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<span class="kw">let </span><font color="Maroon" title="c1">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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">rseq</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st <font color="Maroon" title="c1">seq</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b1">rseq</font> holds <br/>( <font color="Maroon" title="c1">seq</font> is  <a href="rinfsup2.html#V2" title="RINFSUP2:attr.2">bounded_above</a>  iff <font color="Olive" title="b1">rseq</font> is  <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">rseq</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">seq</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c2">rseq</font> implies ( <font color="Maroon" title="c1">seq</font> is  <a href="rinfsup2.html#V2" title="RINFSUP2:attr.2">bounded_above</a>  iff <font color="Maroon" title="c2">rseq</font> is  <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a>  ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:19"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
<font color="Maroon" title="c1">seq</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c2">rseq</font>
 ; <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">seq</font> is  <a href="rinfsup2.html#V2" title="RINFSUP2:attr.2">bounded_above</a>  iff <font color="Maroon" title="c2">rseq</font> is  <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a>  )</span><br/>

<div><span class="kw">hereby </span> <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="c2">rseq</font> is  <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a>  implies <font color="Maroon" title="c1">seq</font> is  <a href="rinfsup2.html#V2" title="RINFSUP2:attr.2">bounded_above</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:19_1"/>
<font color="Maroon" title="c1">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="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">rseq</font> is  <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a> </span><br/><a NAME="E2:19_1"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">rseq</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="#E1:19"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <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">p</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="E4:19_1"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
for <font color="Olive" title="b1">y</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>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">rseq</font> holds <br/><font color="Olive" title="b1">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#D1" title="SEQ_4:def.1">SEQ_4:def 1</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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> ; <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="c2">rseq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">p</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span><br/><a NAME="E1:19_1_1"/>
<font color="Maroon" title="c2">rseq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">rseq</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></span>;<br/><a NAME="E2:19_1_1"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">rseq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c3">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:19_1"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><span class="kw">hence </span><a NAME="E3:19_1_1"/>
<font color="Maroon" title="c2">rseq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">p</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
 ; <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="E6:19_1"/>
<font color="Maroon" title="c2">rseq</font> is  <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_2.html#D3" title="SEQ_2:def.3">SEQ_2:def 3</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">assume </span><a NAME="E3:19"/>
<font color="Maroon" title="c2">rseq</font> is  <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</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">seq</font> is  <a href="rinfsup2.html#V2" title="RINFSUP2:attr.2">bounded_above</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">q</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="E5:19"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
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 <font color="Maroon" title="c2">rseq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_2.html#D3" title="SEQ_2:def.3">SEQ_2:def 3</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><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="Red">::  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#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">seq</font> 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">q</font> )</span><br/><span class="kw">assume </span><a NAME="E1:19_2"/>
<font color="Maroon" title="c4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">seq</font>
 ; <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="c4">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">q</font></span><br/><a NAME="E2:19_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</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> <font color="Maroon" title="c2">rseq</font> &amp; <font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">rseq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E3:19_2"/>
<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">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:19"><span class="lab"><font color="Green" title="E18">A3</font></span></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>
<a NAME="E7:19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A4</font></span>: 
<font color="Maroon" title="c3">q</font> is    <a href="xxreal_2.html#M1" title="XXREAL_2:mode.1">UpperBound</a> of  <a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">seq</font>
 
<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="E8:19"/>
<font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:19"/>
 <a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">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="txt" href="#E7:19"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="supinf_1.html#D11" title="SUPINF_1:def.11">SUPINF_1:def 11</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="rinfsup2.html#D4" target="_self" title="RINFSUP2:def.4">RINFSUP2:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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