<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">seq</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span><br/> for <font color="Olive" title="b2">x0</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span><br/> for <font color="Olive" title="b3">y0</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span>  st <font color="Olive" title="b2">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y0</font> &amp; ( for <font color="Olive" title="b4">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font>  ex <font color="Olive" title="b5">seqi</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</a> <font color="Olive" title="b4">i</font></span>)</span> st <br/>( <font color="Olive" title="b5">seqi</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp; <font color="Olive" title="b3">y0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Olive" title="b5">seqi</font> &amp; ( for <font color="Olive" title="b6">m</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>   ex <font color="Olive" title="b7">seqm</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> st <br/>( <font color="Olive" title="b7">seqm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b6">m</font> &amp; <font color="Olive" title="b5">seqi</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">seqm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> ) ) ) ) holds <br/>( <font color="Olive" title="b1">seq</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Olive" title="b1">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x0</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">seq</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x0</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span><br/> for <font color="Olive" title="b2">y0</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span>  st <font color="Olive" title="b1">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y0</font> &amp; ( for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font>  ex <font color="Olive" title="b4">seqi</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</a> <font color="Olive" title="b3">i</font></span>)</span> st <br/>( <font color="Olive" title="b4">seqi</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp; <font color="Olive" title="b2">y0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Olive" title="b4">seqi</font> &amp; ( for <font color="Olive" title="b5">m</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>   ex <font color="Olive" title="b6">seqm</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> st <br/>( <font color="Olive" title="b6">seqm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b5">m</font> &amp; <font color="Olive" title="b4">seqi</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">seqm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> ) ) ) ) holds <br/>( <font color="Maroon" title="c2">seq</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c2">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x0</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">x0</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">y0</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span>  st <font color="Maroon" title="c3">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y0</font> &amp; ( for <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font>  ex <font color="Olive" title="b3">seqi</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</a> <font color="Olive" title="b2">i</font></span>)</span> st <br/>( <font color="Olive" title="b3">seqi</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp; <font color="Olive" title="b1">y0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Olive" title="b3">seqi</font> &amp; ( for <font color="Olive" title="b4">m</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>   ex <font color="Olive" title="b5">seqm</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> st <br/>( <font color="Olive" title="b5">seqm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b4">m</font> &amp; <font color="Olive" title="b3">seqi</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">seqm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> ) ) ) ) holds <br/>( <font color="Maroon" title="c2">seq</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c2">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x0</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">y0</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y0</font> &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="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font>  ex <font color="Olive" title="b2">seqi</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</a> <font color="Olive" title="b1">i</font></span>)</span> st <br/>( <font color="Olive" title="b2">seqi</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp; <font color="Maroon" title="c4">y0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Olive" title="b2">seqi</font> &amp; ( for <font color="Olive" title="b3">m</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>   ex <font color="Olive" title="b4">seqm</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> st <br/>( <font color="Olive" title="b4">seqm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b3">m</font> &amp; <font color="Olive" title="b2">seqi</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">seqm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) ) ) ) implies ( <font color="Maroon" title="c2">seq</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c2">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x0</font> ) )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:43"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
<font color="Maroon" title="c3">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y0</font>
 <span class="kw">and </span><br/><a NAME="E2:43"/><span class="lab"><font color="Green" title="E33">A2</font></span>: 
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="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font>  ex <font color="Olive" title="b2">seqi</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</a> <font color="Olive" title="b1">i</font></span>)</span> st <br/>( <font color="Olive" title="b2">seqi</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp; <font color="Maroon" title="c4">y0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Olive" title="b2">seqi</font> &amp; ( for <font color="Olive" title="b3">m</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>   ex <font color="Olive" title="b4">seqm</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> st <br/>( <font color="Olive" title="b4">seqm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b3">m</font> &amp; <font color="Olive" title="b2">seqi</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">seqm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">seq</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c2">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x0</font> )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">rseqi</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> ex <font color="Olive" title="b2">seqi</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</a> $1</span>)</span> st <br/>( <font color="Olive" title="b1">rseqi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; <font color="Olive" title="b2">seqi</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp; <font color="Olive" title="b1">rseqi</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">rseqi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">rseqi</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">seqi</font> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <span class="p3">(<span class="default"><a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Olive" title="b2">seqi</font></span>)</span></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> &amp; ( for <font color="Olive" title="b3">m</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>   ex <font color="Olive" title="b4">seqm</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> st <br/>( <font color="Olive" title="b4">seqm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b3">m</font> &amp; <font color="Olive" title="b2">seqi</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">seqm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 ) ) );<br/>
<a NAME="E3:43"/><span class="lab"><font color="Green" title="E34">A3</font></span>: 
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="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font>  ex <font color="Olive" title="b2">yyseqi</font> being   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">yyseqi</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">yyseqi</font> being   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">i</font>,<font color="Olive" title="b1">yyseqi</font>]</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">seqi</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</a> <font color="Maroon" title="c5">i</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E2:43_1"/><span class="lab"><font color="Green" title="E35">A4</font></span>: 
<font color="Maroon" title="c6">seqi</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a> 
 <span class="kw">and </span><br/><a NAME="E3:43_1"/>
<font color="Maroon" title="c4">y0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c6">seqi</font>
 <span class="kw">and </span><br/><a NAME="E4:43_1"/><span class="lab"><font color="Green" title="E36">A5</font></span>: 
for <font color="Olive" title="b1">m</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>   ex <font color="Olive" title="b2">Sm</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> st <br/>( <font color="Olive" title="b2">Sm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">m</font> &amp; <font color="Maroon" title="c6">seqi</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Sm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E33">A2</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">rseqi</font> = <span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">seqi</font> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <span class="p3">(<span class="default"><a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c6">seqi</font></span>)</span></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span>;<br/>
<a NAME="E5:43_1"/><span class="lab"><font color="Green" title="E37">A6</font></span>: 
<span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">seqi</font> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <span class="p3">(<span class="default"><a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c6">seqi</font></span>)</span></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> is    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></span>;<br/>
<a NAME="E6:43_1"/>
( <span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">seqi</font> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <span class="p3">(<span class="default"><a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c6">seqi</font></span>)</span></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> 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> <span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">seqi</font> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <span class="p3">(<span class="default"><a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c6">seqi</font></span>)</span></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_1"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="prvect_2.html#E31"><span class="lab"><font color="Green" title="E29">Lm8</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:43_1"/>
 ex <font color="Olive" title="b1">yyseqi</font> being   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">i</font>,<font color="Olive" title="b1">yyseqi</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_1"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E4:43_1"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="txt" href="#E5:43_1"><span class="lab"><font color="Green" title="E37">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c5">yyseq</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span><span class="kw"> such that </span><br/><a NAME="E5:43"/><span class="lab"><font color="Green" title="E35">A7</font></span>: 
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="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c5">yyseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E34">A3</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c6">I</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</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>  ;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</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> ,   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c6">I</font>] <span class="kw">means </span>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="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c5">yyseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font>;<br/>
<a NAME="E7:43"/><span class="lab"><font color="Green" title="E36">A8</font></span>: 
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>   ex <font color="Olive" title="b2">yseqk</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c6">I</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">yseqk</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">yseqk</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c6">I</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">k</font>,<font color="Olive" title="b1">yseqk</font>]</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">yyseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> );<br/>
<a NAME="E1:43_2"/><span class="lab"><font color="Green" title="E37">A9</font></span>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">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#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">I</font> holds <br/> ex <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">j</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">I</font> implies  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c8">j</font>,<font color="Olive" title="b1">x</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:43_2_1"/>
<font color="Maroon" title="c8">j</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">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c8">j</font>,<font color="Olive" title="b1">x</font>]</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">i</font> = <font color="Maroon" title="c8">j</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">take </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">yyseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c8">j</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">yyseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>]</span><br/>

<span class="kw">thus </span><a NAME="E3:43_2_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c8">j</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">yyseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c8">yseqk</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E3:43_2"/><span class="lab"><font color="Green" title="E38">A10</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">yseqk</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="c6">I</font> &amp; ( for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">j</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">I</font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">j</font>,<font color="Maroon" title="c8">yseqk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_1.html#S1" title="FINSEQ_1:sch.1">FINSEQ_1:sch 1</a>(<a class="txt" href="#E1:43_2"><span class="lab"><font color="Green" title="E37">A9</font></span></a>);<br/></span>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">j</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">yseqk</font> implies <font color="Maroon" title="c8">yseqk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:43_2_2"/>
<font color="Maroon" title="c9">j</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">yseqk</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">yseqk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E3:43_2_2"/>
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j</font>
 <span class="kw">and </span><br/><a NAME="E4:43_2_2"/><span class="lab"><font color="Green" title="E39">A11</font></span>: 
<font color="Maroon" title="c8">yseqk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">yyseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_2"><span class="lab"><font color="Green" title="E38">A10</font></span></a></span>;<br/><a NAME="E5:43_2_2"/>
<font color="Maroon" title="c5">yyseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">i</font> is   <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#K1" title="NUMBERS:func.1">REAL</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T121" title="FUNCT_2:th.121">FUNCT_2:121</a></span>;<br/><span class="kw">hence </span><a NAME="E6:43_2_2"/>
<font color="Maroon" title="c8">yseqk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN: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="#E4:43_2_2"><span class="lab"><font color="Green" title="E39">A11</font></span></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">yyy</font> = <font color="Maroon" title="c8">yseqk</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T14" title="FINSEQ_2:th.14">FINSEQ_2:14</a></span>;<br/>
<a NAME="E6:43_2"/>
<font color="Maroon" title="c9">yyy</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">yyy</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">yseqk</font> = <font color="Maroon" title="c8">yseqk</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c6">I</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_2"><span class="lab"><font color="Green" title="E38">A10</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">j</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">yseqk</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">yyseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font></span><br/><a NAME="E1:43_2_3"/>
<font color="Maroon" title="c11">j</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">G</font>
 ;<br/><a NAME="E2:43_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c11">j</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">I</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E3:43_2_3"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j</font> &amp; <font color="Maroon" title="c10">yseqk</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">yyseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_2"><span class="lab"><font color="Green" title="E38">A10</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:43_2_3"/>
<font color="Maroon" title="c10">yseqk</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">yyseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E9:43_2"/>
 ex <font color="Olive" title="b1">yseqk</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c6">I</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">k</font>,<font color="Olive" title="b1">yseqk</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c7">yseq</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> ,<span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c6">I</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E9:43"/><span class="lab"><font color="Green" title="E37">A12</font></span>: 
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>2</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c7">yseq</font> <a href="real_ns1.html#K5" title="REAL_NS1:func.5">.</a> <font color="Olive" title="b1">k</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E7:43"><span class="lab"><font color="Green" title="E36">A8</font></span></a>);<br/></span>
<div><span class="lab"><font color="Green" title="E38">A13</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">i0</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">i0</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">I</font> implies  ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> ex <font color="Olive" title="b2">rseqi</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> ex <font color="Olive" title="b3">seqi</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</a> <font color="Olive" title="b1">i</font></span>)</span> st <br/>( ( for <font color="Olive" title="b4">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="Olive" title="b2">rseqi</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">yseq</font> <a href="real_ns1.html#K5" title="REAL_NS1:func.5">.</a> <font color="Olive" title="b4">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i0</font> ) &amp; <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i0</font> &amp; <font color="Olive" title="b3">seqi</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp; <font color="Olive" title="b2">rseqi</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="b2">rseqi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c6">I</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b2">rseqi</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">seqi</font> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <span class="p3">(<span class="default"><a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Olive" title="b3">seqi</font></span>)</span></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> &amp; ( for <font color="Olive" title="b4">m</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>   ex <font color="Olive" title="b5">seqm</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> st <br/>( <font color="Olive" title="b5">seqm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b4">m</font> &amp; <font color="Olive" title="b3">seqi</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">seqm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) ) ) )</span><br/><span class="kw">assume </span><a NAME="E1:43_3"/><span class="lab"><font color="Green" title="E39">A14</font></span>: 
<font color="Maroon" title="c8">i0</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">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> ex <font color="Olive" title="b2">rseqi</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> ex <font color="Olive" title="b3">seqi</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</a> <font color="Olive" title="b1">i</font></span>)</span> st <br/>( ( for <font color="Olive" title="b4">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="Olive" title="b2">rseqi</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">yseq</font> <a href="real_ns1.html#K5" title="REAL_NS1:func.5">.</a> <font color="Olive" title="b4">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i0</font> ) &amp; <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i0</font> &amp; <font color="Olive" title="b3">seqi</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp; <font color="Olive" title="b2">rseqi</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="b2">rseqi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c6">I</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b2">rseqi</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">seqi</font> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <span class="p3">(<span class="default"><a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Olive" title="b3">seqi</font></span>)</span></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> &amp; ( for <font color="Olive" title="b4">m</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>   ex <font color="Olive" title="b5">seqm</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> st <br/>( <font color="Olive" title="b5">seqm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b4">m</font> &amp; <font color="Olive" title="b3">seqi</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">seqm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) ) )</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">i</font> = <font color="Maroon" title="c8">i0</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c10">i</font> = <font color="Maroon" title="c9">i</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">rseqi</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> ex <font color="Olive" title="b2">seqi</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</a> <font color="Maroon" title="c10">i</font></span>)</span> st <br/>( ( 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>  holds  <font color="Olive" title="b1">rseqi</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">yseq</font> <a href="real_ns1.html#K5" title="REAL_NS1:func.5">.</a> <font color="Olive" title="b3">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i0</font> ) &amp; <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i0</font> &amp; <font color="Olive" title="b2">seqi</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp; <font color="Olive" title="b1">rseqi</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">rseqi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c6">I</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Olive" title="b1">rseqi</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">seqi</font> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <span class="p3">(<span class="default"><a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Olive" title="b2">seqi</font></span>)</span></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> &amp; ( for <font color="Olive" title="b3">m</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>   ex <font color="Olive" title="b4">seqm</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> st <br/>( <font color="Olive" title="b4">seqm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b3">m</font> &amp; <font color="Olive" title="b2">seqi</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">seqm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> ) ) )</span><br/><span class="kw">consider </span><font color="Maroon" title="c11">rseqi</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>, <font color="Maroon" title="c12">seqi</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</a> <font color="Maroon" title="c10">i</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:43_3"/><span class="lab"><font color="Green" title="E40">A15</font></span>: 
( <font color="Maroon" title="c11">rseqi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">yyseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c12">seqi</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp; <font color="Maroon" title="c11">rseqi</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="c11">rseqi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c11">rseqi</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">seqi</font> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <span class="p3">(<span class="default"><a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c12">seqi</font></span>)</span></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> &amp; ( for <font color="Olive" title="b1">m</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>   ex <font color="Olive" title="b2">seqm</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> st <br/>( <font color="Olive" title="b2">seqm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">m</font> &amp; <font color="Maroon" title="c12">seqi</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">seqm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43"><span class="lab"><font color="Green" title="E35">A7</font></span></a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c13">rseqi</font> = <font color="Maroon" title="c11">rseqi</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">seqi</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</a> <font color="Maroon" title="c10">i</font></span>)</span> st <br/>( ( for <font color="Olive" title="b2">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" title="c13">rseqi</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">yseq</font> <a href="real_ns1.html#K5" title="REAL_NS1:func.5">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i0</font> ) &amp; <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i0</font> &amp; <font color="Olive" title="b1">seqi</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp; <font color="Maroon" title="c13">rseqi</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="c13">rseqi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c6">I</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c13">rseqi</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">seqi</font> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <span class="p3">(<span class="default"><a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Olive" title="b1">seqi</font></span>)</span></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> &amp; ( for <font color="Olive" title="b2">m</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>   ex <font color="Olive" title="b3">seqm</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> st <br/>( <font color="Olive" title="b3">seqm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b2">m</font> &amp; <font color="Olive" title="b1">seqi</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">seqm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> ) ) )</span><br/><span class="kw">take </span><font color="Maroon" title="c14">seqi</font> = <font color="Maroon" title="c12">seqi</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( 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" title="c13">rseqi</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">yseq</font> <a href="real_ns1.html#K5" title="REAL_NS1:func.5">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i0</font> ) &amp; <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i0</font> &amp; <font color="Maroon" title="c14">seqi</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp; <font color="Maroon" title="c13">rseqi</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="c13">rseqi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c6">I</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c13">rseqi</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">seqi</font> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <span class="p3">(<span class="default"><a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c14">seqi</font></span>)</span></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> &amp; ( for <font color="Olive" title="b1">m</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>   ex <font color="Olive" title="b2">seqm</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> st <br/>( <font color="Olive" title="b2">seqm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">m</font> &amp; <font color="Maroon" title="c14">seqi</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">seqm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> ) ) )</span><br/><span class="kw">thus </span><a NAME="E5:43_3"/>
( ( 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" title="c13">rseqi</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">yseq</font> <a href="real_ns1.html#K5" title="REAL_NS1:func.5">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i0</font> ) &amp; <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i0</font> &amp; <font color="Maroon" title="c14">seqi</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp; <font color="Maroon" title="c13">rseqi</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="c13">rseqi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c6">I</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c13">rseqi</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">seqi</font> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <span class="p3">(<span class="default"><a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c14">seqi</font></span>)</span></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> &amp; ( for <font color="Olive" title="b1">m</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>   ex <font color="Olive" title="b2">seqm</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> st <br/>( <font color="Olive" title="b2">seqm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">m</font> &amp; <font color="Maroon" title="c14">seqi</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">seqm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:43"><span class="lab"><font color="Green" title="E37">A12</font></span></a>, <a class="txt" href="#E1:43_3"><span class="lab"><font color="Green" title="E39">A14</font></span></a>, <a class="txt" href="#E4:43_3"><span class="lab"><font color="Green" title="E40">A15</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E11:43"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c6">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c6">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" title="REAL_NS1:def.4">REAL_NS1:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">xseq</font> = <font color="Maroon" title="c7">yseq</font> as   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c6">I</font></span>)</span> ;<br/>
<a NAME="E13:43"/>
<font color="Maroon" title="c8">xseq</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c7">yseq</font>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">xseq</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c6">I</font></span>)</span>, <font color="Maroon" title="c10">yseq</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> ,<span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c6">I</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E15:43"/><span class="lab"><font color="Green" title="E39">A16</font></span>: 
<font color="Maroon" title="c9">xseq</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c10">yseq</font>
 <span class="kw">and </span><br/><a NAME="E16:43"/><span class="lab"><font color="Green" title="E40">A17</font></span>: 
for <font color="Olive" title="b1">i0</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">i0</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">I</font> holds <br/> ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> ex <font color="Olive" title="b3">rseqi</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> ex <font color="Olive" title="b4">seqi</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</a> <font color="Olive" title="b2">i</font></span>)</span> st <br/>( ( for <font color="Olive" title="b5">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="Olive" title="b3">rseqi</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">yseq</font> <a href="real_ns1.html#K5" title="REAL_NS1:func.5">.</a> <font color="Olive" title="b5">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i0</font> ) &amp; <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">i0</font> &amp; <font color="Olive" title="b4">seqi</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp; <font color="Olive" title="b3">rseqi</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="b3">rseqi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c6">I</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b3">rseqi</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b4">seqi</font> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <span class="p3">(<span class="default"><a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Olive" title="b4">seqi</font></span>)</span></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> &amp; ( for <font color="Olive" title="b5">m</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>   ex <font color="Olive" title="b6">seqm</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> st <br/>( <font color="Olive" title="b6">seqm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b5">m</font> &amp; <font color="Olive" title="b4">seqi</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">seqm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:43"><span class="lab"><font color="Green" title="E38">A13</font></span></a></span>;<br/>
<a NAME="E17:43"/><span class="lab"><font color="Green" title="E41">A18</font></span>: 
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="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">I</font> holds <br/> ex <font color="Olive" title="b2">rseqi</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> st <br/>( ( 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>  holds  <font color="Olive" title="b2">rseqi</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">yseq</font> <a href="real_ns1.html#K5" title="REAL_NS1:func.5">.</a> <font color="Olive" title="b3">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) &amp; <font color="Olive" title="b2">rseqi</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp; <span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c6">I</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Olive" title="b2">rseqi</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">i0</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">i0</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">I</font> implies  ex <font color="Olive" title="b1">rseqi</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> st <br/>( ( for <font color="Olive" title="b2">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="Olive" title="b1">rseqi</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">yseq</font> <a href="real_ns1.html#K5" title="REAL_NS1:func.5">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">i0</font> ) &amp; <font color="Olive" title="b1">rseqi</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp; <span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c6">I</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Olive" title="b1">rseqi</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:43_4"/>
<font color="Maroon" title="c11">i0</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">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">rseqi</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> st <br/>( ( for <font color="Olive" title="b2">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="Olive" title="b1">rseqi</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">yseq</font> <a href="real_ns1.html#K5" title="REAL_NS1:func.5">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">i0</font> ) &amp; <font color="Olive" title="b1">rseqi</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp; <span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c6">I</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Olive" title="b1">rseqi</font> )</span><br/>

<a NAME="E2:43_4"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> ex <font color="Olive" title="b2">rseqi</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> ex <font color="Olive" title="b3">seqi</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</a> <font color="Olive" title="b1">i</font></span>)</span> st <br/>( ( for <font color="Olive" title="b4">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="Olive" title="b2">rseqi</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">yseq</font> <a href="real_ns1.html#K5" title="REAL_NS1:func.5">.</a> <font color="Olive" title="b4">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">i0</font> ) &amp; <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i0</font> &amp; <font color="Olive" title="b3">seqi</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp; <font color="Olive" title="b2">rseqi</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="b2">rseqi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c6">I</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b2">rseqi</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">seqi</font> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <span class="p3">(<span class="default"><a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Olive" title="b3">seqi</font></span>)</span></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> &amp; ( for <font color="Olive" title="b4">m</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>   ex <font color="Olive" title="b5">seqm</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> st <br/>( <font color="Olive" title="b5">seqm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b4">m</font> &amp; <font color="Olive" title="b3">seqi</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">seqm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:43"><span class="lab"><font color="Green" title="E40">A17</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:43_4"/>
 ex <font color="Olive" title="b1">rseqi</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> st <br/>( ( for <font color="Olive" title="b2">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="Olive" title="b1">rseqi</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">yseq</font> <a href="real_ns1.html#K5" title="REAL_NS1:func.5">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">i0</font> ) &amp; <font color="Olive" title="b1">rseqi</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp; <span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c6">I</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Olive" title="b1">rseqi</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E18:43"/><span class="lab"><font color="Green" title="E42">A19</font></span>: 
 <a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#G1" title="NORMSP_1:aggr.1">NORMSTR</a>(# <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="prvect_2.html#K8" title="PRVECT_2:func.8">zeros</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"><a href="prvect_1.html#K14" title="PRVECT_1:func.14">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_2.html#K6" title="PRVECT_2:func.6">addop</a> <font color="Maroon" title="c1">G</font></span>)</span></span><a href="prvect_1.html#K14" title="PRVECT_1:func.14">:]</a></span>,<span class="p1"><a href="prvect_2.html#K2" title="PRVECT_2:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_2.html#K9" title="PRVECT_2:func.9">multop</a> <font color="Maroon" title="c1">G</font></span>)</span></span><a href="prvect_2.html#K2" title="PRVECT_2:func.2">:]</a></span>,<span class="p1">(<span class="default"><a href="prvect_2.html#K13" title="PRVECT_2:func.13">productnorm</a> <font color="Maroon" title="c1">G</font></span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_2.html#T6" target="_self" title="PRVECT_2:th.6">Th6</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  implies <span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">xseq</font> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c6">I</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <font color="Maroon" title="c3">x0</font></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:43_5"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">xseq</font> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c6">I</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <font color="Maroon" title="c3">x0</font></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">i</font> = <font color="Maroon" title="c11">x</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>  ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">seqimx0</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">x0</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:43"><span class="lab"><font color="Green" title="E42">A19</font></span></a></span>;<br/><div><span class="lab"><font color="Green" title="E43">A20</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5_1"><span class="kw">now </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c14">mx0</font> =  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c3">x0</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:43"><span class="lab"><font color="Green" title="E42">A19</font></span></a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c15">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">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> <span class="p1">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c13">seqimx0</font></span>)</span> implies <span class="p1">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c13">seqimx0</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">yseq</font> <a href="real_ns1.html#K5" title="REAL_NS1:func.5">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">k</font> )</span><br/><span class="kw">assume </span><a NAME="E2:43_5_1"/><span class="lab"><font color="Green" title="E44">A21</font></span>: 
<font color="Maroon" title="c15">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> <span class="p1">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c13">seqimx0</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c13">seqimx0</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">yseq</font> <a href="real_ns1.html#K5" title="REAL_NS1:func.5">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">k</font></span><br/><a NAME="E3:43_5_1"/><span class="lab"><font color="Green" title="E45">A22</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c13">seqimx0</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_2.html#D11" target="_self" title="PRVECT_2:def.11">Def11</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">k0</font> = <font color="Maroon" title="c15">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_5_1"><span class="lab"><font color="Green" title="E44">A21</font></span></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></span>;<br/><a NAME="E5:43_5_1"/>
<font color="Maroon" title="c15">k</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">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_5_1"><span class="lab"><font color="Green" title="E44">A21</font></span></a>, <a class="txt" href="#E3:43_5_1"><span class="lab"><font color="Green" title="E45">A22</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">k1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c18">rseqk1i</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>, <font color="Maroon" title="c19">seqk1i</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</a> <font color="Maroon" title="c17">k1</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E7:43_5_1"/><span class="lab"><font color="Green" title="E46">A23</font></span>: 
for <font color="Olive" title="b1">j</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="c18">rseqk1i</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">yseq</font> <a href="real_ns1.html#K5" title="REAL_NS1:func.5">.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">k</font>
 <span class="kw">and </span><br/><a NAME="E8:43_5_1"/><span class="lab"><font color="Green" title="E47">A24</font></span>: 
<font color="Maroon" title="c17">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">k</font>
 <span class="kw">and </span><br/><a NAME="E9:43_5_1"/>
<font color="Maroon" title="c19">seqk1i</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a> 
 <span class="kw">and </span><br/><a NAME="E10:43_5_1"/>
<font color="Maroon" title="c18">rseqk1i</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 <span class="kw">and </span><br/><a NAME="E11:43_5_1"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c18">rseqk1i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c6">I</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">k1</font>
 <span class="kw">and </span><br/><a NAME="E12:43_5_1"/><span class="lab"><font color="Green" title="E48">A25</font></span>: 
<font color="Maroon" title="c18">rseqk1i</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">seqk1i</font> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <span class="p3">(<span class="default"><a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c19">seqk1i</font></span>)</span></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span>
 <span class="kw">and </span><br/><a NAME="E13:43_5_1"/><span class="lab"><font color="Green" title="E49">A26</font></span>: 
for <font color="Olive" title="b1">m</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>   ex <font color="Olive" title="b2">seqk1m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> st <br/>( <font color="Olive" title="b2">seqk1m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">m</font> &amp; <font color="Maroon" title="c19">seqk1i</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">seqk1m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">k1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:43"><span class="lab"><font color="Green" title="E40">A17</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c20">seqk0</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</a> <font color="Maroon" title="c16">k0</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E15:43_5_1"/>
<font color="Maroon" title="c20">seqk0</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a> 
 <span class="kw">and </span><br/><a NAME="E16:43_5_1"/><span class="lab"><font color="Green" title="E50">A27</font></span>: 
<font color="Maroon" title="c4">y0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c20">seqk0</font>
 <span class="kw">and </span><br/><a NAME="E17:43_5_1"/><span class="lab"><font color="Green" title="E51">A28</font></span>: 
for <font color="Olive" title="b1">m</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>   ex <font color="Olive" title="b2">seqm0</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> st <br/>( <font color="Olive" title="b2">seqm0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">m</font> &amp; <font color="Maroon" title="c20">seqk0</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">seqm0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k0</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E33">A2</font></span></a></span>;<br/><a NAME="E18:43_5_1"/><span class="lab"><font color="Green" title="E52">A29</font></span>: 
 ex <font color="Olive" title="b1">seqm0</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> st <br/>( <font color="Olive" title="b1">seqm0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c20">seqk0</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">seqm0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k0</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:43_5_1"><span class="lab"><font color="Green" title="E51">A28</font></span></a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c21">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  implies <font color="Maroon" title="c19">seqk1i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">seqk0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:43_5_1_1"/>
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">seqk1i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">seqk0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c22">m</font> = <font color="Maroon" title="c21">x</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>  ;<br/><a NAME="E3:43_5_1_1"/>
(  ex <font color="Olive" title="b1">seqk1m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> st <br/>( <font color="Olive" title="b1">seqk1m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c22">m</font> &amp; <font color="Maroon" title="c19">seqk1i</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c22">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">seqk1m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">k1</font> ) &amp;  ex <font color="Olive" title="b1">seqm0</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> st <br/>( <font color="Olive" title="b1">seqm0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c22">m</font> &amp; <font color="Maroon" title="c20">seqk0</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c22">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">seqm0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k0</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:43_5_1"><span class="lab"><font color="Green" title="E51">A28</font></span></a>, <a class="txt" href="#E13:43_5_1"><span class="lab"><font color="Green" title="E49">A26</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:43_5_1_1"/>
<font color="Maroon" title="c19">seqk1i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">seqk0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:43_5_1"><span class="lab"><font color="Green" title="E47">A24</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E20:43_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A30</font></span>: 
<font color="Maroon" title="c19">seqk1i</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c20">seqk0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:43_5_1"><span class="lab"><font color="Green" title="E47">A24</font></span></a>, <a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</a></span>;<br/><a NAME="E21:43_5_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_2.html#D4" target="_self" title="PRVECT_2:def.4">Def4</a></span>;<br/><a NAME="E22:43_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A31</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></span>;<br/><a NAME="E23:43_5_1"/>
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c3">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">x0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T29" title="RLVECT_1:th.29">RLVECT_1:29</a></span>;<br/><a NAME="E24:43_5_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">mx0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c20">seqk0</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E18:43"><span class="lab"><font color="Green" title="E42">A19</font></span></a>, <a class="txt" href="#E16:43_5_1"><span class="lab"><font color="Green" title="E50">A27</font></span></a>, <a class="txt" href="#E22:43_5_1"><span class="lab"><font color="Green" title="E54">A31</font></span></a>, <a class="txt" href="prvect_2.html#E18"><span class="lab"><font color="Green" title="E16">Lm4</font></span></a></span>;<br/><a NAME="E25:43_5_1"/><span class="kw">then </span>
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c20">seqk0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">mx0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T29" title="RLVECT_1:th.29">RLVECT_1:29</a></span>;<br/><a NAME="E26:43_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A32</font></span>: 
<font color="Maroon" title="c13">seqimx0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">seqk0</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c20">seqk0</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:43"><span class="lab"><font color="Green" title="E42">A19</font></span></a>, <a class="txt" href="#E18:43_5_1"><span class="lab"><font color="Green" title="E52">A29</font></span></a>, <a class="txt" href="#E22:43_5_1"><span class="lab"><font color="Green" title="E54">A31</font></span></a>, <a class="txt" href="prvect_2.html#E17"><span class="lab"><font color="Green" title="E15">Lm3</font></span></a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c13">seqimx0</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">k</font> = 
the <a href="normsp_1.html#U1" title="NORMSP_1:sel.1">norm</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</a> <font color="Maroon" title="c16">k0</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">seqimx0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_2.html#D11" target="_self" title="PRVECT_2:def.11">Def11</a></span>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">seqk0</font> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <span class="p4">(<span class="default"><a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c20">seqk0</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c12">i</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:43_5_1"><span class="lab"><font color="Green" title="E55">A32</font></span></a>, <a class="ref" href="normsp_1.html#D7" title="NORMSP_1:def.7">NORMSP_1:def 7</a></span>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">seqk1i</font> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <span class="p3">(<span class="default"><a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c19">seqk1i</font></span>)</span></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:43_5_1"><span class="lab"><font color="Green" title="E47">A24</font></span></a>, <a class="txt" href="#E20:43_5_1"><span class="lab"><font color="Green" title="E53">A30</font></span></a>, <a class="ref" href="normsp_1.html#D10" title="NORMSP_1:def.10">NORMSP_1:def 10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">yseq</font> <a href="real_ns1.html#K5" title="REAL_NS1:func.5">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43_5_1"><span class="lab"><font color="Green" title="E46">A23</font></span></a>, <a class="txt" href="#E12:43_5_1"><span class="lab"><font color="Green" title="E48">A25</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E5:43_5"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">yseq</font> <a href="real_ns1.html#K5" title="REAL_NS1:func.5">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">I</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>;<br/><a NAME="E6:43_5"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">yseq</font> <a href="real_ns1.html#K5" title="REAL_NS1:func.5">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c13">seqimx0</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_2.html#D11" target="_self" title="PRVECT_2:def.11">Def11</a></span>;<br/><a NAME="E7:43_5"/><span class="kw">then </span>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">yseq</font> <a href="real_ns1.html#K5" title="REAL_NS1:func.5">.</a> <font color="Maroon" title="c12">i</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> <span class="p1">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c13">seqimx0</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></span>;<br/><a NAME="E8:43_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A33</font></span>: 
<font color="Maroon" title="c10">yseq</font> <a href="real_ns1.html#K5" title="REAL_NS1:func.5">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c13">seqimx0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_5"><span class="lab"><font color="Green" title="E43">A20</font></span></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></span>;<br/><span class="kw">thus </span><span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">xseq</font> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c6">I</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">x</font> = 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">xseq</font> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p5">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c6">I</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c12">i</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#D10" title="NORMSP_1:def.10">NORMSP_1:def 10</a></span>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c6">I</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#D7" title="NORMSP_1:def.7">NORMSP_1:def 7</a></span>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">xseq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c12">i</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T26" title="RLVECT_1:th.26">RLVECT_1:26</a></span>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">yseq</font> <a href="real_ns1.html#K5" title="REAL_NS1:func.5">.</a> <font color="Maroon" title="c12">i</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:43"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="ref" href="real_ns1.html#T1" title="REAL_NS1:th.1">REAL_NS1:1</a></span>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">x0</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:43_5"><span class="lab"><font color="Green" title="E44">A33</font></span></a>, <a class="ref" href="prvect_2.html#T7" target="_self" title="PRVECT_2:th.7">Th7</a></span>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <font color="Maroon" title="c3">x0</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c12">i</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#D7" title="NORMSP_1:def.7">NORMSP_1:def 7</a></span>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <font color="Maroon" title="c3">x0</font></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#D10" title="NORMSP_1:def.10">NORMSP_1:def 10</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E20:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A34</font></span>: 
<span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">xseq</font> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c6">I</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <font color="Maroon" title="c3">x0</font></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</a></span>;<br/>
<a NAME="E21:43"/>
 <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c6">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c6">I</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" title="REAL_NS1:def.4">REAL_NS1:def 4</a></span>;<br/>
<a NAME="E22:43"/><span class="kw">then </span>
( <font color="Maroon" title="c9">xseq</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c9">xseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c6">I</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:43"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="txt" href="#E17:43"><span class="lab"><font color="Green" title="E41">A18</font></span></a>, <a class="ref" href="real_ns1.html#T11" title="REAL_NS1:th.11">REAL_NS1:11</a></span>;<br/>
<a NAME="E23:43"/><span class="kw">then </span>
( <span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <font color="Maroon" title="c3">x0</font></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> 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> <span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <font color="Maroon" title="c3">x0</font></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:43"><span class="lab"><font color="Green" title="E43">A34</font></span></a>, <a class="txt" href="prvect_2.html#E31"><span class="lab"><font color="Green" title="E29">Lm8</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E24:43"/>
( <font color="Maroon" title="c2">seq</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c2">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x0</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="prvect_2.html#E31"><span class="lab"><font color="Green" title="E29">Lm8</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
