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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">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#V3" title="CLVECT_2:attr.3">bounded</a> &amp; <font color="Olive" title="b2">seq2</font> is <a href="clvect_2.html#V3" title="CLVECT_2:attr.3">bounded</a> holds <br/><font color="Olive" title="b1">seq1</font> <a href="csspace.html#K16" title="CSSPACE:func.16">+</a> <font color="Olive" title="b2">seq2</font> is <a href="clvect_2.html#V3" title="CLVECT_2:attr.3">bounded</a></span><br/><b>let </b><font color="Maroon" title="c2">seq1</font>, <font color="Maroon" title="c3">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="c2">seq1</font> is <a href="clvect_2.html#V3" title="CLVECT_2:attr.3">bounded</a> &amp; <font color="Maroon" title="c3">seq2</font> is <a href="clvect_2.html#V3" title="CLVECT_2:attr.3">bounded</a> implies <font color="Maroon" title="c2">seq1</font> <a href="csspace.html#K16" title="CSSPACE:func.16">+</a> <font color="Maroon" title="c3">seq2</font> is <a href="clvect_2.html#V3" title="CLVECT_2:attr.3">bounded</a> )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:84"/><i><font color="Green" title="E44">A1</font></i>: 
<font color="Maroon" title="c2">seq1</font> is <a href="clvect_2.html#V3" title="CLVECT_2:attr.3">bounded</a>
 <b>and </b><br/><a NAME="E2:84"/><i><font color="Green" title="E45">A2</font></i>: 
<font color="Maroon" title="c3">seq2</font> is <a href="clvect_2.html#V3" title="CLVECT_2:attr.3">bounded</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">seq1</font> <a href="csspace.html#K16" title="CSSPACE:func.16">+</a> <font color="Maroon" title="c3">seq2</font> is <a href="clvect_2.html#V3" title="CLVECT_2:attr.3">bounded</a></span><br/>

<b>consider </b><font color="Maroon" title="c4">M1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:84"/><i><font color="Green" title="E46">A3</font></i>: 
<font color="Maroon" title="c4">M1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E5:84"/><i><font color="Green" title="E47">A4</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">M1</font>
 <b>by </b><i><a class="txt" href="#E1:84"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="clvect_2.html#D10" target="_self" title="CLVECT_2:def.10">Def10</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">M2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E7:84"/><i><font color="Green" title="E48">A5</font></i>: 
<font color="Maroon" title="c5">M2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E8:84"/><i><font color="Green" title="E49">A6</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">M2</font>
 <b>by </b><i><a class="txt" href="#E2:84"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="clvect_2.html#D10" target="_self" title="CLVECT_2:def.10">Def10</a></i>;<br/>
<a NAME="E9:84"/><i><font color="Green" title="E50">A7</font></i>: 
<font color="Maroon" title="c4">M1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c5">M2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E4:84"><i><font color="Green" title="E46">A3</font></i></a>, <a class="txt" href="#E7:84"><i><font color="Green" title="E48">A5</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="csspace.html#K16" title="CSSPACE:func.16">+</a> <font color="Maroon" title="c3">seq2</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">M1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c5">M2</font></span><br/><a NAME="E1:84_1"/><i><font color="Green" title="E51">A8</font></i>: 
<span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">M1</font>
 <b>by </b><i><a class="txt" href="#E5:84"><i><font color="Green" title="E47">A4</font></i></a></i>;<br/><a NAME="E2:84_1"/>
<span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">M2</font>
 <b>by </b><i><a class="txt" href="#E8:84"><i><font color="Green" title="E49">A6</font></i></a></i>;<br/><a NAME="E3:84_1"/><b>then </b><i><font color="Green" title="E52">A9</font></i>: 
<span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">M1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c5">M2</font>
 <b>by </b><i><a class="txt" href="#E1:84_1"><i><font color="Green" title="E51">A8</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E4:84_1"/>
<span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="csspace.html#K16" title="CSSPACE:func.16">+</a> <font color="Maroon" title="c3">seq2</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span>
 <b>by </b><i><a class="ref" href="normsp_1.html#D5" title="NORMSP_1:def.5">NORMSP_1:def 5</a></i>;<br/><a NAME="E5:84_1"/><b>then </b>
<span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="csspace.html#K16" title="CSSPACE:func.16">+</a> <font color="Maroon" title="c3">seq2</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span>
 <b>by </b><i><a class="ref" href="csspace.html#T48" title="CSSPACE:th.48">CSSPACE:48</a></i>;<br/><b>hence </b><a NAME="E6:84_1"/>
<span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="csspace.html#K16" title="CSSPACE:func.16">+</a> <font color="Maroon" title="c3">seq2</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">M1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c5">M2</font>
 <b>by </b><i><a class="txt" href="#E3:84_1"><i><font color="Green" title="E52">A9</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:84"/>
<font color="Maroon" title="c2">seq1</font> <a href="csspace.html#K16" title="CSSPACE:func.16">+</a> <font color="Maroon" title="c3">seq2</font> is <a href="clvect_2.html#V3" title="CLVECT_2:attr.3">bounded</a>
 <b>by </b><i><a class="txt" href="#E9:84"><i><font color="Green" title="E50">A7</font></i></a>, <a class="ref" href="clvect_2.html#D10" target="_self" title="CLVECT_2:def.10">Def10</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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