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<span class="kw">let </span><font color="Maroon" title="c1">c</font> be   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">seq1</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><br/>  for <font color="Olive" title="b2">seq</font> being   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>  st <font color="Olive" title="b2">seq</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a>  &amp; <font color="Olive" title="b1">seq1</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  holds <br/> for <font color="Olive" title="b3">rseq</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st (  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="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Olive" title="b3">rseq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b2">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">i</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c1">c</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b2">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">i</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c1">c</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">seq1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b4">i</font></span>)</span> ) holds <br/>( <font color="Olive" title="b3">rseq</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">rseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Olive" title="b2">seq</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c1">c</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Olive" title="b2">seq</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c1">c</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Olive" title="b1">seq1</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">seq1</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">seq</font> being   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>  st <font color="Olive" title="b1">seq</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a>  &amp; <font color="Maroon" title="c2">seq1</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  holds <br/> for <font color="Olive" title="b2">rseq</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st (  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="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Olive" title="b2">rseq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b1">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">i</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c1">c</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b1">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">i</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c1">c</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font></span>)</span> ) holds <br/>( <font color="Olive" title="b2">rseq</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">rseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Olive" title="b1">seq</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c1">c</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Olive" title="b1">seq</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c1">c</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">seq1</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">seq</font> be   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">seq</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a>  &amp; <font color="Maroon" title="c2">seq1</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  implies  for <font color="Olive" title="b1">rseq</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st (  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="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Olive" title="b1">rseq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c1">c</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c1">c</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font></span>)</span> ) holds <br/>( <font color="Olive" title="b1">rseq</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">rseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c1">c</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c1">c</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">seq1</font></span>)</span> ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:26"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
<font color="Maroon" title="c3">seq</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a> 
 <span class="kw">and </span><br/><a NAME="E2:26"/><span class="lab"><font color="Green" title="E27">A2</font></span>: 
<font color="Maroon" title="c2">seq1</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">rseq</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st (  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="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Olive" title="b1">rseq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c1">c</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c1">c</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font></span>)</span> ) holds <br/>( <font color="Olive" title="b1">rseq</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">rseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c1">c</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c1">c</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">seq1</font></span>)</span> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">cseq</font> = <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c1">c</font> as   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a> ;<br/>
<a NAME="E4:26"/><span class="lab"><font color="Green" title="E28">A3</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c4">cseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>;<br/>
<a NAME="E5:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A4</font></span>: 
<font color="Maroon" title="c4">cseq</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_2.html#T9" title="COMSEQ_2:th.9">COMSEQ_2:9</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">s</font> = <font color="Maroon" title="c3">seq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c4">cseq</font> as   <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a>  <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E26">A1</font></span></a></span>;<br/>
<a NAME="E7:26"/><span class="lab"><font color="Green" title="E30">A5</font></span>: 
<span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c5">s</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 
;<br/>
<a NAME="E8:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A6</font></span>: 
<span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c4">cseq</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c4">cseq</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_2.html#T14" title="SEQ_2:th.14">SEQ_2:14</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">rseq</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  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>  holds  <font color="Maroon" title="c6">rseq</font> <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> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c1">c</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c1">c</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> ) implies ( <font color="Maroon" title="c6">rseq</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="c6">rseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c1">c</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c1">c</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">seq1</font></span>)</span> ) )</span><br/>

<span class="kw">assume </span><a NAME="E9:26"/><span class="lab"><font color="Green" title="E32">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="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c6">rseq</font> <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> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c1">c</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c1">c</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">rseq</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="c6">rseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c1">c</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c1">c</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">seq1</font></span>)</span> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">i</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="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">rseq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c4">cseq</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c4">cseq</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span> <a href="series_1.html#K1" title="SERIES_1:func.1">+</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span><br/><span class="kw">thus </span><font color="Maroon" title="c6">rseq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c1">c</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c1">c</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:26"><span class="lab"><font color="Green" title="E32">A7</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">cseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">i</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c1">c</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">cseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">i</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">cseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">i</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="valued_1.html#K31" title="VALUED_1:func.31">-</a> <font color="Maroon" title="c4">cseq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">i</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">cseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">i</font></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T8" title="VALUED_1:th.8">VALUED_1:8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="valued_1.html#K31" title="VALUED_1:func.31">-</a> <font color="Maroon" title="c4">cseq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">i</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="valued_1.html#K31" title="VALUED_1:func.31">-</a> <font color="Maroon" title="c4">cseq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">i</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T8" title="VALUED_1:th.8">VALUED_1:8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="series_1.html#K1" title="SERIES_1:func.1">+</a> <span class="p5">(<span class="default"><a href="valued_1.html#K31" title="VALUED_1:func.31">-</a> <font color="Maroon" title="c4">cseq</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">i</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="valued_1.html#K31" title="VALUED_1:func.31">-</a> <font color="Maroon" title="c4">cseq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">i</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T1" title="VALUED_1:th.1">VALUED_1:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c4">cseq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">i</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="series_1.html#K1" title="SERIES_1:func.1">+</a> <span class="p5">(<span class="default"><a href="valued_1.html#K31" title="VALUED_1:func.31">-</a> <font color="Maroon" title="c4">cseq</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">i</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T1" title="VALUED_1:th.1">VALUED_1:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c4">cseq</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c4">cseq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">i</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T18" title="VALUED_1:th.18">VALUED_1:18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c4">cseq</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c4">cseq</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T18" title="VALUED_1:th.18">VALUED_1:18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c4">cseq</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c4">cseq</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T8" title="SEQ_1:th.8">SEQ_1:8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c4">cseq</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c4">cseq</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span> <a href="series_1.html#K1" title="SERIES_1:func.1">+</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T7" title="SEQ_1:th.7">SEQ_1:7</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E11:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A8</font></span>: 
<font color="Maroon" title="c6">rseq</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c4">cseq</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c4">cseq</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span> <a href="series_1.html#K1" title="SERIES_1:func.1">+</a> <font color="Maroon" title="c2">seq1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T63" title="FUNCT_2:th.63">FUNCT_2:63</a></span>;<br/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c4">cseq</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c4">cseq</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c4">cseq</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c4">cseq</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="ref" href="seq_2.html#T15" title="SEQ_2:th.15">SEQ_2:15</a></span>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c4">cseq</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c4">cseq</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E5:26"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="ref" href="comseq_2.html#T27" title="COMSEQ_2:th.27">COMSEQ_2:27</a></span>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c4">cseq</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c4">cseq</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E5:26"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="ref" href="comseq_2.html#T27" title="COMSEQ_2:th.27">COMSEQ_2:27</a></span>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c1">c</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c4">cseq</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="ref" href="comseq_2.html#T10" title="COMSEQ_2:th.10">COMSEQ_2:10</a></span>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c1">c</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c1">c</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="ref" href="comseq_2.html#T10" title="COMSEQ_2:th.10">COMSEQ_2:10</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E13:26"/>
( <font color="Maroon" title="c6">rseq</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="c6">rseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c1">c</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c1">c</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">seq1</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E11:26"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E31">A6</font></span></a>, <a class="ref" href="seq_2.html#T5" title="SEQ_2:th.5">SEQ_2:5</a>, <a class="ref" href="seq_2.html#T6" title="SEQ_2:th.6">SEQ_2:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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