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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font> being   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">P</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">S</font><br/>  for <font color="Olive" title="b3">F</font> being   <a href="measur11.html#V3" title="MEASUR11:attr.3">summable</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c1">X</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>)  st (  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="b4">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b3">F</font> holds <br/><font color="Olive" title="b3">F</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b4">n</font> is <font color="Olive" title="b2">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) holds <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="b4">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b3">F</font> holds <br/><span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Olive" title="b3">F</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b4">n</font> is <font color="Olive" title="b2">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">P</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font><br/>  for <font color="Olive" title="b2">F</font> being   <a href="measur11.html#V3" title="MEASUR11:attr.3">summable</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c1">X</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>)  st (  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="b3">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b2">F</font> holds <br/><font color="Olive" title="b2">F</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b3">n</font> is <font color="Olive" title="b1">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) holds <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="b3">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b2">F</font> holds <br/><span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Olive" title="b2">F</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b3">n</font> is <font color="Olive" title="b1">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">P</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" 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="measur11.html#V3" title="MEASUR11:attr.3">summable</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c1">X</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>)  st (  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="b2">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">F</font> holds <br/><font color="Olive" title="b1">F</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b2">n</font> is <font color="Maroon" title="c3">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) holds <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="b2">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">F</font> holds <br/><span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Olive" title="b1">F</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b2">n</font> is <font color="Maroon" title="c3">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">F</font> be   <a href="measur11.html#V3" title="MEASUR11:attr.3">summable</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c1">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">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">F</font> holds <br/><font color="Maroon" title="c4">F</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b1">n</font> is <font color="Maroon" title="c3">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) implies  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">F</font> holds <br/><span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b1">n</font> is <font color="Maroon" title="c3">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  )</span><br/>







<span class="kw">assume </span><a NAME="E1:92"/><span class="lab"><font color="Green" title="E62">A1</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="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">F</font> holds <br/><font color="Maroon" title="c4">F</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b1">n</font> is <font color="Maroon" title="c3">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</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">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">F</font> holds <br/><span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b1">n</font> is <font color="Maroon" title="c3">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> </span><br/>

<a NAME="E2:92"/><span class="lab"><font color="Green" title="E63">A2</font></span>: 
<font color="Maroon" title="c3">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font>
 
;<br/>
<a NAME="E3:92"/><span class="lab"><font color="Green" title="E64">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measur11.html#D11" target="_self" onmouseover="rs('measur11/D11')" onmouseout="rh()">DEF13</a></span>;<br/>
<a NAME="E4:92"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A4</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T29" onmouseover="rs('finseq_3/T29')" onmouseout="rh()">FINSEQ_3:29</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span>( $1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">F</font> implies <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> $1 is <font color="Maroon" title="c3">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  );<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:92_1"/>
( <font color="Maroon" title="c4">F</font> is  <a href="measur11.html#V1" title="MEASUR11:attr.1">without_+infty-valued</a>  or <font color="Maroon" title="c4">F</font> is  <a href="measur11.html#V2" title="MEASUR11:attr.2">without_-infty-valued</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="measur11.html#D10" target="_self" onmouseover="rs('measur11/D10')" onmouseout="rh()">DEF12</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1"><span class="kw">suppose </span></a><a NAME="E1:92_1_1"/><span class="lab"><font color="Green" title="E66">A5</font></span>: 
<font color="Maroon" title="c4">F</font> is  <a href="measur11.html#V1" title="MEASUR11:attr.1">without_+infty-valued</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">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">F</font> holds <br/><span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b1">n</font> is <font color="Maroon" title="c3">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> </span><br/><div class="add"><a NAME="E2:92_1_1"/><span class="lab"><font color="Green" title="E67">A6</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T24" onmouseover="rs('finseq_3/T24')" onmouseout="rh()">FINSEQ_3:24</a></span>;<br/><a NAME="E3:92_1_1"/><span class="lab"><font color="Green" title="E68">A7</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">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="92_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">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">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:92_1_1_1"/><span class="lab"><font color="Green" title="E69">A8</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">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"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">assume </span><a NAME="E2:92_1_1_1"/><span class="lab"><font color="Green" title="E70">A9</font></span>: 
<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">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">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <font color="Maroon" title="c3">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> </span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:92_1_1_1_1"/>
( <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c5">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:92_1_1_1_1_1"/><span class="lab"><font color="Green" title="E71">A10</font></span>: 
<font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <font color="Maroon" title="c3">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> </span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:92"><span class="lab"><font color="Green" title="E65">A4</font></span></a>, <a class="txt" href="#E2:92_1_1_1"><span class="lab"><font color="Green" title="E70">A9</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="measur11.html#D11" target="_self" onmouseover="rs('measur11/D11')" onmouseout="rh()">DEF13</a></span>
<br/>.= 
<font color="Maroon" title="c4">F</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:92_1_1_1"><span class="lab"><font color="Green" title="E70">A9</font></span></a>, <a class="txt" href="#E1:92_1_1_1_1_1"><span class="lab"><font color="Green" title="E71">A10</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:92_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <font color="Maroon" title="c3">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E2:92_1_1_1"><span class="lab"><font color="Green" title="E70">A9</font></span></a>, <a class="txt" href="#E1:92_1_1_1_1_1"><span class="lab"><font color="Green" title="E71">A10</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:92_1_1_1_1_2"/><span class="lab"><font color="Green" title="E71">A11</font></span>: 
<font color="Maroon" title="c5">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <font color="Maroon" title="c3">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> </span><br/><div class="add"><a NAME="E2:92_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A12</font></span>: 
<font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" onmouseover="rs('nat_1/T14')" onmouseout="rh()">NAT_1:14</a></span>;<br/><a NAME="E3:92_1_1_1_1_2"/>
<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:92_1_1_1"><span class="lab"><font color="Green" title="E70">A9</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E4:92_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A13</font></span>: 
<font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E5:92_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A15</font></span>: 
( <font color="Maroon" title="c4">F</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> &amp; <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:92"><span class="lab"><font color="Green" title="E65">A4</font></span></a>, <a class="txt" href="#E2:92_1_1_1"><span class="lab"><font color="Green" title="E70">A9</font></span></a>, <a class="txt" href="#E2:92_1_1_1_1_2"><span class="lab"><font color="Green" title="E72">A12</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/><a NAME="E6:92_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A16</font></span>: 
<span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92_1_1_1_1_2"><span class="lab"><font color="Green" title="E71">A11</font></span></a>, <a class="txt" href="#E4:92_1_1_1_1_2"><span class="lab"><font color="Green" title="E73">A13</font></span></a>, <a class="ref" href="nat_1.html#T14" onmouseover="rs('nat_1/T14')" onmouseout="rh()">NAT_1:14</a>, <a class="ref" href="measur11.html#D11" target="_self" onmouseover="rs('measur11/D11')" onmouseout="rh()">DEF13</a></span>;<br/><a NAME="E7:92_1_1_1_1_2"/>
 <a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c4">F</font> is  <a href="measur11.html#V1" title="MEASUR11:attr.1">without_+infty-valued</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92_1_1"><span class="lab"><font color="Green" title="E66">A5</font></span></a>, <a class="ref" href="measur11.html#T60" target="_self" onmouseover="rs('measur11/T60')" onmouseout="rh()">Th56</a></span>;<br/><a NAME="E8:92_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A17</font></span>: 
( <font color="Maroon" title="c4">F</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="mesfunc5.html#V4" title="MESFUNC5:attr.4">V179</a>() &amp; <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c5">n</font> is <a href="mesfunc5.html#V4" title="MESFUNC5:attr.4">V179</a>() )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92_1_1"><span class="lab"><font color="Green" title="E66">A5</font></span></a>, <a class="txt" href="#E2:92_1_1_1"><span class="lab"><font color="Green" title="E70">A9</font></span></a>, <a class="txt" href="#E2:92_1_1_1_1_2"><span class="lab"><font color="Green" title="E72">A12</font></span></a>, <a class="txt" href="#E5:92_1_1_1_1_2"><span class="lab"><font color="Green" title="E74">A15</font></span></a>, <a class="txt" href="#E4:92_1_1_1_1_2"><span class="lab"><font color="Green" title="E73">A13</font></span></a>, <a class="txt" href="#E3:92"><span class="lab"><font color="Green" title="E64">A3</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E9:92_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A19</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></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#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc9.html#T1" onmouseover="rs('mesfunc9/T1')" onmouseout="rh()">MESFUNC9:1</a></span>;<br/><a NAME="E10:92_1_1_1_1_2"/><span class="lab"><font color="Green" title="E78">A18</font></span>: 
( <font color="Maroon" title="c3">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c5">n</font></span>)</span> &amp; <font color="Maroon" title="c3">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:92"><span class="lab"><font color="Green" title="E63">A2</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><a NAME="E11:92_1_1_1_1_2"/>
<font color="Maroon" title="c4">F</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <font color="Maroon" title="c3">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:92_1_1_1"><span class="lab"><font color="Green" title="E70">A9</font></span></a>, <a class="txt" href="#E1:92"><span class="lab"><font color="Green" title="E62">A1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E12:92_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <font color="Maroon" title="c3">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92_1_1_1"><span class="lab"><font color="Green" title="E69">A8</font></span></a>, <a class="txt" href="#E2:92_1_1_1_1_2"><span class="lab"><font color="Green" title="E72">A12</font></span></a>, <a class="txt" href="#E4:92_1_1_1_1_2"><span class="lab"><font color="Green" title="E73">A13</font></span></a>, <a class="txt" href="#E6:92_1_1_1_1_2"><span class="lab"><font color="Green" title="E75">A16</font></span></a>, <a class="txt" href="#E8:92_1_1_1_1_2"><span class="lab"><font color="Green" title="E76">A17</font></span></a>, <a class="txt" href="#E10:92_1_1_1_1_2"><span class="lab"><font color="Green" title="E78">A18</font></span></a>, <a class="txt" href="#E9:92_1_1_1_1_2"><span class="lab"><font color="Green" title="E77">A19</font></span></a>, <a class="ref" href="measur11.html#T65" target="_self" onmouseover="rs('measur11/T65')" onmouseout="rh()">Th61</a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a>, <a class="ref" href="xboole_1.html#T19" onmouseover="rs('xboole_1/T19')" onmouseout="rh()">XBOOLE_1:19</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="E4:92_1_1"/>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E2:92_1_1"><span class="lab"><font color="Green" title="E67">A6</font></span></a>, <a class="txt" href="#E3:92_1_1"><span class="lab"><font color="Green" title="E68">A7</font></span></a>);<br/></span><span class="kw">hence </span><a NAME="E5:92_1_1"/>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">F</font> holds <br/><span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b1">n</font> is <font color="Maroon" title="c3">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</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"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_2"><span class="kw">suppose </span></a><a NAME="E1:92_1_2"/><span class="lab"><font color="Green" title="E66">A19</font></span>: 
<font color="Maroon" title="c4">F</font> is  <a href="measur11.html#V2" title="MEASUR11:attr.2">without_-infty-valued</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">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">F</font> holds <br/><span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b1">n</font> is <font color="Maroon" title="c3">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> </span><br/><div class="add"><a NAME="E2:92_1_2"/><span class="lab"><font color="Green" title="E67">A20</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T24" onmouseover="rs('finseq_3/T24')" onmouseout="rh()">FINSEQ_3:24</a></span>;<br/><a NAME="E3:92_1_2"/><span class="lab"><font color="Green" title="E68">A21</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">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="92_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">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">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:92_1_2_1"/><span class="lab"><font color="Green" title="E69">A24</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">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"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">assume </span><a NAME="E2:92_1_2_1"/><span class="lab"><font color="Green" title="E70">A25</font></span>: 
<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">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">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <font color="Maroon" title="c3">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> </span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:92_1_2_1_1"/>
( <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c5">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:92_1_2_1_1_1"/><span class="lab"><font color="Green" title="E71">A26</font></span>: 
<font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <font color="Maroon" title="c3">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> </span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:92_1_2_1"><span class="lab"><font color="Green" title="E70">A25</font></span></a>, <a class="txt" href="#E4:92"><span class="lab"><font color="Green" title="E65">A4</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="measur11.html#D11" target="_self" onmouseover="rs('measur11/D11')" onmouseout="rh()">DEF13</a></span>
<br/>.= 
<font color="Maroon" title="c4">F</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:92_1_2_1"><span class="lab"><font color="Green" title="E70">A25</font></span></a>, <a class="txt" href="#E1:92_1_2_1_1_1"><span class="lab"><font color="Green" title="E71">A26</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:92_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <font color="Maroon" title="c3">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E2:92_1_2_1"><span class="lab"><font color="Green" title="E70">A25</font></span></a>, <a class="txt" href="#E1:92_1_2_1_1_1"><span class="lab"><font color="Green" title="E71">A26</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:92_1_2_1_1_2"/><span class="lab"><font color="Green" title="E71">A27</font></span>: 
<font color="Maroon" title="c5">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <font color="Maroon" title="c3">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> </span><br/><div class="add"><a NAME="E2:92_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A28</font></span>: 
<font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" onmouseover="rs('nat_1/T14')" onmouseout="rh()">NAT_1:14</a></span>;<br/><a NAME="E3:92_1_2_1_1_2"/>
<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:92_1_2_1"><span class="lab"><font color="Green" title="E70">A25</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E4:92_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A29</font></span>: 
<font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E5:92_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A30</font></span>: 
( <font color="Maroon" title="c4">F</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> &amp; <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:92"><span class="lab"><font color="Green" title="E65">A4</font></span></a>, <a class="txt" href="#E2:92_1_2_1"><span class="lab"><font color="Green" title="E70">A25</font></span></a>, <a class="txt" href="#E2:92_1_2_1_1_2"><span class="lab"><font color="Green" title="E72">A28</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/><a NAME="E6:92_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A31</font></span>: 
<span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92_1_2_1_1_2"><span class="lab"><font color="Green" title="E71">A27</font></span></a>, <a class="txt" href="#E4:92_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A29</font></span></a>, <a class="ref" href="measur11.html#D11" target="_self" onmouseover="rs('measur11/D11')" onmouseout="rh()">DEF13</a>, <a class="ref" href="nat_1.html#T14" onmouseover="rs('nat_1/T14')" onmouseout="rh()">NAT_1:14</a></span>;<br/><a NAME="E7:92_1_2_1_1_2"/>
 <a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c4">F</font> is  <a href="measur11.html#V2" title="MEASUR11:attr.2">without_-infty-valued</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92_1_2"><span class="lab"><font color="Green" title="E66">A19</font></span></a>, <a class="ref" href="measur11.html#T61" target="_self" onmouseover="rs('measur11/T61')" onmouseout="rh()">Th57</a></span>;<br/><a NAME="E8:92_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A32</font></span>: 
( <font color="Maroon" title="c4">F</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">V178</a>() &amp; <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c5">n</font> is <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">V178</a>() )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92_1_2"><span class="lab"><font color="Green" title="E66">A19</font></span></a>, <a class="txt" href="#E2:92_1_2_1"><span class="lab"><font color="Green" title="E70">A25</font></span></a>, <a class="txt" href="#E4:92_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A29</font></span></a>, <a class="txt" href="#E3:92"><span class="lab"><font color="Green" title="E64">A3</font></span></a>, <a class="txt" href="#E2:92_1_2_1_1_2"><span class="lab"><font color="Green" title="E72">A28</font></span></a>, <a class="txt" href="#E5:92_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A30</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E9:92_1_2_1_1_2"/>
<font color="Maroon" title="c4">F</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <font color="Maroon" title="c3">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:92_1_2_1"><span class="lab"><font color="Green" title="E70">A25</font></span></a>, <a class="txt" href="#E1:92"><span class="lab"><font color="Green" title="E62">A1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E10:92_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <font color="Maroon" title="c3">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:92_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A31</font></span></a>, <a class="txt" href="#E8:92_1_2_1_1_2"><span class="lab"><font color="Green" title="E76">A32</font></span></a>, <a class="txt" href="#E4:92_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A29</font></span></a>, <a class="txt" href="#E1:92_1_2_1"><span class="lab"><font color="Green" title="E69">A24</font></span></a>, <a class="txt" href="#E2:92_1_2_1_1_2"><span class="lab"><font color="Green" title="E72">A28</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a>, <a class="ref" href="mesfunc5.html#T31" onmouseover="rs('mesfunc5/T31')" onmouseout="rh()">MESFUNC5:31</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="E4:92_1_2"/>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E2:92_1_2"><span class="lab"><font color="Green" title="E67">A20</font></span></a>, <a class="txt" href="#E3:92_1_2"><span class="lab"><font color="Green" title="E68">A21</font></span></a>);<br/></span><span class="kw">hence </span><a NAME="E5:92_1_2"/>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">F</font> holds <br/><span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b1">n</font> is <font color="Maroon" title="c3">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</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"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
