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<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a>  <a href="rlvect_1.html#L1" title="RLVECT_1:struct.1">LoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">p</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c3">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>assume </b><a NAME="E1:2"/><i><font color="Green" title="E2">A1</font></i>: 
for <font color="Olive" title="b1">k</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">k</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">p</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">n</font> holds <br/><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">L</font>
 ;<br/>

<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">p</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">L</font><br/> 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>   st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; ( for <font color="Olive" title="b3">k</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="b3">k</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="Olive" title="b1">p</font> &amp; <font color="Olive" title="b3">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b2">n</font> holds <br/><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">L</font> ) holds <br/> <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Olive" title="b2">n</font></span>)</span>;<br/>
<a NAME="E2:2"/><i><font color="Green" title="E3">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="finseq_1.html#T79" title="FINSEQ_1:th.79">FINSEQ_1:79</a></i>;<br/>
<div><i><font color="Green" title="E4">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">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="E1:2_1"/><i><font color="Green" title="E5">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>]
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">p</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c6">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>assume </b><a NAME="E1:2_1_1"/><i><font color="Green" title="E6">A5</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; ( for <font color="Olive" title="b1">l</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">l</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="c5">p</font> &amp; <font color="Olive" title="b1">l</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">n</font> holds <br/><font color="Maroon" title="c5">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">L</font> ) )
 ;<br/><b>set </b><font color="Maroon" title="c7">q</font> = <font color="Maroon" title="c5">p</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">k</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c8">q</font> = <font color="Maroon" title="c5">p</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">k</font></span>)</span> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="finseq_1.html#T23" title="FINSEQ_1:th.23">FINSEQ_1:23</a></i>;<br/><a NAME="E3:2_1_1"/><i><font color="Green" title="E7">A6</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="groeb_2.html#E1:2_1_1"><i><font color="Green" title="E6">A5</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="E4:2_1_1"/><i><font color="Green" title="E8">A7</font></i>: 
<font color="Maroon" title="c4">k</font> <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="c5">p</font>
 <b>by </b><i><a class="txt" href="groeb_2.html#E1:2_1_1"><i><font color="Green" title="E6">A5</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E5:2_1_1"/><b>then </b><i><font color="Green" title="E9">A8</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T21" title="FINSEQ_1:th.21">FINSEQ_1:21</a></i>;<br/><a NAME="E6:2_1_1"/><i><font color="Green" title="E10">A9</font></i>: 
<font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="E7:2_1_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">k</font>
 <b>by </b><i><a class="txt" href="groeb_2.html#E4:2_1_1"><i><font color="Green" title="E8">A7</font></i></a>, <a class="ref" href="finseq_1.html#T21" title="FINSEQ_1:th.21">FINSEQ_1:21</a></i>;<br/><a NAME="E8:2_1_1"/><b>then </b><i><font color="Green" title="E11">A10</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">p</font>
 <b>by </b><i><a class="txt" href="groeb_2.html#E3:2_1_1"><i><font color="Green" title="E7">A6</font></i></a>, <a class="txt" href="groeb_2.html#E6:2_1_1"><i><font color="Green" title="E10">A9</font></i></a>, <a class="ref" href="finseq_1.html#T7" title="FINSEQ_1:th.7">FINSEQ_1:7</a></i>;<br/><a NAME="E9:2_1_1"/><i><font color="Green" title="E12">A11</font></i>: 
<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">p</font>
 <b>by </b><i><a class="txt" href="groeb_2.html#E3:2_1_1"><i><font color="Green" title="E7">A6</font></i></a>, <a class="ref" href="finseq_1.html#T6" title="FINSEQ_1:th.6">FINSEQ_1:6</a></i>;<br/><b>then </b><i><font color="Green" title="E13">A12</font></i>: <font color="Maroon" title="c8">q</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> = 
<font color="Maroon" title="c8">q</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c5">p</font>
<b>by </b><i><a class="txt" href="groeb_2.html#E1:2_1_1"><i><font color="Green" title="E6">A5</font></i></a>, <a class="ref" href="finseq_3.html#T61" title="FINSEQ_3:th.61">FINSEQ_3:61</a></i>
;<br/><a NAME="E11:2_1_1"/><i><font color="Green" title="E14">A13</font></i>: 
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">k</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_1_1_2_1"/>
( <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">n</font> or <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_1_1"><b>case </b></a><a NAME="E1:2_1_1_2_1_1"/>
<font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">n</font>
 ;<br/><div class="add"><a NAME="E2:2_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E15">A14</font></i>: 
<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:2_1_1_2_1_1"/><i><font color="Green" title="E16">A15</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>;<br/><div><i><font color="Green" title="E17">A16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:2_1_1_2_1_1_1"/>
<font color="Maroon" title="c9">u</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">n</font></span>)</span>
 ;<br/><a NAME="E2:2_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E18">A17</font></i>: 
<font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>;<br/><a NAME="E3:2_1_1_2_1_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">p</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T89" title="RELAT_1:th.89">RELAT_1:89</a></i>;<br/><b>hence </b><a NAME="E4:2_1_1_2_1_1_1"/>
<font color="Maroon" title="c9">u</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="c5">p</font>
 <b>by </b><i><a class="txt" href="groeb_2.html#E2:2_1_1_2_1_1_1"><i><font color="Green" title="E18">A17</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:2_1_1_2_1_1_2"/><i><font color="Green" title="E18">A18</font></i>: 
<font color="Maroon" title="c9">u</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="c5">p</font>
 ;<br/><a NAME="E2:2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E19">A19</font></i>: 
<font color="Maroon" title="c9">u</font> <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"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="groeb_2.html#E1:2_1_1"><i><font color="Green" title="E6">A5</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">u'</font> = <font color="Maroon" title="c9">u</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="txt" href="groeb_2.html#E1:2_1_1_2_1_1_2"><i><font color="Green" title="E18">A18</font></i></a></i>;<br/><a NAME="E4:2_1_1_2_1_1_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">u'</font> &amp; <font color="Maroon" title="c10">u'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="groeb_2.html#E2:2_1_1_2_1_1_2"><i><font color="Green" title="E19">A19</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:2_1_1_2_1_1_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">u'</font> &amp; <font color="Maroon" title="c10">u'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font> )
 <b>by </b><i><a class="txt" href="groeb_2.html#E2:2_1_1_2_1_1"><i><font color="Green" title="E15">A14</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">u'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">n</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="E7:2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">u'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_2.html#E1:2_1_1_2_1_1_2"><i><font color="Green" title="E18">A18</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E8:2_1_1_2_1_1_2"/>
<font color="Maroon" title="c9">u</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_2.html#E3:2_1_1_2_1_1"><i><font color="Green" title="E16">A15</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/></div><b>end;</b></div><a NAME="E6:2_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E18">A20</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">n</font></span>)</span> <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="c5">p</font>
 <b>by </b><i><a class="txt" href="groeb_2.html#E4:2_1_1_2_1_1"><i><font color="Green" title="E17">A16</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/><a NAME="E7:2_1_1_2_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/><a NAME="E8:2_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c5">p</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font>
 <b>by </b><i><a class="txt" href="groeb_2.html#E3:2_1_1_2_1_1"><i><font color="Green" title="E16">A15</font></i></a>, <a class="txt" href="groeb_2.html#E6:2_1_1_2_1_1"><i><font color="Green" title="E18">A20</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/><b>hence </b><a NAME="E9:2_1_1_2_1_1"/>
 <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <font color="Maroon" title="c5">p</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_1_2"><b>case </b></a><a NAME="E1:2_1_1_2_1_2"/><i><font color="Green" title="E15">A21</font></i>: 
<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font>
 ;<br/><div class="add"><div><i><font color="Green" title="E16">A22</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">l</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="E1:2_1_1_2_1_2_1"/><i><font color="Green" title="E17">A23</font></i>: 
( <font color="Maroon" title="c9">l</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="c8">q</font> &amp; <font color="Maroon" title="c9">l</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">n</font> )
 ;<br/><a NAME="E2:2_1_1_2_1_2_1"/><b>then </b><i><font color="Green" title="E18">A24</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="groeb_2.html#E1:2_1_1"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="groeb_2.html#E8:2_1_1"><i><font color="Green" title="E11">A10</font></i></a></i>;<br/><b>thus </b><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">l</font> = 
<font color="Maroon" title="c8">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">l</font>
<b>by </b><i><a class="txt" href="groeb_2.html#E1:2_1_1_2_1_2_1"><i><font color="Green" title="E17">A23</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c5">p</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">l</font>
<b>by </b><i><a class="txt" href="groeb_2.html#E11:2_1_1"><i><font color="Green" title="E14">A13</font></i></a>, <a class="txt" href="groeb_2.html#E1:2_1_1_2_1_2_1"><i><font color="Green" title="E17">A23</font></i></a>, <a class="ref" href="finseq_4.html#T85" title="FINSEQ_4:th.85">FINSEQ_4:85</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">L</font>
<b>by </b><i><a class="txt" href="groeb_2.html#E8:2_1_1"><i><font color="Green" title="E11">A10</font></i></a>, <a class="txt" href="groeb_2.html#E1:2_1_1_2_1_2_1"><i><font color="Green" title="E17">A23</font></i></a>, <a class="txt" href="groeb_2.html#E2:2_1_1_2_1_2_1"><i><font color="Green" title="E18">A24</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/></div><b>end;</b></div><a NAME="E3:2_1_1_2_1_2"/>
<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="txt" href="groeb_2.html#E1:2_1_1_2_1_2"><i><font color="Green" title="E15">A21</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then </b><i><font color="Green" title="E17">A25</font></i>:  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">L</font> = 
<font color="Maroon" title="c5">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="groeb_2.html#E1:2_1_1"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="groeb_2.html#E9:2_1_1"><i><font color="Green" title="E12">A11</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">p</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="groeb_2.html#E9:2_1_1"><i><font color="Green" title="E12">A11</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/><b>thus </b> <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <font color="Maroon" title="c5">p</font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <font color="Maroon" title="c8">q</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="groeb_2.html#E10:2_1_1"><i><font color="Green" title="E13">A12</font></i></a>, <a class="ref" href="rlvect_1.html#T58" title="RLVECT_1:th.58">RLVECT_1:58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <font color="Maroon" title="c8">q</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T61" title="RLVECT_1:th.61">RLVECT_1:61</a></i>
<br/>.= 
 <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <font color="Maroon" title="c8">q</font>
<b>by </b><i><a class="txt" href="groeb_2.html#E4:2_1_1_2_1_2"><i><font color="Green" title="E17">A25</font></i></a>, <a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></i>
<br/>.= 
 <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">n</font></span>)</span>
<b>by </b><i><a class="txt" href="groeb_2.html#E1:2_1"><i><font color="Green" title="E5">A4</font></i></a>, <a class="txt" href="groeb_2.html#E5:2_1_1"><i><font color="Green" title="E9">A8</font></i></a>, <a class="txt" href="groeb_2.html#E2:2_1_1_2_1_2"><i><font color="Green" title="E16">A22</font></i></a></i>
<br/>.= 
 <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">n</font></span>)</span>
<b>by </b><i><a class="txt" href="groeb_2.html#E11:2_1_1"><i><font color="Green" title="E14">A13</font></i></a>, <a class="txt" href="groeb_2.html#E1:2_1_1_2_1_2"><i><font color="Green" title="E15">A21</font></i></a>, <a class="ref" href="groeb_2.html#T2" target="_self" title="GROEB_2:th.2">Th2</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E13:2_1_1"/>
 <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">n</font></span>)</span>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E4:2"/><i><font color="Green" title="E5">A26</font></i>: 
for <font color="Olive" title="b1">k</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">k</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="groeb_2.html#E2:2"><i><font color="Green" title="E3">A2</font></i></a>, <a class="txt" href="groeb_2.html#E3:2"><i><font color="Green" title="E4">A3</font></i></a>);<br/></i>
<b>consider </b><font color="Maroon" title="c4">k</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> <b> such that </b><br/><a NAME="E6:2"/><i><font color="Green" title="E6">A27</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font>
 ;<br/>
<b>thus </b><a NAME="E7:2"/>
 <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_2.html#E1:2"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="groeb_2.html#E4:2"><i><font color="Green" title="E5">A26</font></i></a>, <a class="txt" href="groeb_2.html#E6:2"><i><font color="Green" title="E6">A27</font></i></a></i>;<br/>


</div>
