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<b>let </b><font color="Maroon" title="c1">F</font>, <font color="Maroon" title="c2">G</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>  st ( ( <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) or for <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">F</font> holds <br/><font color="Maroon" title="c1">F</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  or for <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">F</font> holds <br/> <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">F</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">i</font> ) &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">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> <font color="Maroon" title="c2">G</font> &amp; ( for <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font> holds <br/><font color="Maroon" title="c2">G</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">i</font></span>)</span> ) holds <br/> <a href="convfun1.html#K5" title="CONVFUN1:func.5">Sum</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><a href="convfun1.html#K5" title="CONVFUN1:func.5">Sum</a> <font color="Maroon" title="c1">F</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">a</font> be   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) or for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">F</font> holds <br/><font color="Maroon" title="c1">F</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  or for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">F</font> holds <br/> <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">F</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">i</font> ) &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">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> <font color="Maroon" title="c2">G</font> &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font> holds <br/><font color="Maroon" title="c2">G</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">i</font></span>)</span> ) implies  <a href="convfun1.html#K5" title="CONVFUN1:func.5">Sum</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><a href="convfun1.html#K5" title="CONVFUN1:func.5">Sum</a> <font color="Maroon" title="c1">F</font></span>)</span> )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:10"/><i><font color="Green" title="E11">A1</font></i>: 
( ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) or for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">F</font> holds <br/><font color="Maroon" title="c1">F</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  or for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">F</font> holds <br/> <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">F</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">i</font> )
 <b>and </b><br/><a NAME="E2:10"/><i><font color="Green" title="E12">A2</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">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> <font color="Maroon" title="c2">G</font>
 <b>and </b><br/><a NAME="E3:10"/><i><font color="Green" title="E13">A3</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font> holds <br/><font color="Maroon" title="c2">G</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">i</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="convfun1.html#K5" title="CONVFUN1:func.5">Sum</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><a href="convfun1.html#K5" title="CONVFUN1:func.5">Sum</a> <font color="Maroon" title="c1">F</font></span>)</span></span><br/>

<b>consider </b><font color="Maroon" title="c4">f</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="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <b> such that </b><br/><a NAME="E5:10"/><i><font color="Green" title="E14">A4</font></i>: 
(  <a href="convfun1.html#K5" title="CONVFUN1:func.5">Sum</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">F</font></span>)</span> &amp; <font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp; ( for <font color="Olive" title="b1">i</font> being   <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>   st <font color="Olive" title="b1">i</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="c1">F</font> holds <br/><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> ) )
 <b>by </b><i><a class="ref" href="convfun1.html#D5" title="CONVFUN1:def.5">CONVFUN1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">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="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <b> such that </b><br/><a NAME="E7:10"/><i><font color="Green" title="E15">A5</font></i>: 
(  <a href="convfun1.html#K5" title="CONVFUN1:func.5">Sum</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font></span>)</span> &amp; <font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp; ( for <font color="Olive" title="b1">i</font> being   <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>   st <font color="Olive" title="b1">i</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="c2">G</font> holds <br/><font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> ) )
 <b>by </b><i><a class="ref" href="convfun1.html#D5" title="CONVFUN1:def.5">CONVFUN1:def 5</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( $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="c2">G</font> implies <font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> $1</span>)</span> );<br/>
<a NAME="E8:10"/><i><font color="Green" title="E16">A6</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="txt" href="mesfunc3.html#E5:10"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="mesfunc3.html#E7:10"><i><font color="Green" title="E15">A5</font></i></a>, <a class="ref" href="extreal1.html#D1" title="EXTREAL1:def.1">EXTREAL1:def 1</a></i>;<br/>
<a NAME="E9:10"/><i><font color="Green" title="E17">A7</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">i</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:10_1"/><i><font color="Green" title="E18">A8</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">i</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1]</span><br/>

<b>assume </b><a NAME="E2:10_1"/><i><font color="Green" title="E19">A9</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</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="c2">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span></span>)</span></span><br/>

<b>reconsider </b><font color="Maroon" title="c7">i</font> = <font color="Maroon" title="c6">i</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="E4:10_1"/><i><font color="Green" title="E20">A10</font></i>: 
<font color="Maroon" title="c7">i</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="c2">G</font>
 
<b>by </b><i><a class="txt" href="mesfunc3.html#E2:10_1"><i><font color="Green" title="E19">A9</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E5:10_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E6:10_1"/><b>then </b><i><font color="Green" title="E21">A11</font></i>: 
<font color="Maroon" title="c7">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font>
 
<b>by </b><i><a class="txt" href="mesfunc3.html#E2:10_1"><i><font color="Green" title="E19">A9</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E7:10_1"/><b>then </b>
<font color="Maroon" title="c7">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">F</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc3.html#E2:10"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:10_1"/><b>then </b>
<font color="Maroon" title="c7">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</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="c1">F</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E9:10_1"/><b>then </b><i><font color="Green" title="E22">A12</font></i>: 
<font color="Maroon" title="c7">i</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="c1">F</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<i><font color="Green" title="E23">A13</font></i>: <font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc3.html#E7:10"><i><font color="Green" title="E15">A5</font></i></a>, <a class="txt" href="mesfunc3.html#E4:10_1"><i><font color="Green" title="E20">A10</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">i</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc3.html#E3:10"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="mesfunc3.html#E1:10_1"><i><font color="Green" title="E18">A8</font></i></a>, <a class="txt" href="mesfunc3.html#E2:10_1"><i><font color="Green" title="E19">A9</font></i></a>, <a class="txt" href="mesfunc3.html#E6:10_1"><i><font color="Green" title="E21">A11</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_1_2_1"/>
( ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) or for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">F</font> holds <br/><font color="Maroon" title="c1">F</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  or for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">F</font> holds <br/> <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">F</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">i</font> )
 </span><b>by </b><i><a class="txt" href="mesfunc3.html#E1:10"><i><font color="Green" title="E11">A1</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_2_1_1"><b>case </b></a><a NAME="E1:10_1_2_1_1"/>
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:10_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc3.html#E10:10_1"><i><font color="Green" title="E23">A13</font></i></a>, <a class="ref" href="extreal1.html#T29" title="EXTREAL1:th.29">EXTREAL1:29</a></i>;<br/><b>hence </b><a NAME="E3:10_1_2_1_1"/>
<font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc3.html#E5:10"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="mesfunc3.html#E9:10_1"><i><font color="Green" title="E22">A12</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_2_1_2"><b>case </b></a><a NAME="E1:10_1_2_1_2"/><i><font color="Green" title="E24">A14</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">F</font> holds <br/><font color="Maroon" title="c1">F</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span></span>)</span></span><br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( $1 <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="c1">F</font> implies <font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> $1 <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>  );<br/><a NAME="E2:10_1_2_1_2"/><i><font color="Green" title="E25">A15</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <b>by </b><i><a class="txt" href="mesfunc3.html#E5:10"><i><font color="Green" title="E14">A4</font></i></a></i>;<br/><a NAME="E3:10_1_2_1_2"/><i><font color="Green" title="E26">A16</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_2_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">i</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:10_1_2_1_2_1"/><i><font color="Green" title="E27">A17</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">i</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1]</span><br/>

<b>assume </b><a NAME="E2:10_1_2_1_2_1"/><i><font color="Green" title="E28">A18</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <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="c1">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</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><br/>

<b>reconsider </b><font color="Maroon" title="c9">i</font> = <font color="Maroon" title="c8">i</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="E4:10_1_2_1_2_1"/>
<font color="Maroon" title="c9">i</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="c1">F</font>
 
<b>by </b><i><a class="txt" href="mesfunc3.html#E2:10_1_2_1_2_1"><i><font color="Green" title="E28">A18</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E5:10_1_2_1_2_1"/><b>then </b><i><font color="Green" title="E29">A19</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc3.html#E5:10"><i><font color="Green" title="E14">A4</font></i></a></i>;<br/>
<a NAME="E6:10_1_2_1_2_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; <font color="Maroon" title="c9">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</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="c1">F</font> )
 
<b>by </b><i><a class="txt" href="mesfunc3.html#E2:10_1_2_1_2_1"><i><font color="Green" title="E28">A18</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E7:10_1_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c9">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">F</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E8:10_1_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c1">F</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 
<b>by </b><i><a class="txt" href="mesfunc3.html#E1:10_1_2_1_2"><i><font color="Green" title="E24">A14</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:10_1_2_1_2_1"/>
<font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</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> 
 <b>by </b><i><a class="txt" href="mesfunc3.html#E1:10_1_2_1_2_1"><i><font color="Green" title="E27">A17</font></i></a>, <a class="txt" href="mesfunc3.html#E2:10_1_2_1_2_1"><i><font color="Green" title="E28">A18</font></i></a>, <a class="txt" href="mesfunc3.html#E5:10_1_2_1_2_1"><i><font color="Green" title="E29">A19</font></i></a>, <a class="ref" href="extreal1.html#T15" title="EXTREAL1:th.15">EXTREAL1:15</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E4:10_1_2_1_2"/>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="mesfunc3.html#E2:10_1_2_1_2"><i><font color="Green" title="E25">A15</font></i></a>, <a class="txt" href="mesfunc3.html#E3:10_1_2_1_2"><i><font color="Green" title="E26">A16</font></i></a>);<br/></i><a NAME="E5:10_1_2_1_2"/><b>then </b>
( ( <font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  or <font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  ) &amp; <font color="Maroon" title="c1">F</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 <b>by </b><i><a class="txt" href="mesfunc3.html#E2:10"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="mesfunc3.html#E6:10_1"><i><font color="Green" title="E21">A11</font></i></a>, <a class="txt" href="mesfunc3.html#E9:10_1"><i><font color="Green" title="E22">A12</font></i></a>, <a class="txt" href="mesfunc3.html#E1:10_1_2_1_2"><i><font color="Green" title="E24">A14</font></i></a></i>;<br/><a NAME="E6:10_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc3.html#E10:10_1"><i><font color="Green" title="E23">A13</font></i></a>, <a class="ref" href="extreal1.html#T29" title="EXTREAL1:th.29">EXTREAL1:29</a></i>;<br/><b>hence </b><a NAME="E7:10_1_2_1_2"/>
<font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc3.html#E5:10"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="mesfunc3.html#E9:10_1"><i><font color="Green" title="E22">A12</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_2_1_3"><b>case </b></a><a NAME="E1:10_1_2_1_3"/><i><font color="Green" title="E24">A20</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">F</font> holds <br/> <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">F</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span></span>)</span></span><br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( $1 <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="c1">F</font> implies  <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">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> $1 );<br/><a NAME="E2:10_1_2_1_3"/><i><font color="Green" title="E25">A21</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <b>by </b><i><a class="txt" href="mesfunc3.html#E5:10"><i><font color="Green" title="E14">A4</font></i></a></i>;<br/><a NAME="E3:10_1_2_1_3"/><i><font color="Green" title="E26">A22</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_2_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">i</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:10_1_2_1_3_1"/><i><font color="Green" title="E27">A23</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">i</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1]</span><br/>

<b>assume </b><a NAME="E2:10_1_2_1_3_1"/><i><font color="Green" title="E28">A24</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <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="c1">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <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">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span></span><br/>

<b>reconsider </b><font color="Maroon" title="c9">i</font> = <font color="Maroon" title="c8">i</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="E4:10_1_2_1_3_1"/>
<font color="Maroon" title="c9">i</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="c1">F</font>
 
<b>by </b><i><a class="txt" href="mesfunc3.html#E2:10_1_2_1_3_1"><i><font color="Green" title="E28">A24</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E5:10_1_2_1_3_1"/><b>then </b><i><font color="Green" title="E29">A25</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc3.html#E5:10"><i><font color="Green" title="E14">A4</font></i></a></i>;<br/>
<a NAME="E6:10_1_2_1_3_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; <font color="Maroon" title="c9">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</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="c1">F</font> )
 
<b>by </b><i><a class="txt" href="mesfunc3.html#E2:10_1_2_1_3_1"><i><font color="Green" title="E28">A24</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E7:10_1_2_1_3_1"/><b>then </b>
<font color="Maroon" title="c9">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">F</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E8:10_1_2_1_3_1"/><b>then </b>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">F</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc3.html#E1:10_1_2_1_3"><i><font color="Green" title="E24">A20</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:10_1_2_1_3_1"/>
 <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">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc3.html#E1:10_1_2_1_3_1"><i><font color="Green" title="E27">A23</font></i></a>, <a class="txt" href="mesfunc3.html#E2:10_1_2_1_3_1"><i><font color="Green" title="E28">A24</font></i></a>, <a class="txt" href="mesfunc3.html#E5:10_1_2_1_3_1"><i><font color="Green" title="E29">A25</font></i></a>, <a class="ref" href="extreal1.html#T14" title="EXTREAL1:th.14">EXTREAL1:14</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E4:10_1_2_1_3"/>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="mesfunc3.html#E2:10_1_2_1_3"><i><font color="Green" title="E25">A21</font></i></a>, <a class="txt" href="mesfunc3.html#E3:10_1_2_1_3"><i><font color="Green" title="E26">A22</font></i></a>);<br/></i><a NAME="E5:10_1_2_1_3"/><b>then </b>
( (  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">i</font> or <font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  ) &amp;  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">F</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="mesfunc3.html#E2:10"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="mesfunc3.html#E6:10_1"><i><font color="Green" title="E21">A11</font></i></a>, <a class="txt" href="mesfunc3.html#E9:10_1"><i><font color="Green" title="E22">A12</font></i></a>, <a class="txt" href="mesfunc3.html#E1:10_1_2_1_3"><i><font color="Green" title="E24">A20</font></i></a></i>;<br/><a NAME="E6:10_1_2_1_3"/><b>then </b>
<font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc3.html#E10:10_1"><i><font color="Green" title="E23">A13</font></i></a>, <a class="ref" href="extreal1.html#T29" title="EXTREAL1:th.29">EXTREAL1:29</a></i>;<br/><b>hence </b><a NAME="E7:10_1_2_1_3"/>
<font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc3.html#E5:10"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="mesfunc3.html#E9:10_1"><i><font color="Green" title="E22">A12</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E12:10_1"/>
<font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E10:10"/><i><font color="Green" title="E18">A26</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="mesfunc3.html#E8:10"><i><font color="Green" title="E16">A6</font></i></a>, <a class="txt" href="mesfunc3.html#E9:10"><i><font color="Green" title="E17">A7</font></i></a>);<br/></i>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">F</font></span>)</span> = 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font>
<b>by </b><i><a class="txt" href="mesfunc3.html#E2:10"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/>
<b>then </b><font color="Maroon" title="c3">a</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><a href="convfun1.html#K5" title="CONVFUN1:func.5">Sum</a> <font color="Maroon" title="c1">F</font></span>)</span> = 
<font color="Maroon" title="c3">a</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc3.html#E5:10"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="finseq_1.html#T8" title="FINSEQ_1:th.8">FINSEQ_1:8</a></i>
<br/>.= 
<font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc3.html#E10:10"><i><font color="Green" title="E18">A26</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E13:10"/>
 <a href="convfun1.html#K5" title="CONVFUN1:func.5">Sum</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><a href="convfun1.html#K5" title="CONVFUN1:func.5">Sum</a> <font color="Maroon" title="c1">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc3.html#E7:10"><i><font color="Green" title="E15">A5</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
