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<b>let </b><font color="Maroon" title="c1">CNS</font> be   <a href="clvect_1.html#NM5" title="CLVECT_1:NM.5">ComplexNormSpace</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">h</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">CNS</font>, <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <br/> for <font color="Olive" title="b2">Cseq</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">CNS</font><br/> for <font color="Olive" title="b3">Ns</font> being <a href="seqm_3.html#V1" title="SEQM_3:attr.1">increasing</a> <a href="seqm_3.html#NM1" title="SEQM_3:NM.1">Seq_of_Nat</a>  st  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">Cseq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b1">h</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="ncfcont1.html#K7" title="NCFCONT1:func.7">*</a> <font color="Olive" title="b2">Cseq</font></span>)</span> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Olive" title="b3">Ns</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">h</font> <a href="ncfcont1.html#K7" title="NCFCONT1:func.7">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">Cseq</font> <a href="lopban_3.html#K4" title="LOPBAN_3:func.4">*</a> <font color="Olive" title="b3">Ns</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">h</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">CNS</font>, <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</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">Cseq</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">CNS</font><br/> for <font color="Olive" title="b2">Ns</font> being <a href="seqm_3.html#V1" title="SEQM_3:attr.1">increasing</a> <a href="seqm_3.html#NM1" title="SEQM_3:NM.1">Seq_of_Nat</a>  st  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">Cseq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">h</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="ncfcont1.html#K7" title="NCFCONT1:func.7">*</a> <font color="Olive" title="b1">Cseq</font></span>)</span> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Olive" title="b2">Ns</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c2">h</font> <a href="ncfcont1.html#K7" title="NCFCONT1:func.7">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">Cseq</font> <a href="lopban_3.html#K4" title="LOPBAN_3:func.4">*</a> <font color="Olive" title="b2">Ns</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">Cseq</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">CNS</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">Ns</font> being <a href="seqm_3.html#V1" title="SEQM_3:attr.1">increasing</a> <a href="seqm_3.html#NM1" title="SEQM_3:NM.1">Seq_of_Nat</a>  st  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">Cseq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">h</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="ncfcont1.html#K7" title="NCFCONT1:func.7">*</a> <font color="Maroon" title="c3">Cseq</font></span>)</span> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Olive" title="b1">Ns</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c2">h</font> <a href="ncfcont1.html#K7" title="NCFCONT1:func.7">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="lopban_3.html#K4" title="LOPBAN_3:func.4">*</a> <font color="Olive" title="b1">Ns</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">Ns</font> be  <a href="seqm_3.html#V1" title="SEQM_3:attr.1">increasing</a> <a href="seqm_3.html#NM1" title="SEQM_3:NM.1">Seq_of_Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">Cseq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">h</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="ncfcont1.html#K7" title="NCFCONT1:func.7">*</a> <font color="Maroon" title="c3">Cseq</font></span>)</span> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c4">Ns</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c2">h</font> <a href="ncfcont1.html#K7" title="NCFCONT1:func.7">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="lopban_3.html#K4" title="LOPBAN_3:func.4">*</a> <font color="Maroon" title="c4">Ns</font></span>)</span> )</span><br/>







<b>assume </b><a NAME="E1:37"/><i><font color="Green" title="E36">A1</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">Cseq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="ncfcont1.html#K7" title="NCFCONT1:func.7">*</a> <font color="Maroon" title="c3">Cseq</font></span>)</span> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c4">Ns</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c2">h</font> <a href="ncfcont1.html#K7" title="NCFCONT1:func.7">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="lopban_3.html#K4" title="LOPBAN_3:func.4">*</a> <font color="Maroon" title="c4">Ns</font></span>)</span></span><br/>

<a NAME="E2:37"/>
<font color="Maroon" title="c3">Cseq</font> <a href="lopban_3.html#K4" title="LOPBAN_3:func.4">*</a> <font color="Maroon" title="c4">Ns</font> is    <a href="bhsp_3.html#M1" title="BHSP_3:mode.1">subsequence</a> of <font color="Maroon" title="c3">Cseq</font>
 
<b>by </b><i><a class="ref" href="bhsp_3.html#D4" title="BHSP_3:def.4">BHSP_3:def 4</a></i>;<br/>
<a NAME="E3:37"/><b>then </b><i><font color="Green" title="E37">A2</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="lopban_3.html#K4" title="LOPBAN_3:func.4">*</a> <font color="Maroon" title="c4">Ns</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">Cseq</font>
 
<b>by </b><i><a class="ref" href="ncfcont1.html#T9" target="_self" title="NCFCONT1:th.9">Th9</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="ncfcont1.html#K7" title="NCFCONT1:func.7">*</a> <font color="Maroon" title="c3">Cseq</font></span>)</span> <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">*</a> <font color="Maroon" title="c4">Ns</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">Cseq</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">Ns</font>
<b>by </b><i><a class="txt" href="ncfcont1.html#E1:37"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="ncfcont1.html#D12" target="_self" title="NCFCONT1:def.12">Def12</a></i>
<br/>.= 
<font color="Maroon" title="c2">h</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="lopban_3.html#K4" title="LOPBAN_3:func.4">*</a> <font color="Maroon" title="c4">Ns</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>
<br/>.= 
<font color="Maroon" title="c2">h</font> <a href="ncfcont1.html#K7" title="NCFCONT1:func.7">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="lopban_3.html#K4" title="LOPBAN_3:func.4">*</a> <font color="Maroon" title="c4">Ns</font></span>)</span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E1:37"><i><font color="Green" title="E36">A1</font></i></a>, <a class="txt" href="ncfcont1.html#E3:37"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="ncfcont1.html#D12" target="_self" title="NCFCONT1:def.12">Def12</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</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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