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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</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>, <font color="Olive" title="b3">seq3</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="bhsp_3.html#M1" title="BHSP_3:mode.1">subsequence</a> of <font color="Olive" title="b2">seq2</font> &amp; <font color="Olive" title="b2">seq2</font> is    <a href="bhsp_3.html#M1" title="BHSP_3:mode.1">subsequence</a> of <font color="Olive" title="b3">seq3</font> holds <br/><font color="Olive" title="b1">seq1</font> is    <a href="bhsp_3.html#M1" title="BHSP_3:mode.1">subsequence</a> of <font color="Olive" title="b3">seq3</font></span><br/><b>let </b><font color="Maroon" title="c2">seq1</font>, <font color="Maroon" title="c3">seq2</font>, <font color="Maroon" title="c4">seq3</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="bhsp_3.html#M1" title="BHSP_3:mode.1">subsequence</a> of <font color="Maroon" title="c3">seq2</font> &amp; <font color="Maroon" title="c3">seq2</font> is    <a href="bhsp_3.html#M1" title="BHSP_3:mode.1">subsequence</a> of <font color="Maroon" title="c4">seq3</font> implies <font color="Maroon" title="c2">seq1</font> is    <a href="bhsp_3.html#M1" title="BHSP_3:mode.1">subsequence</a> of <font color="Maroon" title="c4">seq3</font> )</span><br/>







<b>assume </b><b>that </b><br/><a NAME="E1:35"/><i><font color="Green" title="E14">A1</font></i>: 
<font color="Maroon" title="c2">seq1</font> is    <a href="bhsp_3.html#M1" title="BHSP_3:mode.1">subsequence</a> of <font color="Maroon" title="c3">seq2</font>
 <b>and </b><br/><a NAME="E2:35"/><i><font color="Green" title="E15">A2</font></i>: 
<font color="Maroon" title="c3">seq2</font> is    <a href="bhsp_3.html#M1" title="BHSP_3:mode.1">subsequence</a> of <font color="Maroon" title="c4">seq3</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="bhsp_3.html#M1" title="BHSP_3:mode.1">subsequence</a> of <font color="Maroon" title="c4">seq3</font></span><br/>

<b>consider </b><font color="Maroon" title="c5">Nseq1</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><b> such that </b><br/><a NAME="E4:35"/><i><font color="Green" title="E16">A3</font></i>: 
<font color="Maroon" title="c2">seq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">seq2</font> <a href="bhsp_3.html#K1" title="BHSP_3:func.1">*</a> <font color="Maroon" title="c5">Nseq1</font>
 <b>by </b><i><a class="txt" href="bhsp_3.html#E1:35"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="bhsp_3.html#D4" target="_self" title="BHSP_3:def.4">Def4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">Nseq2</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><b> such that </b><br/><a NAME="E6:35"/><i><font color="Green" title="E17">A4</font></i>: 
<font color="Maroon" title="c3">seq2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">seq3</font> <a href="bhsp_3.html#K1" title="BHSP_3:func.1">*</a> <font color="Maroon" title="c6">Nseq2</font>
 <b>by </b><i><a class="txt" href="bhsp_3.html#E2:35"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="bhsp_3.html#D4" target="_self" title="BHSP_3:def.4">Def4</a></i>;<br/>
<b>take </b><font color="Maroon" title="c7">Nseq</font> = <font color="Maroon" title="c6">Nseq2</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c5">Nseq1</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="bhsp_3.html#D4" target="_self" title="BHSP_3:def.4">BHSP_3:def 4</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">seq3</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">Nseq</font></span><br/>

<b>thus </b><a NAME="E7:35"/>
<font color="Maroon" title="c2">seq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">seq3</font> <a href="bhsp_3.html#K1" title="BHSP_3:func.1">*</a> <font color="Maroon" title="c7">Nseq</font>
 <b>by </b><i><a class="txt" href="bhsp_3.html#E4:35"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="bhsp_3.html#E6:35"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</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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