<?xml version="1.0"?>
<div class="add">

<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="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a> &amp; <font color="Maroon" title="c2">seq1</font> is <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a> implies <font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">seq1</font> is <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a> ) &amp; ( <font color="Maroon" title="c1">seq</font> is <a href="valued_0.html#V8" title="VALUED_0:attr.8">decreasing</a> &amp; <font color="Maroon" title="c2">seq1</font> is <a href="valued_0.html#V8" title="VALUED_0:attr.8">decreasing</a> implies <font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">seq1</font> is <a href="valued_0.html#V8" title="VALUED_0:attr.8">decreasing</a> ) &amp; ( <font color="Maroon" title="c1">seq</font> is <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a> &amp; <font color="Maroon" title="c2">seq1</font> is <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a> implies <font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">seq1</font> is <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a> ) &amp; ( <font color="Maroon" title="c1">seq</font> is <a href="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</a> &amp; <font color="Maroon" title="c2">seq1</font> is <a href="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</a> implies <font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">seq1</font> is <a href="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</a> ) )</span><br/>



<b>thus </b><a NAME="E1:50"/>
( <font color="Maroon" title="c1">seq</font> is <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a> &amp; <font color="Maroon" title="c2">seq1</font> is <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a> implies <font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">seq1</font> is <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</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="valued_0.html#V8" title="VALUED_0:attr.8">decreasing</a> &amp; <font color="Maroon" title="c2">seq1</font> is <a href="valued_0.html#V8" title="VALUED_0:attr.8">decreasing</a> implies <font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">seq1</font> is <a href="valued_0.html#V8" title="VALUED_0:attr.8">decreasing</a> ) &amp; ( <font color="Maroon" title="c1">seq</font> is <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a> &amp; <font color="Maroon" title="c2">seq1</font> is <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a> implies <font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">seq1</font> is <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a> ) &amp; ( <font color="Maroon" title="c1">seq</font> is <a href="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</a> &amp; <font color="Maroon" title="c2">seq1</font> is <a href="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</a> implies <font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">seq1</font> is <a href="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</a> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:50_1"/><i><font color="Green" title="E36">A1</font></i>: 
<font color="Maroon" title="c1">seq</font> is <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a>
 <b>and </b><br/><a NAME="E2:50_1"/><i><font color="Green" title="E37">A2</font></i>: 
<font color="Maroon" title="c2">seq1</font> is <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</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> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">seq1</font> is <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a></span><br/>

<b>let </b><font color="Maroon" title="c3">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#D11" target="_self" title="SEQM_3:def.11">SEQM_3:def 11</a> <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="c1">seq</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">seq1</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="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/>

<a NAME="E3:50_1"/><i><font color="Green" title="E38">A3</font></i>: 
<font color="Maroon" title="c1">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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">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="#E1:50_1"><i><font color="Green" title="E36">A1</font></i></a>, <a class="txt" href="seqm_3.html#E10"><i><font color="Green" title="E10">Lm6</font></i></a></i>;<br/>
<a NAME="E4:50_1"/>
<font color="Maroon" title="c2">seq1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c3">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="#E2:50_1"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="seqm_3.html#E10"><i><font color="Green" title="E10">Lm6</font></i></a></i>;<br/>
<a NAME="E5:50_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:50_1"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<a NAME="E6:50_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T11" title="SEQ_1:th.11">SEQ_1:11</a></i>;<br/>
<b>hence </b><a NAME="E7:50_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">seq1</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="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#T11" title="SEQ_1:th.11">SEQ_1:11</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>thus </b><a NAME="E2:50"/>
( <font color="Maroon" title="c1">seq</font> is <a href="valued_0.html#V8" title="VALUED_0:attr.8">decreasing</a> &amp; <font color="Maroon" title="c2">seq1</font> is <a href="valued_0.html#V8" title="VALUED_0:attr.8">decreasing</a> implies <font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">seq1</font> is <a href="valued_0.html#V8" title="VALUED_0:attr.8">decreasing</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="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a> &amp; <font color="Maroon" title="c2">seq1</font> is <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a> implies <font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">seq1</font> is <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a> ) &amp; ( <font color="Maroon" title="c1">seq</font> is <a href="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</a> &amp; <font color="Maroon" title="c2">seq1</font> is <a href="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</a> implies <font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">seq1</font> is <a href="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</a> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:50_2"/><i><font color="Green" title="E36">A4</font></i>: 
<font color="Maroon" title="c1">seq</font> is <a href="valued_0.html#V8" title="VALUED_0:attr.8">decreasing</a>
 <b>and </b><br/><a NAME="E2:50_2"/><i><font color="Green" title="E37">A5</font></i>: 
<font color="Maroon" title="c2">seq1</font> is <a href="valued_0.html#V8" title="VALUED_0:attr.8">decreasing</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> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">seq1</font> is <a href="valued_0.html#V8" title="VALUED_0:attr.8">decreasing</a></span><br/>

<b>let </b><font color="Maroon" title="c3">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#D12" target="_self" title="SEQM_3:def.12">SEQM_3:def 12</a> <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="c1">seq</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">seq1</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="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/>

<a NAME="E3:50_2"/><i><font color="Green" title="E38">A6</font></i>: 
<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">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="txt" href="#E1:50_2"><i><font color="Green" title="E36">A4</font></i></a>, <a class="txt" href="seqm_3.html#E11"><i><font color="Green" title="E11">Lm7</font></i></a></i>;<br/>
<a NAME="E4:50_2"/>
<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="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="c3">n</font>
 
<b>by </b><i><a class="txt" href="#E2:50_2"><i><font color="Green" title="E37">A5</font></i></a>, <a class="txt" href="seqm_3.html#E11"><i><font color="Green" title="E11">Lm7</font></i></a></i>;<br/>
<a NAME="E5:50_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:50_2"><i><font color="Green" title="E38">A6</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<a NAME="E6:50_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">seq1</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="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T11" title="SEQ_1:th.11">SEQ_1:11</a></i>;<br/>
<b>hence </b><a NAME="E7:50_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">seq1</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="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="ref" href="seq_1.html#T11" title="SEQ_1:th.11">SEQ_1:11</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>thus </b><a NAME="E3:50"/>
( <font color="Maroon" title="c1">seq</font> is <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a> &amp; <font color="Maroon" title="c2">seq1</font> is <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a> implies <font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">seq1</font> is <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</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="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</a> &amp; <font color="Maroon" title="c2">seq1</font> is <a href="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</a> implies <font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">seq1</font> is <a href="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:50_3"/><i><font color="Green" title="E36">A7</font></i>: 
<font color="Maroon" title="c1">seq</font> is <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a>
 <b>and </b><br/><a NAME="E2:50_3"/><i><font color="Green" title="E37">A8</font></i>: 
<font color="Maroon" title="c2">seq1</font> is <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</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> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">seq1</font> is <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a></span><br/>

<b>let </b><font color="Maroon" title="c3">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#D13" target="_self" title="SEQM_3:def.13">SEQM_3:def 13</a> <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="c1">seq</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">seq1</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="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/>

<a NAME="E3:50_3"/><i><font color="Green" title="E38">A9</font></i>: 
<font color="Maroon" title="c1">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font> <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">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="#E1:50_3"><i><font color="Green" title="E36">A7</font></i></a>, <a class="txt" href="seqm_3.html#E12"><i><font color="Green" title="E12">Lm8</font></i></a></i>;<br/>
<a NAME="E4:50_3"/>
<font color="Maroon" title="c2">seq1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">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="#E2:50_3"><i><font color="Green" title="E37">A8</font></i></a>, <a class="txt" href="seqm_3.html#E12"><i><font color="Green" title="E12">Lm8</font></i></a></i>;<br/>
<a NAME="E5:50_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:50_3"><i><font color="Green" title="E38">A9</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E6:50_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T11" title="SEQ_1:th.11">SEQ_1:11</a></i>;<br/>
<b>hence </b><a NAME="E7:50_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">seq1</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="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#T11" title="SEQ_1:th.11">SEQ_1:11</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>assume </b><b>that </b><br/><a NAME="E4:50"/><i><font color="Green" title="E36">A10</font></i>: 
<font color="Maroon" title="c1">seq</font> is <a href="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</a>
 <b>and </b><br/><a NAME="E5:50"/><i><font color="Green" title="E37">A11</font></i>: 
<font color="Maroon" title="c2">seq1</font> is <a href="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</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> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">seq1</font> is <a href="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</a></span><br/>

<b>let </b><font color="Maroon" title="c3">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"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">seq1</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="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/>

<a NAME="E6:50"/><i><font color="Green" title="E38">A12</font></i>: 
<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">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> <font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="txt" href="#E4:50"><i><font color="Green" title="E36">A10</font></i></a>, <a class="txt" href="seqm_3.html#E13"><i><font color="Green" title="E13">Lm9</font></i></a></i>;<br/>
<a NAME="E7:50"/>
<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="c3">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="c3">n</font>
 
<b>by </b><i><a class="txt" href="#E5:50"><i><font color="Green" title="E37">A11</font></i></a>, <a class="txt" href="seqm_3.html#E13"><i><font color="Green" title="E13">Lm9</font></i></a></i>;<br/>
<a NAME="E8:50"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E6:50"><i><font color="Green" title="E38">A12</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E9:50"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">seq1</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="c3">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T11" title="SEQ_1:th.11">SEQ_1:11</a></i>;<br/>
<b>hence </b><a NAME="E10:50"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">seq1</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="c3">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="ref" href="seq_1.html#T11" title="SEQ_1:th.11">SEQ_1:11</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>
