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; (  for <font color="Olive" title="b6">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b4">E</font> holds <br/><font color="Olive" title="b2">f</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b6">x</font> is  <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</a>  ) &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">E</font> &amp; (  for <font color="Olive" title="b6">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b5">G</font> holds <br/><font color="Olive" title="b3">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b6">x</font></span>)</span> ) ) holds <br/> for <font color="Olive" title="b5">e</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">e</font> holds <br/> ex <font color="Olive" title="b6">F</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font> st <br/>( <font color="Olive" title="b6">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">E</font> &amp; <font color="Olive" title="b1">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Olive" title="b6">F</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">e</font> &amp; (  for <font color="Olive" title="b7">p</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b7">p</font> holds <br/> ex <font color="Olive" title="b8">N</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b9">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b8">N</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b9">n</font> holds <br/> for <font color="Olive" title="b10">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b10">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b6">F</font> holds <br/><span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b2">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b9">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b10">x</font></span>)</span> <a href="xxreal_3.html#K10" title="XXREAL_3:func.10">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b10">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b7">p</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">M</font> be   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c3">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> being   <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a>  <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a><br/>  for <font color="Olive" title="b2">g</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a><br/>  for <font color="Olive" title="b3">E</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font>  st  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">E</font> &amp; (  for <font color="Olive" title="b4">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Olive" title="b1">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b4">n</font> is <font color="Olive" title="b3">E</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) &amp; <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">E</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  &amp; (  for <font color="Olive" title="b4">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  ex <font color="Olive" title="b5">L</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font> st <br/>( <font color="Olive" title="b5">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">E</font> &amp; <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Olive" title="b5">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; (  for <font color="Olive" title="b6">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b5">L</font> holds <br/><span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b4">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b6">x</font></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  ) ) ) &amp;  ex <font color="Olive" title="b4">G</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font> st <br/>( <font color="Olive" title="b4">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">E</font> &amp; <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Olive" title="b4">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; (  for <font color="Olive" title="b5">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b5">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b3">E</font> holds <br/><font color="Olive" title="b1">f</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b5">x</font> is  <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</a>  ) &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">E</font> &amp; (  for <font color="Olive" title="b5">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b5">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b4">G</font> holds <br/><font color="Olive" title="b2">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b5">x</font></span>)</span> ) ) holds <br/> for <font color="Olive" title="b4">e</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">e</font> holds <br/> ex <font color="Olive" title="b5">F</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font> st <br/>( <font color="Olive" title="b5">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">E</font> &amp; <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Olive" title="b5">F</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">e</font> &amp; (  for <font color="Olive" title="b6">p</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b6">p</font> holds <br/> ex <font color="Olive" title="b7">N</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b8">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b7">N</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b8">n</font> holds <br/> for <font color="Olive" title="b9">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b9">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b5">F</font> holds <br/><span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b1">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b8">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b9">x</font></span>)</span> <a href="xxreal_3.html#K10" title="XXREAL_3:func.10">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b9">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b6">p</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">f</font> be   <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a>  <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</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">g</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a><br/>  for <font color="Olive" title="b2">E</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font>  st  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">E</font> &amp; (  for <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b3">n</font> is <font color="Olive" title="b2">E</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) &amp; <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">E</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  &amp; (  for <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  ex <font color="Olive" title="b4">L</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font> st <br/>( <font color="Olive" title="b4">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">E</font> &amp; <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Olive" title="b4">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; (  for <font color="Olive" title="b5">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b5">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b4">L</font> holds <br/><span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b3">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b5">x</font></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  ) ) ) &amp;  ex <font color="Olive" title="b3">G</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font> st <br/>( <font color="Olive" title="b3">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">E</font> &amp; <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Olive" title="b3">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; (  for <font color="Olive" title="b4">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">E</font> holds <br/><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b4">x</font> is  <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</a>  ) &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">E</font> &amp; (  for <font color="Olive" title="b4">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b3">G</font> holds <br/><font color="Olive" title="b1">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b4">x</font></span>)</span> ) ) holds <br/> for <font color="Olive" title="b3">e</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">e</font> holds <br/> ex <font color="Olive" title="b4">F</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font> st <br/>( <font color="Olive" title="b4">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">E</font> &amp; <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Olive" title="b4">F</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">e</font> &amp; (  for <font color="Olive" title="b5">p</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">p</font> holds <br/> ex <font color="Olive" title="b6">N</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b7">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b6">N</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b7">n</font> holds <br/> for <font color="Olive" title="b8">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b4">F</font> holds <br/><span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b7">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b8">x</font></span>)</span> <a href="xxreal_3.html#K10" title="XXREAL_3:func.10">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b8">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">p</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">g</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</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">E</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font>  st  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">E</font> &amp; (  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">n</font> is <font color="Olive" title="b1">E</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) &amp; <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">E</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  &amp; (  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  ex <font color="Olive" title="b3">L</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font> st <br/>( <font color="Olive" title="b3">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">E</font> &amp; <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Olive" title="b3">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; (  for <font color="Olive" title="b4">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b3">L</font> holds <br/><span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b4">x</font></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  ) ) ) &amp;  ex <font color="Olive" title="b2">G</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font> st <br/>( <font color="Olive" title="b2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">E</font> &amp; <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Olive" title="b2">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; (  for <font color="Olive" title="b3">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">E</font> holds <br/><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b3">x</font> is  <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</a>  ) &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">E</font> &amp; (  for <font color="Olive" title="b3">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">G</font> holds <br/><font color="Maroon" title="c6">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b3">x</font></span>)</span> ) ) holds <br/> for <font color="Olive" title="b2">e</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">e</font> holds <br/> ex <font color="Olive" title="b3">F</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font> st <br/>( <font color="Olive" title="b3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">E</font> &amp; <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Olive" title="b3">F</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">e</font> &amp; (  for <font color="Olive" title="b4">p</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">p</font> holds <br/> ex <font color="Olive" title="b5">N</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b6">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b5">N</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b6">n</font> holds <br/> for <font color="Olive" title="b7">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b7">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b3">F</font> holds <br/><span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b6">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b7">x</font></span>)</span> <a href="xxreal_3.html#K10" title="XXREAL_3:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b7">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">p</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c7">E</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">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"> (  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">E</font> &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font> is <font color="Maroon" title="c7">E</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) &amp; <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">E</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  ex <font color="Olive" title="b2">L</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font> st <br/>( <font color="Olive" title="b2">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">E</font> &amp; <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Olive" title="b2">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; (  for <font color="Olive" title="b3">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">L</font> holds <br/><span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">x</font></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  ) ) ) &amp;  ex <font color="Olive" title="b1">G</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font> st <br/>( <font color="Olive" title="b1">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">E</font> &amp; <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Olive" title="b1">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">E</font> holds <br/><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b2">x</font> is  <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</a>  ) &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">E</font> &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">G</font> holds <br/><font color="Maroon" title="c6">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b2">x</font></span>)</span> ) ) implies  for <font color="Olive" title="b1">e</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">e</font> holds <br/> ex <font color="Olive" title="b2">F</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font> st <br/>( <font color="Olive" title="b2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">E</font> &amp; <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Olive" title="b2">F</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">e</font> &amp; (  for <font color="Olive" title="b3">p</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">p</font> holds <br/> ex <font color="Olive" title="b4">N</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b5">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b4">N</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">n</font> holds <br/> for <font color="Olive" title="b6">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">F</font> holds <br/><span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b5">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b6">x</font></span>)</span> <a href="xxreal_3.html#K10" title="XXREAL_3:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b6">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">p</font> ) ) )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:40"/><span class="lab"><font color="Green" title="E33">A1</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="c5">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">E</font>
 <span class="kw">and </span><br/><a NAME="E2:40"/><span class="lab"><font color="Green" title="E34">A2</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font> is <font color="Maroon" title="c7">E</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> 
 <span class="kw">and </span><br/><a NAME="E3:40"/><span class="lab"><font color="Green" title="E35">A3</font></span>: 
<font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">E</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
 <span class="kw">and </span><br/><a NAME="E4:40"/><span class="lab"><font color="Green" title="E36">A4</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  ex <font color="Olive" title="b2">L</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font> st <br/>( <font color="Olive" title="b2">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">E</font> &amp; <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Olive" title="b2">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; (  for <font color="Olive" title="b3">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">L</font> holds <br/><span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">x</font></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  ) )
 <span class="kw">and </span><br/><a NAME="E5:40"/><span class="lab"><font color="Green" title="E37">A5</font></span>: 
 ex <font color="Olive" title="b1">G</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font> st <br/>( <font color="Olive" title="b1">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">E</font> &amp; <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Olive" title="b1">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">E</font> holds <br/><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b2">x</font> is  <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</a>  ) &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">E</font> &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">G</font> holds <br/><font color="Maroon" title="c6">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b2">x</font></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">  for <font color="Olive" title="b1">e</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">e</font> holds <br/> ex <font color="Olive" title="b2">F</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font> st <br/>( <font color="Olive" title="b2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">E</font> &amp; <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Olive" title="b2">F</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">e</font> &amp; (  for <font color="Olive" title="b3">p</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">p</font> holds <br/> ex <font color="Olive" title="b4">N</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b5">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b4">N</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">n</font> holds <br/> for <font color="Olive" title="b6">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">F</font> holds <br/><span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b5">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b6">x</font></span>)</span> <a href="xxreal_3.html#K10" title="XXREAL_3:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b6">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">p</font> ) )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span>( $2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">E</font> &amp; <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> $2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; (  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="c2">X</font>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> $2 holds <br/><span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> $1</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  ) );<br/>
<a NAME="E6:40"/><span class="lab"><font color="Green" title="E38">A6</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   ex <font color="Olive" title="b2">Z</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">Z</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40"><span class="lab"><font color="Green" title="E36">A4</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">LN</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <font color="Maroon" title="c3">S</font><span class="kw"> such that </span><br/><a NAME="E8:40"/><span class="lab"><font color="Green" title="E39">A7</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c8">LN</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" onmouseover="rs('funct_2/S3')" onmouseout="rh()">FUNCT_2:sch 3</a>(<a class="txt" href="#E6:40"><span class="lab"><font color="Green" title="E38">A6</font></span></a>);<br/></span>
<a NAME="E9:40"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">LN</font> is   <a href="measure1.html#NM3" title="MEASURE1:NM.3">N_Sub_set_fam</a> of <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure1.html#T23" onmouseover="rs('measure1/T23')" onmouseout="rh()">MEASURE1:23</a></span>;<br/>
<a NAME="E10:40"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">LN</font> is    <a href="measure2.html#M1" title="MEASURE2:mode.1">N_Measure_fam</a> of <font color="Maroon" title="c3">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure2.html#D1" onmouseover="rs('measure2/D1')" onmouseout="rh()">MEASURE2:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">MRLN</font> =  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">LN</font></span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="measure2.html#T2" onmouseover="rs('measure2/T2')" onmouseout="rh()">MEASURE2:2</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c10">e0</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.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"> (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">e0</font> implies  ex <font color="Olive" title="b1">F</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font> st <br/>( <font color="Olive" title="b1">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">E</font> &amp; <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Olive" title="b1">F</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">e0</font> &amp; (  for <font color="Olive" title="b2">p</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">p</font> holds <br/> ex <font color="Olive" title="b3">N</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b4">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">N</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">n</font> holds <br/> for <font color="Olive" title="b5">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b5">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">F</font> holds <br/><span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b4">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b5">x</font></span>)</span> <a href="xxreal_3.html#K10" title="XXREAL_3:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b5">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">p</font> ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E12:40"/><span class="lab"><font color="Green" title="E40">A8</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">e0</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">  ex <font color="Olive" title="b1">F</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font> st <br/>( <font color="Olive" title="b1">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">E</font> &amp; <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Olive" title="b1">F</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">e0</font> &amp; (  for <font color="Olive" title="b2">p</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">p</font> holds <br/> ex <font color="Olive" title="b3">N</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b4">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">N</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">n</font> holds <br/> for <font color="Olive" title="b5">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b5">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">F</font> holds <br/><span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b4">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b5">x</font></span>)</span> <a href="xxreal_3.html#K10" title="XXREAL_3:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b5">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">p</font> ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c11">e</font> = <font color="Maroon" title="c10">e0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">G</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font><span class="kw"> such that </span><br/><a NAME="E14:40"/><span class="lab"><font color="Green" title="E41">A9</font></span>: 
<font color="Maroon" title="c12">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">E</font>
 <span class="kw">and </span><br/><a NAME="E15:40"/><span class="lab"><font color="Green" title="E42">A10</font></span>: 
<font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c12">G</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">and </span><br/><a NAME="E16:40"/><span class="lab"><font color="Green" title="E43">A11</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">E</font> holds <br/><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font> is  <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</a> 
 <span class="kw">and </span><br/><a NAME="E17:40"/><span class="lab"><font color="Green" title="E44">A12</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">E</font>
 <span class="kw">and </span><br/><a NAME="E18:40"/><span class="lab"><font color="Green" title="E45">A13</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c12">G</font> holds <br/><font color="Maroon" title="c6">g</font> <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>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40"><span class="lab"><font color="Green" title="E37">A5</font></span></a></span>;<br/>
<a NAME="E19:40"/>
<font color="Maroon" title="c9">MRLN</font> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <font color="Maroon" title="c12">G</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">L</font> = <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">LN</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c12">G</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">gL</font> = <font color="Maroon" title="c6">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">L</font>;<br/>
<a NAME="E21:40"/><span class="lab"><font color="Green" title="E46">A14</font></span>: 
<font color="Maroon" title="c13">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a></span>;<br/>
<a NAME="E22:40"/><span class="kw">then </span>
<font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c13">L</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:40"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="ref" href="measure1.html#T31" onmouseover="rs('measure1/T31')" onmouseout="rh()">MEASURE1:31</a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/>
<a NAME="E23:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A15</font></span>: 
<font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c13">L</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E24:40"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">g</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c13">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>;<br/>
<a NAME="E25:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A16</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="c6">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:40"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="txt" href="#E17:40"><span class="lab"><font color="Green" title="E44">A12</font></span></a>, <a class="txt" href="#E21:40"><span class="lab"><font color="Green" title="E46">A14</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a>, <a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">X</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> $1</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">L</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">fL</font> being   <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a><span class="kw"> such that </span><br/><a NAME="E27:40"/><span class="lab"><font color="Green" title="E49">A17</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c15">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="seqfunc.html#S1" onmouseover="rs('seqfunc/S1')" onmouseout="rh()">SEQFUNC:sch 1</a>();<br/></span>
<a NAME="E28:40"/>
 for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds   <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></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="c15">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">m</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="40_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">n</font>, <font color="Maroon" title="c17">m</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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">  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c16">n</font></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="c15">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">m</font></span>)</span></span><br/>

<a NAME="E1:40_1"/>
<font color="Maroon" title="c15">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">m</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">m</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:40"><span class="lab"><font color="Green" title="E49">A17</font></span></a></span>;<br/>
<a NAME="E2:40_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A18</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="c15">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">m</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c13">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>;<br/>
<a NAME="E3:40_1"/>
<font color="Maroon" title="c15">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c16">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:40"><span class="lab"><font color="Green" title="E49">A17</font></span></a></span>;<br/>
<a NAME="E4:40_1"/><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="c15">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c16">n</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c13">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:40_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c16">n</font></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="c15">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1"><span class="lab"><font color="Green" title="E50">A18</font></span></a>, <a class="ref" href="mesfunc8.html#D2" target="_self" onmouseover="rs('mesfunc8/D2')" onmouseout="rh()">Def2</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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">fL</font> = <font color="Maroon" title="c15">fL</font> as   <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a>  <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc8.html#D2" target="_self" onmouseover="rs('mesfunc8/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<a NAME="E30:40"/><span class="lab"><font color="Green" title="E50">A19</font></span>: 
<font color="Maroon" title="c13">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:40"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="txt" href="#E21:40"><span class="lab"><font color="Green" title="E46">A14</font></span></a></span>;<br/>
<a NAME="E31:40"/><span class="lab"><font color="Green" title="E51">A20</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">L</font> holds <br/>( <font color="Maroon" title="c16">fL</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font> is  <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">L</font></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>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">fL</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font></span>)</span> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="40_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">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="c17">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">L</font> implies ( <font color="Maroon" title="c16">fL</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font> is  <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">L</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">fL</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font></span>)</span> ) )</span><br/>

<a NAME="E1:40_2"/><span class="lab"><font color="Green" title="E52">A21</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="c16">fL</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E2:40_2"/><span class="lab"><font color="Green" title="E53">A22</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="c5">f</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:40_2"/><span class="lab"><font color="Green" title="E54">A23</font></span>: 
<font color="Maroon" title="c17">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">L</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="c16">fL</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font> is  <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">L</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">fL</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font></span>)</span> )</span><br/>

<a NAME="E4:40_2"/><span class="lab"><font color="Green" title="E55">A24</font></span>: 
 for <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c16">fL</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="40_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c16">fL</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:40_2_1"/>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c16">fL</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">n</font> = <font color="Maroon" title="c18">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  ;<br/>
<a NAME="E3:40_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">L</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c17">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_2"><span class="lab"><font color="Green" title="E54">A23</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="E4:40_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A25</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c17">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:40"><span class="lab"><font color="Green" title="E49">A17</font></span></a></span>;<br/>
<a NAME="E5:40_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c19">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#D13" onmouseover="rs('mesfunc5/D13')" onmouseout="rh()">MESFUNC5:def 13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:40_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">fL</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_2_1"><span class="lab"><font color="Green" title="E56">A25</font></span></a>, <a class="ref" href="mesfunc5.html#D13" onmouseover="rs('mesfunc5/D13')" onmouseout="rh()">MESFUNC5:def 13</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>
<a NAME="E5:40_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A26</font></span>: 
<font color="Maroon" title="c16">fL</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T12" onmouseover="rs('funct_2/T12')" onmouseout="rh()">FUNCT_2:12</a></span>;<br/>
<a NAME="E6:40_2"/>
<font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font> is  <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:40"><span class="lab"><font color="Green" title="E43">A11</font></span></a>, <a class="txt" href="#E30:40"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E3:40_2"><span class="lab"><font color="Green" title="E54">A23</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:40_2"/>
<font color="Maroon" title="c16">fL</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font> is  <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_2"><span class="lab"><font color="Green" title="E52">A21</font></span></a>, <a class="txt" href="#E2:40_2"><span class="lab"><font color="Green" title="E53">A22</font></span></a>, <a class="txt" href="#E4:40_2"><span class="lab"><font color="Green" title="E55">A24</font></span></a>, <a class="ref" href="funct_1.html#T2" onmouseover="rs('funct_1/T2')" onmouseout="rh()">FUNCT_1:2</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">L</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">fL</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font></span>)</span></span><br/>

<a NAME="E8:40_2"/>
<font color="Maroon" title="c13">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a></span>;<br/>
<a NAME="E9:40_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:40"><span class="lab"><font color="Green" title="E45">A13</font></span></a>, <a class="txt" href="#E3:40_2"><span class="lab"><font color="Green" title="E54">A23</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:40_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">L</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">fL</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_2"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="txt" href="#E5:40_2"><span class="lab"><font color="Green" title="E56">A26</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">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span>( $2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">L</font> &amp; <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> $2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">e0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> $1 &amp;  ex <font color="Olive" title="b1">Np</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/> for <font color="Olive" title="b2">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Olive" title="b1">Np</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">k</font> holds <br/> for <font color="Olive" title="b3">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">L</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> $2 holds <br/><span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="xxreal_3.html#K10" title="XXREAL_3:func.10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">L</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> $1 );<br/>
<a NAME="E32:40"/><span class="lab"><font color="Green" title="E52">A27</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds <br/> (  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">L</font> &amp; <font color="Maroon" title="c16">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font> is <font color="Maroon" title="c13">L</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  &amp; <font color="Maroon" title="c16">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font> is  <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a>  )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="40_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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"> (  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">L</font> &amp; <font color="Maroon" title="c16">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">n</font> is <font color="Maroon" title="c13">L</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  &amp; <font color="Maroon" title="c16">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">n</font> is  <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a>  )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c18">n9</font> = <font color="Maroon" title="c17">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E2:40_3"/><span class="lab"><font color="Green" title="E53">A28</font></span>: 
<font color="Maroon" title="c16">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:40"><span class="lab"><font color="Green" title="E49">A17</font></span></a></span>;<br/>
<a NAME="E3:40_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A29</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="c16">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">n</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c13">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>;<br/>
<a NAME="E4:40_3"/><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="c16">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">n9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">E</font> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <font color="Maroon" title="c13">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="mesfunc8.html#D2" target="_self" onmouseover="rs('mesfunc8/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:40_3"/><span class="lab"><font color="Green" title="E55">A30</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="c16">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:40"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="txt" href="#E21:40"><span class="lab"><font color="Green" title="E46">A14</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a>, <a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</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"> ( <font color="Maroon" title="c16">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">n</font> is <font color="Maroon" title="c13">L</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  &amp; <font color="Maroon" title="c16">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">n</font> is  <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a>  )</span><br/>

<a NAME="E6:40_3"/>
<font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">n</font> is <font color="Maroon" title="c13">L</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="txt" href="#E30:40"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="ref" href="mesfunc1.html#T30" onmouseover="rs('mesfunc1/T30')" onmouseout="rh()">MESFUNC1:30</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:40_3"/>
<font color="Maroon" title="c16">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">n</font> is <font color="Maroon" title="c13">L</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_3"><span class="lab"><font color="Green" title="E53">A28</font></span></a>, <a class="txt" href="#E3:40_3"><span class="lab"><font color="Green" title="E54">A29</font></span></a>, <a class="txt" href="#E5:40_3"><span class="lab"><font color="Green" title="E55">A30</font></span></a>, <a class="ref" href="mesfunc5.html#T42" onmouseover="rs('mesfunc5/T42')" onmouseout="rh()">MESFUNC5:42</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"> <font color="Maroon" title="c16">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">n</font> is  <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> </span><br/>

<a NAME="E8:40_3"/>
 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="c2">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="c16">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">n</font></span>)</span> holds <br/><span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="40_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">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="c19">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="c16">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">n</font></span>)</span> implies <span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c19">x</font></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  )</span><br/>

<a NAME="E1:40_3_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">LN</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E2:40_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A31</font></span>: 
<font color="Maroon" title="c8">LN</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c18">n9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">LN</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:40_3_1"/><span class="lab"><font color="Green" title="E57">A32</font></span>: 
<font color="Maroon" title="c19">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="c16">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">n</font></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"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c19">x</font></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span><br/>

<a NAME="E4:40_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">LN</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_3"><span class="lab"><font color="Green" title="E55">A30</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E5:40_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">LN</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c17">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_3_1"><span class="lab"><font color="Green" title="E56">A31</font></span></a>, <a class="ref" href="setfam_1.html#D1" onmouseover="rs('setfam_1/D1')" onmouseout="rh()">SETFAM_1:def 1</a></span>;<br/>
<a NAME="E6:40_3_1"/><span class="kw">then </span>
<span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">n9</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c19">x</font></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:40"><span class="lab"><font color="Green" title="E39">A7</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:40_3_1"/>
<span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c19">x</font></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_3"><span class="lab"><font color="Green" title="E53">A28</font></span></a>, <a class="txt" href="#E3:40_3_1"><span class="lab"><font color="Green" title="E57">A32</font></span></a>, <a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</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="E9:40_3"/>
<font color="Maroon" title="c16">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">n</font> is  <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc2.html#D1" onmouseover="rs('mesfunc2/D1')" onmouseout="rh()">MESFUNC2:def 1</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>
<a NAME="E33:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A33</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="c16">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">L</font>
 
;<br/>
<a NAME="E34:40"/><span class="lab"><font color="Green" title="E54">A34</font></span>: 
 for <font color="Olive" title="b1">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  ex <font color="Olive" title="b2">Hp</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font> ex <font color="Olive" title="b3">Np</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Olive" title="b2">Hp</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">L</font> &amp; <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">Hp</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">e0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">p</font> &amp; (  for <font color="Olive" title="b4">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">Np</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">k</font> holds <br/> for <font color="Olive" title="b5">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b5">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">L</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Olive" title="b2">Hp</font> holds <br/><span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b4">k</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b5">x</font></span>)</span> <a href="xxreal_3.html#K10" title="XXREAL_3:func.10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">L</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b5">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">p</font> ) )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="40_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">p</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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">  ex <font color="Olive" title="b1">Hp</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font> ex <font color="Olive" title="b2">Np</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Olive" title="b1">Hp</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">L</font> &amp; <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">Hp</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">e0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c17">p</font> &amp; (  for <font color="Olive" title="b3">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">Np</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">k</font> holds <br/> for <font color="Olive" title="b4">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">L</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Olive" title="b1">Hp</font> holds <br/><span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b3">k</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b4">x</font></span>)</span> <a href="xxreal_3.html#K10" title="XXREAL_3:func.10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">L</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b4">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c17">p</font> ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c18">p9</font> = <font color="Maroon" title="c17">p</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E2:40_4"/><span class="lab"><font color="Green" title="E55">A35</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">e0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c17">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">e0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c18">p9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T9" onmouseover="rs('seq_1/T9')" onmouseout="rh()">SEQ_1:9</a></span>;<br/>
<a NAME="E3:40_4"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">jj</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c17">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#T6" onmouseover="rs('prepower/T6')" onmouseout="rh()">PREPOWER:6</a></span>;<br/>
<a NAME="E4:40_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A36</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c18">p9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#D1" onmouseover="rs('prepower/D1')" onmouseout="rh()">PREPOWER:def 1</a></span>;<br/>
<a NAME="E5:40_4"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">jj</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c17">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#T6" onmouseover="rs('prepower/T6')" onmouseout="rh()">PREPOWER:6</a></span>;<br/>
<a NAME="E6:40_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A37</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c18">p9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#D1" onmouseover="rs('prepower/D1')" onmouseout="rh()">PREPOWER:def 1</a></span>;<br/>
<a NAME="E7:40_4"/>
<font color="Maroon" title="c10">e0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">e0</font> <a href="xxreal_3.html#K6" title="XXREAL_3:func.6">/</a> 2
 
;<br/>
<a NAME="E8:40_4"/><span class="kw">then </span>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">e0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:40"><span class="lab"><font color="Green" title="E40">A8</font></span></a></span>;<br/>
<a NAME="E9:40_4"/><span class="kw">then </span>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">e0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c17">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_4"><span class="lab"><font color="Green" title="E56">A36</font></span></a>, <a class="txt" href="#E2:40_4"><span class="lab"><font color="Green" title="E55">A35</font></span></a>, <a class="ref" href="xreal_1.html#T129" onmouseover="rs('xreal_1/T129')" onmouseout="rh()">XREAL_1:129</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:40_4"/>
 ex <font color="Olive" title="b1">Hp</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font> ex <font color="Olive" title="b2">Np</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Olive" title="b1">Hp</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">L</font> &amp; <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">Hp</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">e0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c17">p</font> &amp; (  for <font color="Olive" title="b3">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">Np</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">k</font> holds <br/> for <font color="Olive" title="b4">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">L</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Olive" title="b1">Hp</font> holds <br/><span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b3">k</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b4">x</font></span>)</span> <a href="xxreal_3.html#K10" title="XXREAL_3:func.10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">L</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b4">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c17">p</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:40"><span class="lab"><font color="Green" title="E47">A15</font></span></a>, <a class="txt" href="#E32:40"><span class="lab"><font color="Green" title="E52">A27</font></span></a>, <a class="txt" href="#E33:40"><span class="lab"><font color="Green" title="E53">A33</font></span></a>, <a class="txt" href="#E25:40"><span class="lab"><font color="Green" title="E48">A16</font></span></a>, <a class="txt" href="#E31:40"><span class="lab"><font color="Green" title="E51">A20</font></span></a>, <a class="txt" href="#E6:40_4"><span class="lab"><font color="Green" title="E57">A37</font></span></a>, <a class="ref" href="mesfunc8.html#T28" target="_self" onmouseover="rs('mesfunc8/T28')" onmouseout="rh()">Th28</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>
<a NAME="E35:40"/><span class="lab"><font color="Green" title="E55">A38</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   ex <font color="Olive" title="b2">Z</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">Z</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="40_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</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">  ex <font color="Olive" title="b1">Z</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c17">n</font>,<font color="Olive" title="b1">Z</font>]</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c18">n9</font> = <font color="Maroon" title="c17">n</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c19">Z</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font>, <font color="Maroon" title="c20">Np</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:40_5"/><span class="lab"><font color="Green" title="E56">A39</font></span>: 
<font color="Maroon" title="c19">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">L</font>
 <span class="kw">and </span><br/><a NAME="E4:40_5"/><span class="lab"><font color="Green" title="E57">A40</font></span>: 
<font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c19">Z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">e0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c18">n9</font>
 <span class="kw">and </span><br/><a NAME="E5:40_5"/><span class="lab"><font color="Green" title="E58">A41</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c20">Np</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> holds <br/> for <font color="Olive" title="b2">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">L</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c19">Z</font> holds <br/><span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="xxreal_3.html#K10" title="XXREAL_3:func.10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">L</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c18">n9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:40"><span class="lab"><font color="Green" title="E54">A34</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c21">Np9</font> = <font color="Maroon" title="c20">Np</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E7:40_5"/>
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Maroon" title="c21">Np9</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> holds <br/> for <font color="Olive" title="b2">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">L</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c19">Z</font> holds <br/><span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="xxreal_3.html#K10" title="XXREAL_3:func.10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">L</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c18">n9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_5"><span class="lab"><font color="Green" title="E58">A41</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:40_5"/>
 ex <font color="Olive" title="b1">Z</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c17">n</font>,<font color="Olive" title="b1">Z</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_5"><span class="lab"><font color="Green" title="E56">A39</font></span></a>, <a class="txt" href="#E4:40_5"><span class="lab"><font color="Green" title="E57">A40</font></span></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">consider </span><font color="Maroon" title="c17">HP</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <font color="Maroon" title="c3">S</font><span class="kw"> such that </span><br/><a NAME="E37:40"/><span class="lab"><font color="Green" title="E56">A42</font></span>: 
 for <font color="Olive" title="b1">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c17">HP</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">p</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" onmouseover="rs('funct_2/S3')" onmouseout="rh()">FUNCT_2:sch 3</a>(<a class="txt" href="#E35:40"><span class="lab"><font color="Green" title="E55">A38</font></span></a>);<br/></span>
<span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">HP</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> $1</span>)</span>;<br/>
<a NAME="E38:40"/><span class="lab"><font color="Green" title="E57">A43</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   ex <font color="Olive" title="b2">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="40_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</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">  ex <font color="Olive" title="b1">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c18">n</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<a NAME="E1:40_6"/><span class="lab"><font color="Green" title="E58">A44</font></span>: 
 <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">HP</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c18">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure1.html#D2" onmouseover="rs('measure1/D2')" onmouseout="rh()">MEASURE1:def 2</a></span>;<br/>
<a NAME="E2:40_6"/>
<font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">HP</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">e0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c18">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:40"><span class="lab"><font color="Green" title="E56">A42</font></span></a></span>;<br/>
<a NAME="E3:40_6"/><span class="kw">then </span>
<font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">HP</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_6"><span class="lab"><font color="Green" title="E58">A44</font></span></a>, <a class="ref" href="xxreal_0.html#T48" onmouseover="rs('xxreal_0/T48')" onmouseout="rh()">XXREAL_0:48</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:40_6"/>
 ex <font color="Olive" title="b1">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c18">n</font>,<font color="Olive" title="b1">y</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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c18">me</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><span class="kw"> such that </span><br/><a NAME="E40:40"/><span class="lab"><font color="Green" title="E58">A45</font></span>: 
 for <font color="Olive" title="b1">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c18">me</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">p</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" onmouseover="rs('funct_2/S3')" onmouseout="rh()">FUNCT_2:sch 3</a>(<a class="txt" href="#E38:40"><span class="lab"><font color="Green" title="E57">A43</font></span></a>);<br/></span>
<a NAME="E41:40"/><span class="lab"><font color="Green" title="E59">A46</font></span>: 
 for <font color="Olive" title="b1">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <font color="Maroon" title="c18">me</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">e0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">p</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="40_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">p</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</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="c18">me</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c19">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">e0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c19">p</font></span><br/>

<a NAME="E1:40_7"/>
<font color="Maroon" title="c18">me</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c19">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">HP</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c19">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:40"><span class="lab"><font color="Green" title="E58">A45</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:40_7"/>
<font color="Maroon" title="c18">me</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c19">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">e0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c19">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:40"><span class="lab"><font color="Green" title="E56">A42</font></span></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>
<a NAME="E42:40"/><span class="lab"><font color="Green" title="E60">A47</font></span>: 
 for <font color="Olive" title="b1">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds <br/> (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">me</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">p</font> &amp; <font color="Maroon" title="c18">me</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">e0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">p</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="40_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">p</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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"> (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">me</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c19">p</font> &amp; <font color="Maroon" title="c18">me</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c19">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">e0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c19">p</font> )</span><br/>

<a NAME="E1:40_8"/><span class="lab"><font color="Green" title="E61">A48</font></span>: 
<font color="Maroon" title="c19">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E2:40_8"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">HP</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c19">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure1.html#D2" onmouseover="rs('measure1/D2')" onmouseout="rh()">MEASURE1:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:40_8"/>
(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">me</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c19">p</font> &amp; <font color="Maroon" title="c18">me</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c19">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">e0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c19">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:40"><span class="lab"><font color="Green" title="E58">A45</font></span></a>, <a class="txt" href="#E41:40"><span class="lab"><font color="Green" title="E59">A46</font></span></a>, <a class="txt" href="#E1:40_8"><span class="lab"><font color="Green" title="E61">A48</font></span></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>
<a NAME="E43:40"/><span class="kw">then </span>
 for <font color="Olive" title="b1">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds <br/> (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">me</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">p</font> &amp; <font color="Maroon" title="c18">me</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">e0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">p</font> )
 
;<br/>
<a NAME="E44:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A49</font></span>: 
<font color="Maroon" title="c18">me</font> is  <a href="partfun3.html#V4" title="PARTFUN3:attr.4">nonnegative</a> 
 
;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ) <span class="kw">-&gt; </span>  <a href="prob_1.html#M1" title="PROB_1:mode.1">M12</a>(<font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">S</font>) = <font color="Maroon" title="c7">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">LN</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> $1</span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c19">ELN</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <font color="Maroon" title="c3">S</font><span class="kw"> such that </span><br/><a NAME="E46:40"/><span class="lab"><font color="Green" title="E62">A50</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <font color="Maroon" title="c19">ELN</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">n</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S4" onmouseover="rs('funct_2/S4')" onmouseout="rh()">FUNCT_2:sch 4</a>();<br/></span>
<a NAME="E47:40"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c19">ELN</font> is   <a href="measure1.html#NM3" title="MEASURE1:NM.3">N_Sub_set_fam</a> of <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure1.html#T23" onmouseover="rs('measure1/T23')" onmouseout="rh()">MEASURE1:23</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">RELN</font> =  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c19">ELN</font> as    <a href="measure2.html#M1" title="MEASURE2:mode.1">N_Measure_fam</a> of <font color="Maroon" title="c3">S</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="measure2.html#D1" onmouseover="rs('measure2/D1')" onmouseout="rh()">MEASURE2:def 1</a></span>;<br/>
<a NAME="E49:40"/><span class="lab"><font color="Green" title="E63">A51</font></span>: 
<font color="Maroon" title="c7">E</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">HP</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">HP</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a></span>;<br/>
<a NAME="E50:40"/>
 for <font color="Olive" title="b1">A</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c20">RELN</font> holds <br/><font color="Olive" title="b1">A</font> is    <a href="measure1.html#M2" title="MEASURE1:mode.2">measure_zero</a> of <font color="Maroon" title="c4">M</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="40_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c21">A</font> be    <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"> ( <font color="Maroon" title="c21">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c20">RELN</font> implies <font color="Maroon" title="c21">A</font> is    <a href="measure1.html#M2" title="MEASURE1:mode.2">measure_zero</a> of <font color="Maroon" title="c4">M</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:40_9"/>
<font color="Maroon" title="c21">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c20">RELN</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="c21">A</font> is    <a href="measure1.html#M2" title="MEASURE1:mode.2">measure_zero</a> of <font color="Maroon" title="c4">M</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c22">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:40_9"/><span class="lab"><font color="Green" title="E64">A52</font></span>: 
<font color="Maroon" title="c22">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 <span class="kw">and </span><br/><a NAME="E4:40_9"/><span class="lab"><font color="Green" title="E65">A53</font></span>: 
<font color="Maroon" title="c21">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">ELN</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T11" onmouseover="rs('funct_2/T11')" onmouseout="rh()">FUNCT_2:11</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c23">n</font> = <font color="Maroon" title="c22">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_9"><span class="lab"><font color="Green" title="E64">A52</font></span></a></span>;<br/>
<font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">ELN</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">n</font></span>)</span> = 
<font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">LN</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:40"><span class="lab"><font color="Green" title="E62">A50</font></span></a></span>
<br/>.= 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:40"><span class="lab"><font color="Green" title="E39">A7</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E7:40_9"/>
<font color="Maroon" title="c21">A</font> is    <a href="measure1.html#M2" title="MEASURE1:mode.2">measure_zero</a> of <font color="Maroon" title="c4">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_9"><span class="lab"><font color="Green" title="E65">A53</font></span></a>, <a class="ref" href="measure1.html#D7" onmouseover="rs('measure1/D7')" onmouseout="rh()">MEASURE1:def 7</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>
<a NAME="E51:40"/><span class="kw">then </span>
 <a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c20">RELN</font> is    <a href="measure1.html#M2" title="MEASURE1:mode.2">measure_zero</a> of <font color="Maroon" title="c4">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure2.html#T14" onmouseover="rs('measure2/T14')" onmouseout="rh()">MEASURE2:14</a></span>;<br/>
<a NAME="E52:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A54</font></span>: 
<font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c20">RELN</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure1.html#D7" onmouseover="rs('measure1/D7')" onmouseout="rh()">MEASURE1:def 7</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="40_10"><span class="kw">now </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">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">E</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">LN</font></span>)</span></span>)</span> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c20">RELN</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c21">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">E</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">LN</font></span>)</span></span>)</span> implies <font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c20">RELN</font> )</span><br/><span class="kw">assume </span><a NAME="E1:40_10"/><span class="lab"><font color="Green" title="E65">A55</font></span>: 
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">E</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">LN</font></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"> <font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c20">RELN</font></span><br/><a NAME="E2:40_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A56</font></span>: 
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><a NAME="E3:40_10"/>
 not <font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">LN</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_10"><span class="lab"><font color="Green" title="E65">A55</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c22">Y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E5:40_10"/><span class="lab"><font color="Green" title="E67">A57</font></span>: 
<font color="Maroon" title="c22">Y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">LN</font>
 <span class="kw">and </span><br/><a NAME="E6:40_10"/><span class="lab"><font color="Green" title="E68">A58</font></span>: 
 not <font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#D1" onmouseover="rs('setfam_1/D1')" onmouseout="rh()">SETFAM_1:def 1</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c23">m</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E8:40_10"/><span class="lab"><font color="Green" title="E69">A59</font></span>: 
<font color="Maroon" title="c23">m</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="c8">LN</font>
 <span class="kw">and </span><br/><a NAME="E9:40_10"/><span class="lab"><font color="Green" title="E70">A60</font></span>: 
<font color="Maroon" title="c22">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">LN</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_10"><span class="lab"><font color="Green" title="E67">A57</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c24">m</font> = <font color="Maroon" title="c23">m</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:40_10"><span class="lab"><font color="Green" title="E69">A59</font></span></a></span>;<br/><a NAME="E11:40_10"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c19">ELN</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><a NAME="E12:40_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A61</font></span>: 
<font color="Maroon" title="c19">ELN</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c24">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c19">ELN</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/><a NAME="E13:40_10"/>
<font color="Maroon" title="c19">ELN</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">LN</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c24">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:40"><span class="lab"><font color="Green" title="E62">A50</font></span></a></span>;<br/><a NAME="E14:40_10"/><span class="kw">then </span>
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">ELN</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c24">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_10"><span class="lab"><font color="Green" title="E66">A56</font></span></a>, <a class="txt" href="#E6:40_10"><span class="lab"><font color="Green" title="E68">A58</font></span></a>, <a class="txt" href="#E9:40_10"><span class="lab"><font color="Green" title="E70">A60</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E15:40_10"/>
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c20">RELN</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:40_10"><span class="lab"><font color="Green" title="E71">A61</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</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>
<a NAME="E54:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A62</font></span>: 
<font color="Maroon" title="c7">E</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">LN</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c20">RELN</font>
 
;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="40_11"><span class="kw">now </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">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c20">RELN</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">E</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">LN</font></span>)</span></span>)</span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c21">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c20">RELN</font> implies <font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">E</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">LN</font></span>)</span></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:40_11"/>
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c20">RELN</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="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">E</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">LN</font></span>)</span></span>)</span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c22">Y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:40_11"/><span class="lab"><font color="Green" title="E66">A63</font></span>: 
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">Y</font>
 <span class="kw">and </span><br/><a NAME="E4:40_11"/><span class="lab"><font color="Green" title="E67">A64</font></span>: 
<font color="Maroon" title="c22">Y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c20">RELN</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c23">m</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:40_11"/><span class="lab"><font color="Green" title="E68">A65</font></span>: 
<font color="Maroon" title="c23">m</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="c19">ELN</font>
 <span class="kw">and </span><br/><a NAME="E7:40_11"/><span class="lab"><font color="Green" title="E69">A66</font></span>: 
<font color="Maroon" title="c19">ELN</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_11"><span class="lab"><font color="Green" title="E67">A64</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c24">m</font> = <font color="Maroon" title="c23">m</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40_11"><span class="lab"><font color="Green" title="E68">A65</font></span></a></span>;<br/><a NAME="E9:40_11"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">LN</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><a NAME="E10:40_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A67</font></span>: 
<font color="Maroon" title="c8">LN</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c24">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">LN</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/><a NAME="E11:40_11"/><span class="lab"><font color="Green" title="E71">A68</font></span>: 
<font color="Maroon" title="c22">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">LN</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c24">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:40"><span class="lab"><font color="Green" title="E62">A50</font></span></a>, <a class="txt" href="#E7:40_11"><span class="lab"><font color="Green" title="E69">A66</font></span></a></span>;<br/><a NAME="E12:40_11"/><span class="kw">then </span>
 not <font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">LN</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c24">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_11"><span class="lab"><font color="Green" title="E66">A63</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><a NAME="E13:40_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A69</font></span>: 
 not <font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">LN</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:40_11"><span class="lab"><font color="Green" title="E70">A67</font></span></a>, <a class="ref" href="setfam_1.html#D1" onmouseover="rs('setfam_1/D1')" onmouseout="rh()">SETFAM_1:def 1</a></span>;<br/><a NAME="E14:40_11"/>
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_11"><span class="lab"><font color="Green" title="E66">A63</font></span></a>, <a class="txt" href="#E11:40_11"><span class="lab"><font color="Green" title="E71">A68</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E15:40_11"/>
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">E</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">LN</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:40_11"><span class="lab"><font color="Green" title="E72">A69</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</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>
<a NAME="E56:40"/><span class="kw">then </span>
 <a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c20">RELN</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">E</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">LN</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E57:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A70</font></span>: 
 <a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c20">RELN</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">E</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">LN</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:40"><span class="lab"><font color="Green" title="E65">A62</font></span></a>, <a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E58:40"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">HP</font> is   <a href="measure1.html#NM3" title="MEASURE1:NM.3">N_Sub_set_fam</a> of <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure1.html#T23" onmouseover="rs('measure1/T23')" onmouseout="rh()">MEASURE1:23</a></span>;<br/>
<a NAME="E59:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A71</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">HP</font> is    <a href="measure2.html#M1" title="MEASURE2:mode.1">N_Measure_fam</a> of <font color="Maroon" title="c3">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure2.html#D1" onmouseover="rs('measure2/D1')" onmouseout="rh()">MEASURE2:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c21">MRHP</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">HP</font></span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="measure1.html#T24" onmouseover="rs('measure1/T24')" onmouseout="rh()">MEASURE1:24</a></span>;<br/>
<a NAME="E61:40"/>
<font color="Maroon" title="c13">L</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c21">MRHP</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c22">F</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font><span class="kw"> such that </span><br/><a NAME="E63:40"/><span class="lab"><font color="Green" title="E68">A72</font></span>: 
<font color="Maroon" title="c22">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">L</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">HP</font></span>)</span></span>)</span>
 ;<br/>
<a NAME="E64:40"/>
<font color="Maroon" title="c7">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c13">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">LN</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c12">G</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T54" onmouseover="rs('xboole_1/T54')" onmouseout="rh()">XBOOLE_1:54</a></span>;<br/>
<a NAME="E65:40"/><span class="kw">then </span>
<font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c13">L</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="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p2">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c20">RELN</font></span>)</span></span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c12">G</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:40"><span class="lab"><font color="Green" title="E66">A70</font></span></a>, <a class="ref" href="measure1.html#T33" onmouseover="rs('measure1/T33')" onmouseout="rh()">MEASURE1:33</a></span>;<br/>
<a NAME="E66:40"/><span class="kw">then </span>
<font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c13">L</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:40"><span class="lab"><font color="Green" title="E42">A10</font></span></a>, <a class="txt" href="#E52:40"><span class="lab"><font color="Green" title="E64">A54</font></span></a></span>;<br/>
<a NAME="E67:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A73</font></span>: 
<font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c13">L</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="measure1.html#D2" onmouseover="rs('measure1/D2')" onmouseout="rh()">MEASURE1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c23">MRHP</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">HP</font></span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="measure1.html#T24" onmouseover="rs('measure1/T24')" onmouseout="rh()">MEASURE1:24</a></span>;<br/>
<a NAME="E69:40"/><span class="lab"><font color="Green" title="E70">A74</font></span>: 
<font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c13">L</font></span>)</span> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c23">MRHP</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="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c13">L</font></span>)</span></span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c23">MRHP</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure1.html#T33" onmouseover="rs('measure1/T33')" onmouseout="rh()">MEASURE1:33</a></span>;<br/>
<a NAME="E70:40"/>
<font color="Maroon" title="c7">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c22">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c13">L</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">HP</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E63:40"><span class="lab"><font color="Green" title="E68">A72</font></span></a>, <a class="ref" href="xboole_1.html#T52" onmouseover="rs('xboole_1/T52')" onmouseout="rh()">XBOOLE_1:52</a></span>;<br/>
<a NAME="E71:40"/><span class="kw">then </span>
<font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c22">F</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c13">L</font></span>)</span> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c23">MRHP</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:40"><span class="lab"><font color="Green" title="E63">A51</font></span></a>, <a class="ref" href="measure1.html#T31" onmouseover="rs('measure1/T31')" onmouseout="rh()">MEASURE1:31</a>, <a class="ref" href="xboole_1.html#T9" onmouseover="rs('xboole_1/T9')" onmouseout="rh()">XBOOLE_1:9</a></span>;<br/>
<a NAME="E72:40"/><span class="kw">then </span>
<font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c22">F</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="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c13">L</font></span>)</span></span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c23">MRHP</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E69:40"><span class="lab"><font color="Green" title="E70">A74</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E73:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A75</font></span>: 
<font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c22">F</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c23">MRHP</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E67:40"><span class="lab"><font color="Green" title="E69">A73</font></span></a>, <a class="ref" href="xxreal_3.html#T4" onmouseover="rs('xxreal_3/T4')" onmouseout="rh()">XXREAL_3:4</a></span>;<br/>
<a NAME="E74:40"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c18">me</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E75:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A76</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c18">me</font> <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="c4">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c17">HP</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E76:40"/><span class="lab"><font color="Green" title="E73">A77</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">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="c18">me</font> holds <br/><font color="Maroon" title="c18">me</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="c4">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c17">HP</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="40_12"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c24">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="c18">me</font> implies <font color="Maroon" title="c18">me</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c17">HP</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:40_12"/><span class="lab"><font color="Green" title="E74">A78</font></span>: 
<font color="Maroon" title="c24">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="c18">me</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="c18">me</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c17">HP</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font></span><br/>

<a NAME="E2:40_12"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c17">HP</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">HP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:40"><span class="lab"><font color="Green" title="E72">A76</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:40_12"/>
<font color="Maroon" title="c18">me</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c17">HP</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:40"><span class="lab"><font color="Green" title="E58">A45</font></span></a>, <a class="txt" href="#E1:40_12"><span class="lab"><font color="Green" title="E74">A78</font></span></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>
<a NAME="E77:40"/><span class="lab"><font color="Green" title="E74">A79</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a></span>;<br/>
<a NAME="E78:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A80</font></span>: 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a>  is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#T24" onmouseover="rs('series_1/T24')" onmouseout="rh()">SERIES_1:24</a></span>;<br/>
<a NAME="E79:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A81</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">e0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#T10" onmouseover="rs('series_1/T10')" onmouseout="rh()">SERIES_1:10</a></span>;<br/>
<a NAME="E80:40"/><span class="kw">then </span>
<font color="Maroon" title="c18">me</font> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:40"><span class="lab"><font color="Green" title="E60">A47</font></span></a>, <a class="ref" href="series_1.html#T20" onmouseover="rs('series_1/T20')" onmouseout="rh()">SERIES_1:20</a></span>;<br/>
<a NAME="E81:40"/><span class="kw">then </span>
 <a href="series_1.html#K4" title="SERIES_1:func.4">Sum</a> <font color="Maroon" title="c18">me</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K14" title="SUPINF_2:func.14">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c17">HP</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:40"><span class="lab"><font color="Green" title="E61">A49</font></span></a>, <a class="txt" href="#E75:40"><span class="lab"><font color="Green" title="E72">A76</font></span></a>, <a class="txt" href="#E76:40"><span class="lab"><font color="Green" title="E73">A77</font></span></a>, <a class="ref" href="funct_1.html#T2" onmouseover="rs('funct_1/T2')" onmouseout="rh()">FUNCT_1:2</a>, <a class="ref" href="prob_4.html#T12" onmouseover="rs('prob_4/T12')" onmouseout="rh()">PROB_4:12</a></span>;<br/>
<a NAME="E82:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A82</font></span>: 
<font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">HP</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="series_1.html#K4" title="SERIES_1:func.4">Sum</a> <font color="Maroon" title="c18">me</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E59:40"><span class="lab"><font color="Green" title="E67">A71</font></span></a>, <a class="ref" href="measure2.html#T11" onmouseover="rs('measure2/T11')" onmouseout="rh()">MEASURE2:11</a></span>;<br/>
<a NAME="E83:40"/><span class="lab"><font color="Green" title="E78">A83</font></span>: 
 for <font color="Olive" title="b1">q</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">q</font> holds <br/> ex <font color="Olive" title="b2">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">q</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="40_13"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">q</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.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"> (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">q</font> implies  ex <font color="Olive" title="b1">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">q</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:40_13"/><span class="lab"><font color="Green" title="E79">A84</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">q</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">  ex <font color="Olive" title="b1">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">q</font></span><br/>

<a NAME="E2:40_13"/><span class="lab"><font color="Green" title="E80">A85</font></span>: 
 <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></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="txt" href="#E78:40"><span class="lab"><font color="Green" title="E75">A80</font></span></a>, <a class="ref" href="series_1.html#T4" onmouseover="rs('series_1/T4')" onmouseout="rh()">SERIES_1:4</a></span>;<br/>
<a NAME="E3:40_13"/>
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a>  is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:40"><span class="lab"><font color="Green" title="E75">A80</font></span></a>, <a class="ref" href="series_1.html#T4" onmouseover="rs('series_1/T4')" onmouseout="rh()">SERIES_1:4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E5:40_13"/><span class="lab"><font color="Green" title="E81">A86</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c25">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_13"><span class="lab"><font color="Green" title="E79">A84</font></span></a>, <a class="txt" href="#E2:40_13"><span class="lab"><font color="Green" title="E80">A85</font></span></a>, <a class="ref" href="seq_2.html#D7" onmouseover="rs('seq_2/D7')" onmouseout="rh()">SEQ_2:def 7</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c25">p</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="c25">p</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c25">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">q</font> )</span><br/>

<a NAME="E6:40_13"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">jj</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c25">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#T6" onmouseover="rs('prepower/T6')" onmouseout="rh()">PREPOWER:6</a></span>;<br/>
<a NAME="E7:40_13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A87</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c25">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#D1" onmouseover="rs('prepower/D1')" onmouseout="rh()">PREPOWER:def 1</a></span>;<br/>
<a NAME="E8:40_13"/><span class="lab"><font color="Green" title="E83">A88</font></span>: 
<font color="Maroon" title="c25">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E9:40_13"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c25">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_13"><span class="lab"><font color="Green" title="E81">A86</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:40_13"/>
( <font color="Maroon" title="c25">p</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c25">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:40_13"><span class="lab"><font color="Green" title="E82">A87</font></span></a>, <a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a>, <a class="txt" href="#E8:40_13"><span class="lab"><font color="Green" title="E83">A88</font></span></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>
<a NAME="E84:40"/><span class="lab"><font color="Green" title="E79">A89</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c22">F</font> holds <br/> for <font color="Olive" title="b2">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">L</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">HP</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">p</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="40_14"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">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="c24">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c22">F</font> implies  for <font color="Olive" title="b1">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <font color="Maroon" title="c24">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">L</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">HP</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">p</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:40_14"/><span class="lab"><font color="Green" title="E80">A90</font></span>: 
<font color="Maroon" title="c24">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c22">F</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">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <font color="Maroon" title="c24">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">L</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">HP</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">p</font></span>)</span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c25">p</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</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="c24">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">L</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">HP</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c25">p</font></span>)</span></span><br/>

<a NAME="E2:40_14"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c17">HP</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E3:40_14"/><span class="kw">then </span>
<font color="Maroon" title="c17">HP</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c25">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">HP</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/>
<a NAME="E4:40_14"/><span class="kw">then </span>
<font color="Maroon" title="c13">L</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">HP</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">L</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">HP</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c25">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T34" onmouseover="rs('xboole_1/T34')" onmouseout="rh()">XBOOLE_1:34</a>, <a class="ref" href="zfmisc_1.html#T74" onmouseover="rs('zfmisc_1/T74')" onmouseout="rh()">ZFMISC_1:74</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:40_14"/>
<font color="Maroon" title="c24">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">L</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">HP</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c25">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E63:40"><span class="lab"><font color="Green" title="E68">A72</font></span></a>, <a class="txt" href="#E1:40_14"><span class="lab"><font color="Green" title="E80">A90</font></span></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>
<a NAME="E85:40"/><span class="lab"><font color="Green" title="E80">A91</font></span>: 
 for <font color="Olive" title="b1">q</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">q</font> holds <br/> ex <font color="Olive" title="b2">N</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">N</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">n</font> holds <br/> for <font color="Olive" title="b4">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c22">F</font> holds <br/><span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b3">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b4">x</font></span>)</span> <a href="xxreal_3.html#K10" title="XXREAL_3:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b4">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">q</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="40_15"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">q</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.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"> (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">q</font> implies  ex <font color="Olive" title="b1">N</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">N</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">n</font> holds <br/> for <font color="Olive" title="b3">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c22">F</font> holds <br/><span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="xxreal_3.html#K10" title="XXREAL_3:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">q</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:40_15"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">q</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">  ex <font color="Olive" title="b1">N</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">N</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">n</font> holds <br/> for <font color="Olive" title="b3">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c22">F</font> holds <br/><span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="xxreal_3.html#K10" title="XXREAL_3:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">q</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:40_15"/><span class="lab"><font color="Green" title="E81">A92</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c25">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E83:40"><span class="lab"><font color="Green" title="E78">A83</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c26">Np</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E5:40_15"/><span class="lab"><font color="Green" title="E82">A93</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Maroon" title="c26">Np</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> holds <br/> for <font color="Olive" title="b2">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">L</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">HP</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c25">p</font></span>)</span> holds <br/><span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="xxreal_3.html#K10" title="XXREAL_3:func.10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">L</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c25">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:40"><span class="lab"><font color="Green" title="E56">A42</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c26">Np</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">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c26">Np</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">n</font> holds <br/> for <font color="Olive" title="b2">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c22">F</font> holds <br/><span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="xxreal_3.html#K10" title="XXREAL_3:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">q</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"> verum</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c27">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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="c26">Np</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c27">n</font> implies  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="c2">X</font>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c22">F</font> holds <br/><span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c27">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="xxreal_3.html#K10" title="XXREAL_3:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">q</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c28">n9</font> = <font color="Maroon" title="c27">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/><span class="kw">assume </span><a NAME="E2:40_15_1"/><span class="lab"><font color="Green" title="E83">A94</font></span>: 
<font color="Maroon" title="c26">Np</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c27">n</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">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c22">F</font> holds <br/><span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c27">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="xxreal_3.html#K10" title="XXREAL_3:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">q</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"> verum</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c29">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">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="c29">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c22">F</font> implies <span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c27">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c29">x</font></span>)</span> <a href="xxreal_3.html#K10" title="XXREAL_3:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c29">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">q</font> )</span><br/><a NAME="E1:40_15_1_1"/><span class="lab"><font color="Green" title="E84">A95</font></span>: 
<font color="Maroon" title="c16">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c27">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c27">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:40"><span class="lab"><font color="Green" title="E49">A17</font></span></a></span>;<br/><span class="kw">assume </span><a NAME="E2:40_15_1_1"/><span class="lab"><font color="Green" title="E85">A96</font></span>: 
<font color="Maroon" title="c29">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c22">F</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"> <span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c27">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c29">x</font></span>)</span> <a href="xxreal_3.html#K10" title="XXREAL_3:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c29">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">q</font></span><br/><a NAME="E3:40_15_1_1"/><span class="kw">then </span>
<span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c28">n9</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c29">x</font></span>)</span> <a href="xxreal_3.html#K10" title="XXREAL_3:func.10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">L</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c29">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c25">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E84:40"><span class="lab"><font color="Green" title="E79">A89</font></span></a>, <a class="txt" href="#E5:40_15"><span class="lab"><font color="Green" title="E82">A93</font></span></a>, <a class="txt" href="#E2:40_15_1"><span class="lab"><font color="Green" title="E83">A94</font></span></a></span>;<br/><a NAME="E4:40_15_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A97</font></span>: 
<span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">fL</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c27">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c29">x</font></span>)</span> <a href="xxreal_3.html#K10" title="XXREAL_3:func.10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">L</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c29">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_15"><span class="lab"><font color="Green" title="E81">A92</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E5:40_15_1_1"/><span class="lab"><font color="Green" title="E87">A98</font></span>: 
<font color="Maroon" title="c22">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E63:40"><span class="lab"><font color="Green" title="E68">A72</font></span></a>, <a class="ref" href="xboole_1.html#T36" onmouseover="rs('xboole_1/T36')" onmouseout="rh()">XBOOLE_1:36</a></span>;<br/><a NAME="E6:40_15_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">L</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c29">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c29">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_15_1_1"><span class="lab"><font color="Green" title="E85">A96</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/><span class="kw">hence </span><a NAME="E7:40_15_1_1"/>
<span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c27">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c29">x</font></span>)</span> <a href="xxreal_3.html#K10" title="XXREAL_3:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c29">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_15_1_1"><span class="lab"><font color="Green" title="E85">A96</font></span></a>, <a class="txt" href="#E4:40_15_1_1"><span class="lab"><font color="Green" title="E86">A97</font></span></a>, <a class="txt" href="#E1:40_15_1_1"><span class="lab"><font color="Green" title="E84">A95</font></span></a>, <a class="txt" href="#E5:40_15_1_1"><span class="lab"><font color="Green" title="E87">A98</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></div>
<span class="kw">end;</span></div>


</div><span class="kw">end;</span></div>
<a NAME="E86:40"/>
 <a href="series_1.html#K4" title="SERIES_1:func.4">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E77:40"><span class="lab"><font color="Green" title="E74">A79</font></span></a>, <a class="ref" href="series_1.html#T24" onmouseover="rs('series_1/T24')" onmouseout="rh()">SERIES_1:24</a></span>;<br/>
<a NAME="E87:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A99</font></span>: 
 <a href="series_1.html#K4" title="SERIES_1:func.4">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">e0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">e0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:40"><span class="lab"><font color="Green" title="E75">A80</font></span></a>, <a class="ref" href="series_1.html#T10" onmouseover="rs('series_1/T10')" onmouseout="rh()">SERIES_1:10</a></span>;<br/>
<a NAME="E88:40"/>
 <a href="series_1.html#K4" title="SERIES_1:func.4">Sum</a> <font color="Maroon" title="c18">me</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="series_1.html#K4" title="SERIES_1:func.4">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">e0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E79:40"><span class="lab"><font color="Green" title="E76">A81</font></span></a>, <a class="txt" href="#E42:40"><span class="lab"><font color="Green" title="E60">A47</font></span></a>, <a class="ref" href="series_1.html#T20" onmouseover="rs('series_1/T20')" onmouseout="rh()">SERIES_1:20</a></span>;<br/>
<a NAME="E89:40"/><span class="kw">then </span>
<font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c23">MRHP</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">e0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E82:40"><span class="lab"><font color="Green" title="E77">A82</font></span></a>, <a class="txt" href="#E87:40"><span class="lab"><font color="Green" title="E81">A99</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E90:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A100</font></span>: 
<font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c22">F</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">e0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E73:40"><span class="lab"><font color="Green" title="E71">A75</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E91:40"/>
<font color="Maroon" title="c22">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E63:40"><span class="lab"><font color="Green" title="E68">A72</font></span></a>, <a class="ref" href="xboole_1.html#T36" onmouseover="rs('xboole_1/T36')" onmouseout="rh()">XBOOLE_1:36</a></span>;<br/>
<span class="kw">hence </span><a NAME="E92:40"/>
 ex <font color="Olive" title="b1">F</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font> st <br/>( <font color="Olive" title="b1">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">E</font> &amp; <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Olive" title="b1">F</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">e0</font> &amp; (  for <font color="Olive" title="b2">p</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">p</font> holds <br/> ex <font color="Olive" title="b3">N</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b4">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">N</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">n</font> holds <br/> for <font color="Olive" title="b5">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b5">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">F</font> holds <br/><span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b4">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b5">x</font></span>)</span> <a href="xxreal_3.html#K10" title="XXREAL_3:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b5">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">p</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:40"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E90:40"><span class="lab"><font color="Green" title="E82">A100</font></span></a>, <a class="txt" href="#E85:40"><span class="lab"><font color="Green" title="E80">A91</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</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>
