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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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>means </b>for <font color="Olive" title="b1">dseqK</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st ( for <font color="Olive" title="b2">n</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>  holds  <font color="Olive" title="b1">dseqK</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="irrat_1.html#K3" title="IRRAT_1:func.3">cseq</a> <font color="Olive" title="b2">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1 ) holds <br/>( <font color="Olive" title="b1">dseqK</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Olive" title="b1">dseqK</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <a href="irrat_1.html#K5" title="IRRAT_1:func.5">eseq</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1 );<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c1">dseq0</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/><b>assume </b><a NAME="E1:31_1"/><i><font color="Green" title="E29">A1</font></i>: 
for <font color="Olive" title="b1">n</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>  holds  <font color="Maroon" title="c1">dseq0</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="irrat_1.html#K3" title="IRRAT_1:func.3">cseq</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>thus </b><font color="Maroon" title="c1">dseq0</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">n</font> = 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="irrat_1.html#K3" title="IRRAT_1:func.3">cseq</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="irrat_1.html#E1:31_1"><i><font color="Green" title="E29">A1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="irrat_1.html#K3" title="IRRAT_1:func.3">cseq</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="series_1.html#D1" title="SERIES_1:def.1">SERIES_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="irrat_1.html#K2" title="IRRAT_1:func.2">bseq</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">n</font>
<b>by </b><i><a class="ref" href="irrat_1.html#T3" target="_self" title="IRRAT_1:th.3">Th3</a></i>
;<br/></div><b>end;</b></div><a NAME="E3:31_1"/><b>then </b><i><font color="Green" title="E30">A2</font></i>: 
<font color="Maroon" title="c1">dseq0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="irrat_1.html#K2" title="IRRAT_1:func.2">bseq</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/><b>hence </b><a NAME="E4:31_1"/>
<font color="Maroon" title="c1">dseq0</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="ref" href="irrat_1.html#T13" target="_self" title="IRRAT_1:th.13">Th13</a></i>;<br/><b>thus </b> <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c1">dseq0</font> = 
<a href="irrat_1.html#K5" title="IRRAT_1:func.5">eseq</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="irrat_1.html#E3:31_1"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="irrat_1.html#T13" target="_self" title="IRRAT_1:th.13">Th13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <a href="irrat_1.html#K5" title="IRRAT_1:func.5">eseq</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="series_1.html#D1" title="SERIES_1:def.1">SERIES_1:def 1</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E2:31"/><b>then </b><i><font color="Green" title="E29">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c1">K</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:31_2"/><i><font color="Green" title="E30">A4</font></i>: 
for <font color="Olive" title="b1">dseqK</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st ( for <font color="Olive" title="b2">n</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>  holds  <font color="Olive" title="b1">dseqK</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="irrat_1.html#K3" title="IRRAT_1:func.3">cseq</a> <font color="Olive" title="b2">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">K</font> ) holds <br/>( <font color="Olive" title="b1">dseqK</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Olive" title="b1">dseqK</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <a href="irrat_1.html#K5" title="IRRAT_1:func.5">eseq</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">K</font> )
 ;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <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>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  = <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="irrat_1.html#K3" title="IRRAT_1:func.3">cseq</a> $1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">K</font>;<br/><b>consider </b><font color="Maroon" title="c2">dseqK</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><b> such that </b><br/><a NAME="E3:31_2"/><i><font color="Green" title="E31">A5</font></i>: 
for <font color="Olive" title="b1">n</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>  holds  <font color="Maroon" title="c2">dseqK</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1" title="SEQ_1:sch.1">SEQ_1:sch 1</a>();<br/></i><a NAME="E4:31_2"/><i><font color="Green" title="E32">A6</font></i>: 
( <font color="Maroon" title="c2">dseqK</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">dseqK</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <a href="irrat_1.html#K5" title="IRRAT_1:func.5">eseq</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">K</font> )
 <b>by </b><i><a class="txt" href="irrat_1.html#E1:31_2"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="irrat_1.html#E3:31_2"><i><font color="Green" title="E31">A5</font></i></a></i>;<br/><b>let </b><font color="Maroon" title="c3">dseqK1</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/><b>assume </b><a NAME="E5:31_2"/><i><font color="Green" title="E33">A7</font></i>: 
for <font color="Olive" title="b1">n</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>  holds  <font color="Maroon" title="c3">dseqK1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="irrat_1.html#K3" title="IRRAT_1:func.3">cseq</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">K</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>thus </b><font color="Maroon" title="c3">dseqK1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">n</font> = 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="irrat_1.html#K3" title="IRRAT_1:func.3">cseq</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">K</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="irrat_1.html#E5:31_2"><i><font color="Green" title="E33">A7</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="irrat_1.html#K3" title="IRRAT_1:func.3">cseq</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="irrat_1.html#K3" title="IRRAT_1:func.3">cseq</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">K</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="series_1.html#D1" title="SERIES_1:def.1">SERIES_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">dseqK</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="irrat_1.html#K3" title="IRRAT_1:func.3">cseq</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">K</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="irrat_1.html#E3:31_2"><i><font color="Green" title="E31">A5</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">dseqK</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="irrat_1.html#K2" title="IRRAT_1:func.2">bseq</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">K</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">n</font></span>)</span>
<b>by </b><i><a class="ref" href="irrat_1.html#T3" target="_self" title="IRRAT_1:th.3">Th3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">dseqK</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><a href="irrat_1.html#K2" title="IRRAT_1:func.2">bseq</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">K</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">n</font>
<b>by </b><i><a class="ref" href="seq_1.html#T11" title="SEQ_1:th.11">SEQ_1:11</a></i>
;<br/></div><b>end;</b></div><a NAME="E7:31_2"/><b>then </b><i><font color="Green" title="E34">A8</font></i>: 
<font color="Maroon" title="c3">dseqK1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">dseqK</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><a href="irrat_1.html#K2" title="IRRAT_1:func.2">bseq</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">K</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/><a NAME="E8:31_2"/><i><font color="Green" title="E35">A9</font></i>: 
 <a href="irrat_1.html#K2" title="IRRAT_1:func.2">bseq</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">K</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="ref" href="irrat_1.html#T13" target="_self" title="IRRAT_1:th.13">Th13</a></i>;<br/><b>hence </b><a NAME="E9:31_2"/>
<font color="Maroon" title="c3">dseqK1</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="irrat_1.html#E4:31_2"><i><font color="Green" title="E32">A6</font></i></a>, <a class="txt" href="irrat_1.html#E7:31_2"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="seq_2.html#T19" title="SEQ_2:th.19">SEQ_2:19</a></i>;<br/><b>thus </b> <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c3">dseqK1</font> = 
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">dseqK</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="irrat_1.html#K2" title="IRRAT_1:func.2">bseq</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">K</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="irrat_1.html#E4:31_2"><i><font color="Green" title="E32">A6</font></i></a>, <a class="txt" href="irrat_1.html#E7:31_2"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="irrat_1.html#E8:31_2"><i><font color="Green" title="E35">A9</font></i></a>, <a class="ref" href="seq_2.html#T20" title="SEQ_2:th.20">SEQ_2:20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <a href="irrat_1.html#K5" title="IRRAT_1:func.5">eseq</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="irrat_1.html#K5" title="IRRAT_1:func.5">eseq</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">K</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="irrat_1.html#E4:31_2"><i><font color="Green" title="E32">A6</font></i></a>, <a class="ref" href="irrat_1.html#T13" target="_self" title="IRRAT_1:th.13">Th13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <a href="irrat_1.html#K5" title="IRRAT_1:func.5">eseq</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">K</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="series_1.html#D1" title="SERIES_1:def.1">SERIES_1:def 1</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E4:31"/><b>then </b><i><font color="Green" title="E30">A10</font></i>: 
for <font color="Olive" title="b1">n</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="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#K2" title="NAT_1:func.2">+</a> 1]
 
;<br/>
<b>thus </b><a NAME="E5:31"/>
for <font color="Olive" title="b1">n</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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="irrat_1.html#E2:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="irrat_1.html#E4:31"><i><font color="Green" title="E30">A10</font></i></a>);</i><br/>


</div>
