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<span class="kw">let </span><font color="Maroon" title="c1">F</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <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"> ( <font color="Maroon" title="c1">F</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>  &amp;  <a href="supinf_2.html#K14" title="SUPINF_2:func.14">SUM</a> <font color="Maroon" title="c1">F</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  implies  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <font color="Maroon" title="c1">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:8"/><span class="lab"><font color="Green" title="E1">A1</font></span>: 
<font color="Maroon" title="c1">F</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> 
 <span class="kw">and </span><br/><a NAME="E2:8"/><span class="lab"><font color="Green" title="E2">A2</font></span>: 
 <a href="supinf_2.html#K14" title="SUPINF_2:func.14">SUM</a> <font color="Maroon" title="c1">F</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <font color="Maroon" title="c1">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c2">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</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><font color="Maroon" title="c1">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K13" title="SUPINF_2:func.13">Ser</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> $1;<br/>
<a NAME="E3:8"/><span class="lab"><font color="Green" title="E3">A3</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</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="8_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">k</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="c3">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:8_1"/>
<font color="Maroon" title="c1">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K13" title="SUPINF_2:func.13">Ser</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c3">k</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="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">x</font> = <span class="p1">(<span class="default"><a href="supinf_2.html#K13" title="SUPINF_2:func.13">Ser</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c3">k</font> as   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> ;<br/>
<a NAME="E3:8_1"/>
<font color="Maroon" title="c4">x</font> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K13" title="SUPINF_2:func.13">Ser</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</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="ref" href="supinf_2.html#D11" onmouseover="rs('supinf_2/D11')" onmouseout="rh()">SUPINF_2:def 11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:8_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E1">A1</font></span></a>, <a class="ref" href="supinf_2.html#T40" onmouseover="rs('supinf_2/T40')" onmouseout="rh()">SUPINF_2:40</a>, <a class="ref" href="xxreal_3.html#T39" onmouseover="rs('xxreal_3/T39')" onmouseout="rh()">XXREAL_3:39</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:8"/><span class="lab"><font color="Green" title="E4">A4</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="supinf_2.html#D11" onmouseover="rs('supinf_2/D11')" onmouseout="rh()">SUPINF_2:def 11</a></span>;<br/>
<a NAME="E5:8"/>
 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="#E4:8"><span class="lab"><font color="Green" title="E4">A4</font></span></a>, <a class="txt" href="#E3:8"><span class="lab"><font color="Green" title="E3">A3</font></span></a>);<br/></span>
<a NAME="E6:8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E5">A5</font></span>: 
<font color="Maroon" title="c1">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K13" title="SUPINF_2:func.13">Ser</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c2">n</font>
 
;<br/>
<a NAME="E7:8"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K13" title="SUPINF_2:func.13">Ser</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K14" title="SUPINF_2:func.14">SUM</a> <font color="Maroon" title="c1">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T4" onmouseover="rs('funct_2/T4')" onmouseout="rh()">FUNCT_2:4</a>, <a class="ref" href="xxreal_2.html#T4" onmouseover="rs('xxreal_2/T4')" onmouseout="rh()">XXREAL_2:4</a></span>;<br/>
<a NAME="E8:8"/><span class="kw">then </span>
<font color="Maroon" title="c1">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K14" title="SUPINF_2:func.14">SUM</a> <font color="Maroon" title="c1">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:8"><span class="lab"><font color="Green" title="E5">A5</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E9:8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E6">A6</font></span>: 
<font color="Maroon" title="c1">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8"><span class="lab"><font color="Green" title="E2">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E10:8"/><span class="lab"><font color="Green" title="E7">A7</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" onmouseover="rs('xreal_0/D1')" onmouseout="rh()">XREAL_0:def 1</a></span>;<br/>
<a NAME="E11:8"/>
(  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <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="c1">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c2">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E1">A1</font></span></a>, <a class="ref" href="supinf_2.html#T39" onmouseover="rs('supinf_2/T39')" onmouseout="rh()">SUPINF_2:39</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:8"/>
<font color="Maroon" title="c1">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c2">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:8"><span class="lab"><font color="Green" title="E6">A6</font></span></a>, <a class="ref" href="xxreal_0.html#T46" onmouseover="rs('xxreal_0/T46')" onmouseout="rh()">XXREAL_0:46</a>, <a class="txt" href="#E10:8"><span class="lab"><font color="Green" title="E7">A7</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/>


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