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<span class="kw">let </span><font color="Maroon" title="c1">seq</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">seq</font> is <a href="valued_0.html#V10" title="VALUED_0:attr.10">V57</a>() implies  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">R</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">k</font></span>)</span> where <font color="Olive" title="b3">k</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">k</font> </span> } </span>  holds <br/> <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">seq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:33"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
<font color="Maroon" title="c1">seq</font> is <a href="valued_0.html#V10" title="VALUED_0:attr.10">V57</a>()
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">R</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">k</font></span>)</span> where <font color="Olive" title="b3">k</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">k</font> </span> } </span>  holds <br/> <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">seq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">R</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Olive" title="b1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">k</font></span>)</span> where <font color="Olive" title="b2">k</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> </span> } </span>  holds <br/> <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <font color="Olive" title="b1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">seq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span><br/>

<div><span class="lab"><font color="Green" title="E30">A2</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">k</font></span>)</span> where <font color="Olive" title="b2">k</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> </span> } </span>  holds <br/><font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">seq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">r</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> </span> } </span>  implies <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">seq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">n</font> )</span><br/><span class="kw">assume </span><a NAME="E1:33_1"/>
<font color="Maroon" title="c3">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> </span> } </span> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">seq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">r1</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E3:33_1"/><span class="lab"><font color="Green" title="E31">A3</font></span>: 
( <font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r1</font> &amp; <font color="Maroon" title="c4">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> </span> } </span>  )
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c5">k1</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E5:33_1"/><span class="lab"><font color="Green" title="E32">A4</font></span>: 
<font color="Maroon" title="c4">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">seq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k1</font>
 <span class="kw">and </span><br/><a NAME="E6:33_1"/><span class="lab"><font color="Green" title="E33">A5</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33_1"><span class="lab"><font color="Green" title="E31">A3</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c6">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E8:33_1"/><span class="lab"><font color="Green" title="E34">A6</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:33_1"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="ref" href="nat_1.html#T10" onmouseover="rs('nat_1/T10')" onmouseout="rh()">NAT_1:10</a></span>;<br/><span class="kw">thus </span><a NAME="E9:33_1"/>
<font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">seq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E3:33_1"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="txt" href="#E5:33_1"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="txt" href="#E8:33_1"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="ref" href="seqm_3.html#T7" onmouseover="rs('seqm_3/T7')" onmouseout="rh()">SEQM_3:7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">let </span><font color="Maroon" title="c3">R</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> </span> } </span>  implies  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <font color="Maroon" title="c3">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">seq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">n</font> )</span><br/>

<div><span class="lab"><font color="Green" title="E31">A7</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">s</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s</font> holds <br/> ex <font color="Olive" title="b2">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">k</font></span>)</span> where <font color="Olive" title="b3">k</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">k</font> </span> } </span>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">r</font> )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">s</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">s</font> implies  ex <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">k</font></span>)</span> where <font color="Olive" title="b2">k</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> </span> } </span>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> ) )</span><br/><a NAME="E1:33_2"/><span class="lab"><font color="Green" title="E32">A8</font></span>: 
<font color="Maroon" title="c1">seq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> </span> } </span> 
 ;<br/><span class="kw">assume </span><a NAME="E2:33_2"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">s</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">k</font></span>)</span> where <font color="Olive" title="b2">k</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> </span> } </span>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> )</span><br/><span class="kw">hence </span><a NAME="E3:33_2"/>
 ex <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">k</font></span>)</span> where <font color="Olive" title="b2">k</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> </span> } </span>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33_2"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="ref" href="xreal_1.html#T44" onmouseover="rs('xreal_1/T44')" onmouseout="rh()">XREAL_1:44</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">assume </span><a NAME="E4:33"/><span class="lab"><font color="Green" title="E32">A9</font></span>: 
<font color="Maroon" title="c3">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> </span> } </span> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <font color="Maroon" title="c3">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">seq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span><br/>

<a NAME="E5:33"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A10</font></span>: 
<font color="Maroon" title="c3">R</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="setlim_1.html#T1" onmouseover="rs('setlim_1/T1')" onmouseout="rh()">SETLIM_1:1</a></span>;<br/>
<a NAME="E6:33"/>
<font color="Maroon" title="c3">R</font> is  <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E4:33"><span class="lab"><font color="Green" title="E32">A9</font></span></a>, <a class="ref" href="rinfsup1.html#T31" target="_self" onmouseover="rs('rinfsup1/T31')" onmouseout="rh()">Th31</a>, <a class="ref" href="limfunc1.html#T1" onmouseover="rs('limfunc1/T1')" onmouseout="rh()">LIMFUNC1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:33"/>
 <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <font color="Maroon" title="c3">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">seq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33"><span class="lab"><font color="Green" title="E32">A9</font></span></a>, <a class="txt" href="#E5:33"><span class="lab"><font color="Green" title="E33">A10</font></span></a>, <a class="txt" href="#E2:33"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E3:33"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="ref" href="seq_4.html#D1" onmouseover="rs('seq_4/D1')" onmouseout="rh()">SEQ_4:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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