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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">f</font> be  <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a>  <a href="mesfunc8.html#M1" title="MESFUNC8:mode.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> ;<br/><b>let </b><font color="Maroon" title="c4">F</font> be    <a href="prob_1.html#M1" title="PROB_1:mode.1">SetSequence</a> of <font color="Maroon" title="c2">S</font>;<br/><b>let </b><font color="Maroon" title="c5">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>









<b>set </b><font color="Maroon" title="c6">E</font> =  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>;<br/>
<b>assume </b><a NAME="E1:30"/><i><font color="Green" title="E24">AS</font></i>: 
for <font color="Olive" title="b1">n</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <font color="Maroon" title="c4">F</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K16" title="MESFUNC1:func.16">great_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon" title="c7">n</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>reconsider </b><font color="Maroon" title="c8">n'</font> = <font color="Maroon" title="c7">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">f1</font> = <font color="Maroon" title="c3">f</font> <a href="mesfunc8.html#K5" title="MESFUNC8:func.5">^\</a> <font color="Maroon" title="c8">n'</font>;<br/>
<b>set </b><font color="Maroon" title="c10">F1</font> = <font color="Maroon" title="c4">F</font> <a href="kurato_2.html#K3" title="KURATO_2:func.3">^\</a> <font color="Maroon" title="c8">n'</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><a NAME="E1:30_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="kurato_2.html#K3" title="KURATO_2:func.3">^\</a> <font color="Maroon" title="c8">n'</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c11">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="kurato_2.html#D2" title="KURATO_2:def.2">KURATO_2:def 2</a></i>;<br/><b>hence </b><a NAME="E2:30_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="kurato_2.html#K3" title="KURATO_2:func.3">^\</a> <font color="Maroon" title="c8">n'</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="ref" href="prob_1.html#D9" title="PROB_1:def.9">PROB_1:def 9</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:30"/><b>then </b><i><font color="Green" title="E25">B0</font></i>: 
<font color="Maroon" title="c4">F</font> <a href="kurato_2.html#K3" title="KURATO_2:func.3">^\</a> <font color="Maroon" title="c8">n'</font> is    <a href="prob_1.html#M1" title="PROB_1:mode.1">SetSequence</a> of <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#D9" title="PROB_1:def.9">PROB_1:def 9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <b> such that </b><br/><a NAME="E6:30"/><i><font color="Green" title="E26">D1</font></i>: 
( <font color="Maroon" title="c3">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c11">g</font> &amp; <font color="Maroon" title="c3">f</font> <a href="mesfunc8.html#K5" title="MESFUNC8:func.5">^\</a> <font color="Maroon" title="c8">n'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c11">g</font> <a href="kurato_2.html#K3" title="KURATO_2:func.3">^\</a> <font color="Maroon" title="c8">n'</font> )
 ;<br/>
<a NAME="E7:30"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc8.html#K5" title="MESFUNC8:func.5">^\</a> <font color="Maroon" title="c8">n'</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc8.html#E6:30"><i><font color="Green" title="E26">D1</font></i></a>, <a class="ref" href="kurato_2.html#D2" title="KURATO_2:def.2">KURATO_2:def 2</a></i>;<br/>
<a NAME="E8:30"/><b>then </b><i><font color="Green" title="E27">D2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc8.html#K5" title="MESFUNC8:func.5">^\</a> <font color="Maroon" title="c8">n'</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc8.html#E6:30"><i><font color="Green" title="E26">D1</font></i></a>, <a class="ref" href="mesfunc8.html#D2" target="_self" title="MESFUNC8:def.2">defsamedom</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">k</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>reconsider </b><font color="Maroon" title="c13">k'</font> = <font color="Maroon" title="c12">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E2:30_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="kurato_2.html#K3" title="KURATO_2:func.3">^\</a> <font color="Maroon" title="c8">n'</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c13">k'</font></span>)</span>
 <b>by </b><i><a class="ref" href="kurato_2.html#D2" title="KURATO_2:def.2">KURATO_2:def 2</a></i>;<br/><a NAME="E3:30_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="kurato_2.html#K3" title="KURATO_2:func.3">^\</a> <font color="Maroon" title="c8">n'</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K16" title="MESFUNC1:func.16">great_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c13">k'</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc8.html#E1:30"><i><font color="Green" title="E24">AS</font></i></a></i>;<br/><b>hence </b><a NAME="E4:30_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="kurato_2.html#K3" title="KURATO_2:func.3">^\</a> <font color="Maroon" title="c8">n'</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K16" title="MESFUNC1:func.16">great_dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc8.html#K5" title="MESFUNC8:func.5">^\</a> <font color="Maroon" title="c8">n'</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c12">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="kurato_2.html#D2" title="KURATO_2:def.2">KURATO_2:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:30"/><b>then </b><i><font color="Green" title="E28">P1</font></i>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="kurato_2.html#K3" title="KURATO_2:func.3">^\</a> <font color="Maroon" title="c8">n'</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K16" title="MESFUNC1:func.16">great_dom</a> <span class="p2">(<span class="default"><a href="mesfunc8.html#K2" title="MESFUNC8:func.2">sup</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc8.html#K5" title="MESFUNC8:func.5">^\</a> <font color="Maroon" title="c8">n'</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc8.html#E8:30"><i><font color="Green" title="E27">D2</font></i></a>, <a class="ref" href="mesfunc8.html#T15" target="_self" title="MESFUNC8:th.15">Lm4AAA</a>, <a class="txt" href="mesfunc8.html#E4:30"><i><font color="Green" title="E25">B0</font></i></a></i>;<br/>
<a NAME="E11:30"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="kurato_2.html#K3" title="KURATO_2:func.3">^\</a> <font color="Maroon" title="c8">n'</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K2" title="SETLIM_1:func.2">superior_setsequence</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c7">n</font>
 
<b>by </b><i><a class="ref" href="mesfunc8.html#T2" target="_self" title="MESFUNC8:th.2">Lm410</a></i>;<br/>
<b>hence </b><a NAME="E12:30"/>
<span class="p1">(<span class="default"><a href="setlim_1.html#K2" title="SETLIM_1:func.2">superior_setsequence</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K16" title="MESFUNC1:func.16">great_dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc8.html#K4" title="MESFUNC8:func.4">superior_realsequence</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc8.html#E10:30"><i><font color="Green" title="E28">P1</font></i></a>, <a class="ref" href="mesfunc8.html#T9" target="_self" title="MESFUNC8:th.9">superiorf1</a></i>;<br/>


</div>
