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

<span class="kw">let </span><font color="Maroon" title="c1">seq1</font>, <font color="Maroon" title="c2">seq2</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">seq1</font> is  <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a>  &amp; <font color="Maroon" title="c1">seq1</font> is  <a href="rinfsup1.html#NV1" title="RINFSUP1:NV.1">nonnegative</a>  &amp; <font color="Maroon" title="c2">seq2</font> is  <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a>  &amp; <font color="Maroon" title="c2">seq2</font> is  <a href="rinfsup1.html#NV1" title="RINFSUP1:NV.1">nonnegative</a>  implies ( <font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c2">seq2</font> is  <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a>  &amp;  <a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <font color="Maroon" title="c1">seq1</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <font color="Maroon" title="c2">seq2</font></span>)</span> ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:26"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
( <font color="Maroon" title="c1">seq1</font> is  <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a>  &amp; <font color="Maroon" title="c1">seq1</font> is  <a href="rinfsup1.html#NV1" title="RINFSUP1:NV.1">nonnegative</a>  )
 <span class="kw">and </span><br/><a NAME="E2:26"/><span class="lab"><font color="Green" title="E22">A2</font></span>: 
( <font color="Maroon" title="c2">seq2</font> is  <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a>  &amp; <font color="Maroon" title="c2">seq2</font> is  <a href="rinfsup1.html#NV1" title="RINFSUP1:NV.1">nonnegative</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c2">seq2</font> is  <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a>  &amp;  <a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <font color="Maroon" title="c1">seq1</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <font color="Maroon" title="c2">seq2</font></span>)</span> )</span><br/>

<a NAME="E3:26"/>
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">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <font color="Maroon" title="c1">seq1</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <font color="Maroon" title="c2">seq2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</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"><a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <font color="Maroon" title="c1">seq1</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <font color="Maroon" title="c2">seq2</font></span>)</span></span><br/>

<a NAME="E1:26_1"/><span class="lab"><font color="Green" title="E23">A3</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span>)</span> &amp; <font color="Maroon" title="c1">seq1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <font color="Maroon" title="c1">seq1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="ref" href="rinfsup1.html#T7" target="_self" title="RINFSUP1:th.7">Th7</a>, <a class="ref" href="seq_1.html#T12" title="SEQ_1:th.12">SEQ_1:12</a></span>;<br/>
<a NAME="E2:26_1"/><span class="lab"><font color="Green" title="E24">A4</font></span>: 
<font color="Maroon" title="c2">seq2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <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="txt" href="#E2:26"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="ref" href="rinfsup1.html#D3" target="_self" title="RINFSUP1:def.3">Def3</a></span>;<br/>
<a NAME="E3:26_1"/>
( <font color="Maroon" title="c2">seq2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <font color="Maroon" title="c2">seq2</font> &amp; <font color="Maroon" title="c1">seq1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <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="txt" href="#E1:26"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="ref" href="rinfsup1.html#D3" target="_self" title="RINFSUP1:def.3">Def3</a>, <a class="ref" href="rinfsup1.html#T7" target="_self" title="RINFSUP1:th.7">Th7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:26_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</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"><a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <font color="Maroon" title="c1">seq1</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <font color="Maroon" title="c2">seq2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="txt" href="#E2:26_1"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="ref" href="xreal_1.html#T68" title="XREAL_1:th.68">XREAL_1:68</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:26"/>
( <font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c2">seq2</font> is  <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a>  &amp;  <a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <font color="Maroon" title="c1">seq1</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <font color="Maroon" title="c2">seq2</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rinfsup1.html#T9" target="_self" title="RINFSUP1:th.9">Th9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
