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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">k</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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">seq</font> being   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a> holds <br/> ( ( <font color="Olive" title="b1">seq</font> is  <a href="rinfsup2.html#V2" title="RINFSUP2:attr.2">bounded_above</a>  implies <font color="Olive" title="b1">seq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c1">k</font> is  <a href="rinfsup2.html#V2" title="RINFSUP2:attr.2">bounded_above</a>  ) &amp; ( <font color="Olive" title="b1">seq</font> is  <a href="rinfsup2.html#V1" title="RINFSUP2:attr.1">bounded_below</a>  implies <font color="Olive" title="b1">seq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c1">k</font> is  <a href="rinfsup2.html#V1" title="RINFSUP2:attr.1">bounded_below</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">seq</font> be   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c2">seq</font> is  <a href="rinfsup2.html#V2" title="RINFSUP2:attr.2">bounded_above</a>  implies <font color="Maroon" title="c2">seq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c1">k</font> is  <a href="rinfsup2.html#V2" title="RINFSUP2:attr.2">bounded_above</a>  ) &amp; ( <font color="Maroon" title="c2">seq</font> is  <a href="rinfsup2.html#V1" title="RINFSUP2:attr.1">bounded_below</a>  implies <font color="Maroon" title="c2">seq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c1">k</font> is  <a href="rinfsup2.html#V1" title="RINFSUP2:attr.1">bounded_below</a>  ) )</span><br/>



<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">seq</font> is  <a href="rinfsup2.html#V1" title="RINFSUP2:attr.1">bounded_below</a>  implies <font color="Maroon" title="c2">seq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c1">k</font> is  <a href="rinfsup2.html#V1" title="RINFSUP2:attr.1">bounded_below</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:29_1"/>
<font color="Maroon" title="c2">seq</font> is  <a href="rinfsup2.html#V2" title="RINFSUP2:attr.2">bounded_above</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">seq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c1">k</font> is  <a href="rinfsup2.html#V2" title="RINFSUP2:attr.2">bounded_above</a> </span><br/><a NAME="E2:29_1"/><span class="kw">then </span>
 <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c2">seq</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="ref" href="rinfsup2.html#D4" target="_self" title="RINFSUP2:def.4">Def4</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">UB</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E4:29_1"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
<font color="Maroon" title="c3">UB</font> is    <a href="xxreal_2.html#M1" title="XXREAL_2:mode.1">UpperBound</a> of  <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c2">seq</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D10" title="XXREAL_2:def.10">XXREAL_2:def 10</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">r</font> be   <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c1">k</font></span>)</span> implies <font color="Maroon" title="c4">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">UB</font> )</span><br/><span class="kw">assume </span><a NAME="E1:29_1_1"/>
<font color="Maroon" title="c4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c1">k</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">UB</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:29_1_1"/><span class="lab"><font color="Green" title="E26">A2</font></span>: 
<font color="Maroon" title="c5">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c1">k</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:29_1_1"/><span class="lab"><font color="Green" title="E27">A3</font></span>: 
<font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c6">n</font> = <font color="Maroon" title="c5">n</font> as    <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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_1_1"><span class="lab"><font color="Green" title="E26">A2</font></span></a></span>;<br/><a NAME="E6:29_1_1"/>
<font color="Maroon" title="c2">seq</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">UB</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_1"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a>, <a class="ref" href="xxreal_2.html#D1" title="XXREAL_2:def.1">XXREAL_2:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E7:29_1_1"/>
<font color="Maroon" title="c4">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">UB</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_1_1"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></span>; <a class="txt" 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><a NAME="E6:29_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">UB</font> is    <a href="xxreal_2.html#M1" title="XXREAL_2:mode.1">UpperBound</a> of  <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c1">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D1" title="XXREAL_2:def.1">XXREAL_2:def 1</a></span>;<br/><a NAME="E7:29_1"/><span class="kw">then </span>
 <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c1">k</font></span>)</span> 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="ref" href="xxreal_2.html#D10" title="XXREAL_2:def.10">XXREAL_2:def 10</a></span>;<br/><span class="kw">hence </span><a NAME="E8:29_1"/>
<font color="Maroon" title="c2">seq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c1">k</font> is  <a href="rinfsup2.html#V2" title="RINFSUP2:attr.2">bounded_above</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rinfsup2.html#D4" target="_self" title="RINFSUP2:def.4">Def4</a></span>; <a class="txt" 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>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:29_2"/>
<font color="Maroon" title="c2">seq</font> is  <a href="rinfsup2.html#V1" title="RINFSUP2:attr.1">bounded_below</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">seq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c1">k</font> is  <a href="rinfsup2.html#V1" title="RINFSUP2:attr.1">bounded_below</a> </span><br/><a NAME="E2:29_2"/><span class="kw">then </span>
 <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c2">seq</font> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rinfsup2.html#D3" target="_self" title="RINFSUP2:def.3">Def3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">UB</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E4:29_2"/><span class="lab"><font color="Green" title="E25">A4</font></span>: 
<font color="Maroon" title="c3">UB</font> is    <a href="xxreal_2.html#M2" title="XXREAL_2:mode.2">LowerBound</a> of  <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c2">seq</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D9" title="XXREAL_2:def.9">XXREAL_2:def 9</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">r</font> be   <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c1">k</font></span>)</span> implies <font color="Maroon" title="c3">UB</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">r</font> )</span><br/><span class="kw">assume </span><a NAME="E1:29_2_1"/>
<font color="Maroon" title="c4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c1">k</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">UB</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">r</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:29_2_1"/><span class="lab"><font color="Green" title="E26">A5</font></span>: 
<font color="Maroon" title="c5">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c1">k</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:29_2_1"/><span class="lab"><font color="Green" title="E27">A6</font></span>: 
<font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c6">n</font> = <font color="Maroon" title="c5">n</font> as    <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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_2_1"><span class="lab"><font color="Green" title="E26">A5</font></span></a></span>;<br/><a NAME="E6:29_2_1"/>
<font color="Maroon" title="c2">seq</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c2">seq</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></span>;<br/><a NAME="E7:29_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">UB</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">seq</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_2"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="ref" href="xxreal_2.html#D2" title="XXREAL_2:def.2">XXREAL_2:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E8:29_2_1"/>
<font color="Maroon" title="c3">UB</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_2_1"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></span>; <a class="txt" 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><a NAME="E6:29_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">UB</font> is    <a href="xxreal_2.html#M2" title="XXREAL_2:mode.2">LowerBound</a> of  <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c1">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D2" title="XXREAL_2:def.2">XXREAL_2:def 2</a></span>;<br/><a NAME="E7:29_2"/><span class="kw">then </span>
 <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c1">k</font></span>)</span> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D9" title="XXREAL_2:def.9">XXREAL_2:def 9</a></span>;<br/><span class="kw">hence </span><a NAME="E8:29_2"/>
<font color="Maroon" title="c2">seq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c1">k</font> is  <a href="rinfsup2.html#V1" title="RINFSUP2:attr.1">bounded_below</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rinfsup2.html#D3" target="_self" title="RINFSUP2:def.3">Def3</a></span>; <a class="txt" 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>


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