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<b>let </b><font color="Maroon" title="c1">seq</font>, <font color="Maroon" title="c2">seq1</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">seq</font> is <a href="seqm_3.html#V4" title="SEQM_3:attr.4">non-increasing</a> &amp; <font color="Maroon" title="c2">seq1</font> is    <a href="seqm_3.html#M1" title="SEQM_3:mode.1">subsequence</a> of <font color="Maroon" title="c1">seq</font> implies <font color="Maroon" title="c2">seq1</font> is <a href="seqm_3.html#V4" title="SEQM_3:attr.4">non-increasing</a> )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:65"/><i><font color="Green" title="E43">A1</font></i>: 
<font color="Maroon" title="c1">seq</font> is <a href="seqm_3.html#V4" title="SEQM_3:attr.4">non-increasing</a>
 <b>and </b><br/><a NAME="E2:65"/><i><font color="Green" title="E44">A2</font></i>: 
<font color="Maroon" title="c2">seq1</font> is    <a href="seqm_3.html#M1" title="SEQM_3:mode.1">subsequence</a> of <font color="Maroon" title="c1">seq</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="seqm_3.html#V4" title="SEQM_3:attr.4">non-increasing</a></span><br/>

<b>consider </b><font color="Maroon" title="c3">Nseq</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="E4:65"/><i><font color="Green" title="E45">A3</font></i>: 
<font color="Maroon" title="c2">seq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">seq</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">*</a> <font color="Maroon" title="c3">Nseq</font>
 <b>by </b><i><a class="txt" href="seqm_3.html#E2:65"><i><font color="Green" title="E44">A2</font></i></a>, <a class="ref" href="seqm_3.html#D10" target="_self" title="SEQM_3:def.10">Def10</a></i>;<br/>
<b>let </b><font color="Maroon" title="c4">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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="seqm_3.html#D14" target="_self" title="SEQM_3:def.14">SEQM_3:def 14</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">seq1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/>

<a NAME="E5:65"/>
<font color="Maroon" title="c3">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="seqm_3.html#E10"><i><font color="Green" title="E10">Lm6</font></i></a></i>;<br/>
<a NAME="E6:65"/><b>then </b>
<font color="Maroon" title="c1">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="seqm_3.html#E1:65"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="seqm_3.html#T14" target="_self" title="SEQM_3:th.14">Th14</a></i>;<br/>
<a NAME="E7:65"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">*</a> <font color="Maroon" title="c3">Nseq</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seqm_3.html#T31" target="_self" title="SEQM_3:th.31">Th31</a></i>;<br/>
<b>hence </b><a NAME="E8:65"/>
<font color="Maroon" title="c2">seq1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">seq1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="txt" href="seqm_3.html#E4:65"><i><font color="Green" title="E45">A3</font></i></a>, <a class="ref" href="seqm_3.html#T31" target="_self" title="SEQM_3:th.31">Th31</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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