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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="csspace.html#NM1" title="CSSPACE:NM.1">ComplexUnitarySpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">g1</font>, <font color="Olive" title="b2">g2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b3">seq1</font>, <font color="Olive" title="b4">seq2</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b3">seq1</font> is <a href="clvect_2.html#V1" title="CLVECT_2:attr.1">convergent</a> &amp;  <a href="clvect_2.html#K1" title="CLVECT_2:func.1">lim</a> <font color="Olive" title="b3">seq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g1</font> &amp; <font color="Olive" title="b4">seq2</font> is <a href="clvect_2.html#V1" title="CLVECT_2:attr.1">convergent</a> &amp;  <a href="clvect_2.html#K1" title="CLVECT_2:func.1">lim</a> <font color="Olive" title="b4">seq2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g2</font> holds <br/>( <span class="p1"><a href="clvect_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b3">seq1</font> <a href="csspace.html#K16" title="CSSPACE:func.16">+</a> <font color="Olive" title="b4">seq2</font></span>)</span> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">g1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">g2</font></span>)</span></span>)</span></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</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="clvect_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b3">seq1</font> <a href="csspace.html#K16" title="CSSPACE:func.16">+</a> <font color="Olive" title="b4">seq2</font></span>)</span> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">g1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">g2</font></span>)</span></span>)</span></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><b>let </b><font color="Maroon" title="c2">g1</font>, <font color="Maroon" title="c3">g2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">seq1</font>, <font color="Olive" title="b2">seq2</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">seq1</font> is <a href="clvect_2.html#V1" title="CLVECT_2:attr.1">convergent</a> &amp;  <a href="clvect_2.html#K1" title="CLVECT_2:func.1">lim</a> <font color="Olive" title="b1">seq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g1</font> &amp; <font color="Olive" title="b2">seq2</font> is <a href="clvect_2.html#V1" title="CLVECT_2:attr.1">convergent</a> &amp;  <a href="clvect_2.html#K1" title="CLVECT_2:func.1">lim</a> <font color="Olive" title="b2">seq2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g2</font> holds <br/>( <span class="p1"><a href="clvect_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">seq1</font> <a href="csspace.html#K16" title="CSSPACE:func.16">+</a> <font color="Olive" title="b2">seq2</font></span>)</span> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">g1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">g2</font></span>)</span></span>)</span></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</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="clvect_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">seq1</font> <a href="csspace.html#K16" title="CSSPACE:func.16">+</a> <font color="Olive" title="b2">seq2</font></span>)</span> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">g1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">g2</font></span>)</span></span>)</span></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><b>let </b><font color="Maroon" title="c4">seq1</font>, <font color="Maroon" title="c5">seq2</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">seq1</font> is <a href="clvect_2.html#V1" title="CLVECT_2:attr.1">convergent</a> &amp;  <a href="clvect_2.html#K1" title="CLVECT_2:func.1">lim</a> <font color="Maroon" title="c4">seq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g1</font> &amp; <font color="Maroon" title="c5">seq2</font> is <a href="clvect_2.html#V1" title="CLVECT_2:attr.1">convergent</a> &amp;  <a href="clvect_2.html#K1" title="CLVECT_2:func.1">lim</a> <font color="Maroon" title="c5">seq2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g2</font> implies ( <span class="p1"><a href="clvect_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="csspace.html#K16" title="CSSPACE:func.16">+</a> <font color="Maroon" title="c5">seq2</font></span>)</span> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">g1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">g2</font></span>)</span></span>)</span></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</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="clvect_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="csspace.html#K16" title="CSSPACE:func.16">+</a> <font color="Maroon" title="c5">seq2</font></span>)</span> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">g1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">g2</font></span>)</span></span>)</span></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )</span><br/>









<b>assume </b><b>that </b><br/><a NAME="E1:30"/><i><font color="Green" title="E26">A1</font></i>: 
<font color="Maroon" title="c4">seq1</font> is <a href="clvect_2.html#V1" title="CLVECT_2:attr.1">convergent</a>
 <b>and </b><br/><a NAME="E2:30"/><i><font color="Green" title="E27">A2</font></i>: 
 <a href="clvect_2.html#K1" title="CLVECT_2:func.1">lim</a> <font color="Maroon" title="c4">seq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g1</font>
 <b>and </b><br/><a NAME="E3:30"/><i><font color="Green" title="E28">A3</font></i>: 
<font color="Maroon" title="c5">seq2</font> is <a href="clvect_2.html#V1" title="CLVECT_2:attr.1">convergent</a>
 <b>and </b><br/><a NAME="E4:30"/><i><font color="Green" title="E29">A4</font></i>: 
 <a href="clvect_2.html#K1" title="CLVECT_2:func.1">lim</a> <font color="Maroon" title="c5">seq2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="clvect_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="csspace.html#K16" title="CSSPACE:func.16">+</a> <font color="Maroon" title="c5">seq2</font></span>)</span> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">g1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">g2</font></span>)</span></span>)</span></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</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="clvect_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="csspace.html#K16" title="CSSPACE:func.16">+</a> <font color="Maroon" title="c5">seq2</font></span>)</span> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">g1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">g2</font></span>)</span></span>)</span></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<a NAME="E5:30"/><i><font color="Green" title="E30">A5</font></i>: 
<font color="Maroon" title="c4">seq1</font> <a href="csspace.html#K16" title="CSSPACE:func.16">+</a> <font color="Maroon" title="c5">seq2</font> is <a href="clvect_2.html#V1" title="CLVECT_2:attr.1">convergent</a>
 
<b>by </b><i><a class="txt" href="clvect_2.html#E1:30"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="clvect_2.html#E3:30"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="clvect_2.html#T3" target="_self" title="CLVECT_2:th.3">Th3</a></i>;<br/>
<a NAME="E6:30"/>
 <a href="clvect_2.html#K1" title="CLVECT_2:func.1">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="csspace.html#K16" title="CSSPACE:func.16">+</a> <font color="Maroon" title="c5">seq2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">g2</font>
 
<b>by </b><i><a class="txt" href="clvect_2.html#E1:30"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="clvect_2.html#E2:30"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="clvect_2.html#E3:30"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="clvect_2.html#E4:30"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="clvect_2.html#T13" target="_self" title="CLVECT_2:th.13">Th13</a></i>;<br/>
<b>hence </b><a NAME="E7:30"/>
( <span class="p1"><a href="clvect_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="csspace.html#K16" title="CSSPACE:func.16">+</a> <font color="Maroon" title="c5">seq2</font></span>)</span> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">g1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">g2</font></span>)</span></span>)</span></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</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="clvect_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="csspace.html#K16" title="CSSPACE:func.16">+</a> <font color="Maroon" title="c5">seq2</font></span>)</span> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">g1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">g2</font></span>)</span></span>)</span></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="clvect_2.html#E5:30"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="clvect_2.html#T22" target="_self" title="CLVECT_2:th.22">Th22</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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