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<div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  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 <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">Cseq</font> being   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>  st <font color="Olive" title="b2">Cseq</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  &amp; <font color="Olive" title="b1">seq</font> is  <a href="clvect_2.html#V3" title="CLVECT_2:attr.3">bounded</a>  holds <br/><font color="Olive" title="b2">Cseq</font> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <font color="Olive" title="b1">seq</font> is  <a href="clvect_2.html#V3" title="CLVECT_2:attr.3">bounded</a> </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 <font color="Maroon" title="c1">X</font>; <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">Cseq</font> being   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>  st <font color="Olive" title="b1">Cseq</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  &amp; <font color="Maroon" title="c2">seq</font> is  <a href="clvect_2.html#V3" title="CLVECT_2:attr.3">bounded</a>  holds <br/><font color="Olive" title="b1">Cseq</font> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <font color="Maroon" title="c2">seq</font> is  <a href="clvect_2.html#V3" title="CLVECT_2:attr.3">bounded</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">Cseq</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">Cseq</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  &amp; <font color="Maroon" title="c2">seq</font> is  <a href="clvect_2.html#V3" title="CLVECT_2:attr.3">bounded</a>  implies <font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <font color="Maroon" title="c2">seq</font> is  <a href="clvect_2.html#V3" title="CLVECT_2:attr.3">bounded</a>  )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:54"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
<font color="Maroon" title="c3">Cseq</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 <span class="kw">and </span><br/><a NAME="E2:54"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
<font color="Maroon" title="c2">seq</font> is  <a href="clvect_2.html#V3" title="CLVECT_2:attr.3">bounded</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">Cseq</font> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <font color="Maroon" title="c2">seq</font> is  <a href="clvect_2.html#V3" title="CLVECT_2:attr.3">bounded</a> </span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">M1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E4:54"/><span class="lab"><font color="Green" title="E26">A3</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E5:54"/><span class="lab"><font color="Green" title="E27">A4</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 <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">M1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="comseq_2.html#T8" title="COMSEQ_2:th.8">COMSEQ_2:8</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">M2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E7:54"/><span class="lab"><font color="Green" title="E28">A5</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E8:54"/><span class="lab"><font color="Green" title="E29">A6</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  <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">seq</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="clvect_2.html#D10" title="CLVECT_2:def.10">CLVECT_2:def 10</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c6">M</font> = <font color="Maroon" title="c4">M1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c5">M2</font>;<br/><span class="kw">take </span><font color="Maroon" title="c7">M</font> = <font color="Maroon" title="c4">M1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c5">M2</font>; <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="c7">M</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>  &amp; (  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"><span class="p3">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</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="c7">M</font> ) )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c3">Cseq</font> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c8">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="c7">M</font></span><br/><a NAME="E1:54_1_1"/>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T46" title="COMPLEX1:th.46">COMPLEX1:46</a></span>;<br/><a NAME="E2:54_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A7</font></span>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">n</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="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c8">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="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">n</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> <font color="Maroon" title="c5">M2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:54"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/><a NAME="E3:54_1_1"/>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">M1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:54"><span class="lab"><font color="Green" title="E27">A4</font></span></a></span>;<br/><a NAME="E4:54_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A8</font></span>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">n</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> <font color="Maroon" title="c5">M2</font> <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#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c5">M2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:54"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/><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="c3">Cseq</font> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span> = 
<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="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_3.html#D8" target="_self" title="CLVECT_3:def.8">Def9</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"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">n</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="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="csspace.html#T43" title="CSSPACE:th.43">CSSPACE:43</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:54_1_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="c3">Cseq</font> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c8">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="c7">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_1_1"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="txt" href="#E4:54_1_1"><span class="lab"><font color="Green" title="E31">A8</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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><span class="kw">hence </span><a NAME="E2:54_1"/>
( <font color="Maroon" title="c7">M</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>  &amp; (  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"><span class="p3">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</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="c7">M</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:54"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="txt" href="#E7:54"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="ref" href="xreal_1.html#T129" title="XREAL_1:th.129">XREAL_1:129</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>
<span class="kw">hence </span><a NAME="E10:54"/>
<font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <font color="Maroon" title="c2">seq</font> is  <a href="clvect_2.html#V3" title="CLVECT_2:attr.3">bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_2.html#D10" title="CLVECT_2:def.10">CLVECT_2:def 10</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>
