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<span class="kw">let </span><font color="Maroon" title="c1">seq1</font>, <font color="Maroon" title="c2">seq2</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"> ( <font color="Maroon" title="c1">seq1</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>  &amp; <font color="Maroon" title="c2">seq2</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>  &amp; ( 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="c1">seq1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">seq2</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> ) implies  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c1">seq1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c2">seq2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:47"/><span class="lab"><font color="Green" title="E42">A1</font></span>: 
<font color="Maroon" title="c1">seq1</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> 
 <span class="kw">and </span><br/><a NAME="E2:47"/><span class="lab"><font color="Green" title="E43">A2</font></span>: 
<font color="Maroon" title="c2">seq2</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> 
 <span class="kw">and </span><br/><a NAME="E3:47"/><span class="lab"><font color="Green" title="E44">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="c1">seq1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">seq2</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c1">seq1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c2">seq2</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:47_1"/>
( <font color="Maroon" title="c1">seq1</font> is  <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</a>  or <font color="Maroon" title="c1">seq1</font> is  <a href="mesfunc5.html#V8" title="MESFUNC5:attr.8">convergent_to_+infty</a>  or <font color="Maroon" title="c1">seq1</font> is  <a href="mesfunc5.html#V9" title="MESFUNC5:attr.9">convergent_to_-infty</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="ref" href="mesfunc5.html#D11" title="MESFUNC5:def.11">MESFUNC5:def 11</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1"><span class="kw">suppose </span></a><a NAME="E1:47_1_1"/><span class="lab"><font color="Green" title="E45">A4</font></span>: 
<font color="Maroon" title="c1">seq1</font> is  <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c1">seq1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c2">seq2</font></span><br/><div class="add"><a NAME="E2:47_1_1"/><span class="lab"><font color="Green" title="E46">A5</font></span>: 
not <font color="Maroon" title="c2">seq2</font> is  <a href="mesfunc5.html#V9" title="MESFUNC5:attr.9">convergent_to_-infty</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:47_1_1_1"/><span class="lab"><font color="Green" title="E47">A6</font></span>: 
<font color="Maroon" title="c2">seq2</font> is  <a href="mesfunc5.html#V9" title="MESFUNC5:attr.9">convergent_to_-infty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">g</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">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="c3">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Maroon" title="c1">seq1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">g</font> )</span><br/><span class="kw">assume </span><a NAME="E1:47_1_1_1_1"/>
<font color="Maroon" title="c3">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Maroon" title="c1">seq1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">g</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:47_1_1_1_1"/><span class="lab"><font color="Green" title="E48">A7</font></span>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c2">seq2</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_1_1"><span class="lab"><font color="Green" title="E47">A6</font></span></a>, <a class="ref" href="mesfunc5.html#D10" title="MESFUNC5:def.10">MESFUNC5:def 10</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">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="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">m</font> implies <font color="Maroon" title="c1">seq1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">g</font> )</span><br/><a NAME="E1:47_1_1_1_1_1"/>
<font color="Maroon" title="c5">m</font> is    <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="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><a NAME="E2:47_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A8</font></span>: 
<font color="Maroon" title="c1">seq1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">seq2</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47"><span class="lab"><font color="Green" title="E44">A3</font></span></a></span>;<br/><span class="kw">assume </span><a NAME="E3:47_1_1_1_1_1"/>
<font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">m</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">seq1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">g</font></span><br/><a NAME="E4:47_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">seq2</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47_1_1_1_1"><span class="lab"><font color="Green" title="E48">A7</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:47_1_1_1_1_1"/>
<font color="Maroon" title="c1">seq1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47_1_1_1_1_1"><span class="lab"><font color="Green" title="E49">A8</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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="E5:47_1_1_1_1"/>
 ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Maroon" title="c1">seq1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">g</font>
 ; <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="E3:47_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">seq1</font> is  <a href="mesfunc5.html#V9" title="MESFUNC5:attr.9">convergent_to_-infty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#D10" title="MESFUNC5:def.10">MESFUNC5:def 10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:47_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_1"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="ref" href="mesfunc5.html#T57" title="MESFUNC5:th.57">MESFUNC5:57</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_2"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:47_1_1_2"/>
( <font color="Maroon" title="c2">seq2</font> is  <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</a>  or <font color="Maroon" title="c2">seq2</font> is  <a href="mesfunc5.html#V8" title="MESFUNC5:attr.8">convergent_to_+infty</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E2:47_1_1"><span class="lab"><font color="Green" title="E46">A5</font></span></a>, <a class="ref" href="mesfunc5.html#D11" title="MESFUNC5:def.11">MESFUNC5:def 11</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_2_1"><span class="kw">suppose </span></a><a NAME="E1:47_1_1_2_1"/><span class="lab"><font color="Green" title="E47">A9</font></span>: 
<font color="Maroon" title="c2">seq2</font> is  <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c1">seq1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c2">seq2</font></span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c3">k1</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> <span class="kw"> such that </span><br/><a NAME="E3:47_1_1_2_1"/><span class="lab"><font color="Green" title="E48">A10</font></span>: 
<font color="Maroon" title="c1">seq1</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c3">k1</font> is  <a href="rinfsup2.html#V3" title="RINFSUP2:attr.3">bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_1"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="ref" href="rinfsup2.html#T19" target="_self" title="RINFSUP2:th.19">Th19</a></span>;<br/><a NAME="E4:47_1_1_2_1"/>
<font color="Maroon" title="c1">seq1</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c3">k1</font> is  <a href="rinfsup2.html#V2" title="RINFSUP2:attr.2">bounded_above</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47_1_1_2_1"><span class="lab"><font color="Green" title="E48">A10</font></span></a>, <a class="ref" href="rinfsup2.html#D5" target="_self" title="RINFSUP2:def.5">Def5</a></span>;<br/><a NAME="E5:47_1_1_2_1"/><span class="kw">then </span>
 <a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c3">k1</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="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="c4">UB</font> being    <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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c3">k1</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E7:47_1_1_2_1"/><span class="lab"><font color="Green" title="E49">A11</font></span>: 
<font color="Maroon" title="c4">UB</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="supinf_1.html#D11" title="SUPINF_1:def.11">SUPINF_1:def 11</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c5">k2</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> <span class="kw"> such that </span><br/><a NAME="E9:47_1_1_2_1"/><span class="lab"><font color="Green" title="E50">A12</font></span>: 
<font color="Maroon" title="c2">seq2</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">k2</font> is  <a href="rinfsup2.html#V3" title="RINFSUP2:attr.3">bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_1_2_1"><span class="lab"><font color="Green" title="E47">A9</font></span></a>, <a class="ref" href="rinfsup2.html#T19" target="_self" title="RINFSUP2:th.19">Th19</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c6">k</font> =  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c3">k1</font>,<font color="Maroon" title="c5">k2</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>  ;<br/><a NAME="E11:47_1_1_2_1"/><span class="lab"><font color="Green" title="E51">A13</font></span>: 
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c2">seq2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_1_2_1"><span class="lab"><font color="Green" title="E47">A9</font></span></a>, <a class="ref" href="rinfsup2.html#T20" target="_self" title="RINFSUP2:th.20">Th20</a></span>;<br/><a NAME="E12:47_1_1_2_1"/><span class="lab"><font color="Green" title="E52">A14</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c3">k1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_2_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c7">k2</font> = <font color="Maroon" title="c6">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">k1</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="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a>, <a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c8">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c8">y</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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span> implies <font color="Maroon" title="c8">y</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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c3">k1</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E2:47_1_1_2_1_1"/>
<font color="Maroon" title="c8">y</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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</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="c8">y</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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c3">k1</font></span>)</span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:47_1_1_2_1_1"/><span class="lab"><font color="Green" title="E53">A15</font></span>: 
<font color="Maroon" title="c9">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="c1">seq1</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:47_1_1_2_1_1"/><span class="lab"><font color="Green" title="E54">A16</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">n</font> = <font color="Maroon" title="c9">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="#E4:47_1_1_2_1_1"><span class="lab"><font color="Green" title="E53">A15</font></span></a></span>;<br/><a NAME="E7:47_1_1_2_1_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">seq1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:47_1_1_2_1_1"><span class="lab"><font color="Green" title="E54">A16</font></span></a>, <a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></span>;<br/><a NAME="E8:47_1_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">seq1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span>
 ;<br/><a NAME="E9:47_1_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">y</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="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c3">k1</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E10:47_1_1_2_1_1"/>
<font color="Maroon" title="c8">y</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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c3">k1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:47_1_1_2_1"><span class="lab"><font color="Green" title="E52">A14</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</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="E14:47_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A17</font></span>: 
 <a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c3">k1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><a NAME="E15:47_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">UB</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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#T6" title="XXREAL_2:th.6">XXREAL_2:6</a></span>;<br/><a NAME="E16:47_1_1_2_1"/><span class="kw">then </span>
 <a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">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="txt" href="#E7:47_1_1_2_1"><span class="lab"><font color="Green" title="E49">A11</font></span></a>, <a class="ref" href="supinf_1.html#D11" title="SUPINF_1:def.11">SUPINF_1:def 11</a></span>;<br/><a NAME="E17:47_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A18</font></span>: 
<font color="Maroon" title="c1">seq1</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">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>;<br/><a NAME="E18:47_1_1_2_1"/>
<font color="Maroon" title="c1">seq1</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c3">k1</font> is  <a href="rinfsup2.html#V1" title="RINFSUP2:attr.1">bounded_below</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47_1_1_2_1"><span class="lab"><font color="Green" title="E48">A10</font></span></a>, <a class="ref" href="rinfsup2.html#D5" target="_self" title="RINFSUP2:def.5">Def5</a></span>;<br/><a NAME="E19:47_1_1_2_1"/><span class="kw">then </span>
 <a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c3">k1</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="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="c7">LB</font> being    <a href="xxreal_2.html#M2" title="XXREAL_2:mode.2">LowerBound</a> of  <a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c3">k1</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E21:47_1_1_2_1"/><span class="lab"><font color="Green" title="E55">A19</font></span>: 
<font color="Maroon" title="c7">LB</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="supinf_1.html#D12" title="SUPINF_1:def.12">SUPINF_1:def 12</a></span>;<br/><a NAME="E22:47_1_1_2_1"/>
<font color="Maroon" title="c7">LB</font> is    <a href="xxreal_2.html#M2" title="XXREAL_2:mode.2">LowerBound</a> of  <a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:47_1_1_2_1"><span class="lab"><font color="Green" title="E53">A17</font></span></a>, <a class="ref" href="xxreal_2.html#T5" title="XXREAL_2:th.5">XXREAL_2:5</a></span>;<br/><a NAME="E23:47_1_1_2_1"/><span class="kw">then </span>
 <a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">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="txt" href="#E21:47_1_1_2_1"><span class="lab"><font color="Green" title="E55">A19</font></span></a>, <a class="ref" href="supinf_1.html#D12" title="SUPINF_1:def.12">SUPINF_1:def 12</a></span>;<br/><a NAME="E24:47_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">seq1</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">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>;<br/><a NAME="E25:47_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">seq1</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">k</font> is  <a href="rinfsup2.html#V3" title="RINFSUP2:attr.3">bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:47_1_1_2_1"><span class="lab"><font color="Green" title="E54">A18</font></span></a>, <a class="ref" href="rinfsup2.html#D5" target="_self" title="RINFSUP2:def.5">Def5</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">rseq1</font> = <font color="Maroon" title="c1">seq1</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">k</font> as   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="rinfsup2.html#T11" target="_self" title="RINFSUP2:th.11">Th11</a></span>;<br/><a NAME="E27:47_1_1_2_1"/>
<font color="Maroon" title="c2">seq2</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">k2</font> is  <a href="rinfsup2.html#V1" title="RINFSUP2:attr.1">bounded_below</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:47_1_1_2_1"><span class="lab"><font color="Green" title="E50">A12</font></span></a>, <a class="ref" href="rinfsup2.html#D5" target="_self" title="RINFSUP2:def.5">Def5</a></span>;<br/><a NAME="E28:47_1_1_2_1"/><span class="kw">then </span>
 <a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">k2</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="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="c9">LB</font> being    <a href="xxreal_2.html#M2" title="XXREAL_2:mode.2">LowerBound</a> of  <a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">k2</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E30:47_1_1_2_1"/><span class="lab"><font color="Green" title="E56">A20</font></span>: 
<font color="Maroon" title="c9">LB</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="supinf_1.html#D12" title="SUPINF_1:def.12">SUPINF_1:def 12</a></span>;<br/><a NAME="E31:47_1_1_2_1"/><span class="lab"><font color="Green" title="E57">A21</font></span>: 
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c1">seq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_1"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="ref" href="rinfsup2.html#T20" target="_self" title="RINFSUP2:th.20">Th20</a></span>;<br/><a NAME="E32:47_1_1_2_1"/>
<font color="Maroon" title="c2">seq2</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">k2</font> is  <a href="rinfsup2.html#V2" title="RINFSUP2:attr.2">bounded_above</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:47_1_1_2_1"><span class="lab"><font color="Green" title="E50">A12</font></span></a>, <a class="ref" href="rinfsup2.html#D5" target="_self" title="RINFSUP2:def.5">Def5</a></span>;<br/><a NAME="E33:47_1_1_2_1"/><span class="kw">then </span>
 <a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">k2</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="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="c10">UB</font> being    <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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">k2</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E35:47_1_1_2_1"/><span class="lab"><font color="Green" title="E58">A22</font></span>: 
<font color="Maroon" title="c10">UB</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="supinf_1.html#D11" title="SUPINF_1:def.11">SUPINF_1:def 11</a></span>;<br/><a NAME="E36:47_1_1_2_1"/><span class="lab"><font color="Green" title="E59">A23</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">k2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_2_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c11">k3</font> = <font color="Maroon" title="c6">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">k2</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="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a>, <a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c12">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c12">y</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span> implies <font color="Maroon" title="c12">y</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">k2</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E2:47_1_1_2_1_2"/>
<font color="Maroon" title="c12">y</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</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="c12">y</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">k2</font></span>)</span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:47_1_1_2_1_2"/><span class="lab"><font color="Green" title="E60">A24</font></span>: 
<font color="Maroon" title="c13">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">seq2</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:47_1_1_2_1_2"/><span class="lab"><font color="Green" title="E61">A25</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">n</font> = <font color="Maroon" title="c13">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="#E4:47_1_1_2_1_2"><span class="lab"><font color="Green" title="E60">A24</font></span></a></span>;<br/><a NAME="E7:47_1_1_2_1_2"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">seq2</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:47_1_1_2_1_2"><span class="lab"><font color="Green" title="E61">A25</font></span></a>, <a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></span>;<br/><a NAME="E8:47_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">seq2</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">k3</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">n</font></span>)</span></span>)</span>
 ;<br/><a NAME="E9:47_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">k2</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k3</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E10:47_1_1_2_1_2"/>
<font color="Maroon" title="c12">y</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">k2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:47_1_1_2_1"><span class="lab"><font color="Green" title="E59">A23</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</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="E38:47_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A26</font></span>: 
 <a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">k2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><a NAME="E39:47_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">UB</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#T6" title="XXREAL_2:th.6">XXREAL_2:6</a></span>;<br/><a NAME="E40:47_1_1_2_1"/><span class="kw">then </span>
 <a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">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="txt" href="#E35:47_1_1_2_1"><span class="lab"><font color="Green" title="E58">A22</font></span></a>, <a class="ref" href="supinf_1.html#D11" title="SUPINF_1:def.11">SUPINF_1:def 11</a></span>;<br/><a NAME="E41:47_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A27</font></span>: 
<font color="Maroon" title="c2">seq2</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">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>;<br/><a NAME="E42:47_1_1_2_1"/>
<font color="Maroon" title="c9">LB</font> is    <a href="xxreal_2.html#M2" title="XXREAL_2:mode.2">LowerBound</a> of  <a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:47_1_1_2_1"><span class="lab"><font color="Green" title="E60">A26</font></span></a>, <a class="ref" href="xxreal_2.html#T5" title="XXREAL_2:th.5">XXREAL_2:5</a></span>;<br/><a NAME="E43:47_1_1_2_1"/><span class="kw">then </span>
 <a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">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="txt" href="#E30:47_1_1_2_1"><span class="lab"><font color="Green" title="E56">A20</font></span></a>, <a class="ref" href="supinf_1.html#D12" title="SUPINF_1:def.12">SUPINF_1:def 12</a></span>;<br/><a NAME="E44:47_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">seq2</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">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>;<br/><a NAME="E45:47_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">seq2</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">k</font> is  <a href="rinfsup2.html#V3" title="RINFSUP2:attr.3">bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:47_1_1_2_1"><span class="lab"><font color="Green" title="E61">A27</font></span></a>, <a class="ref" href="rinfsup2.html#D5" target="_self" title="RINFSUP2:def.5">Def5</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">rseq2</font> = <font color="Maroon" title="c2">seq2</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">k</font> as   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="rinfsup2.html#T11" target="_self" title="RINFSUP2:th.11">Th11</a></span>;<br/><a NAME="E47:47_1_1_2_1"/><span class="lab"><font color="Green" title="E62">A28</font></span>: 
<font color="Maroon" title="c2">seq2</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">k</font> is  <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_1_2_1"><span class="lab"><font color="Green" title="E47">A9</font></span></a>, <a class="ref" href="rinfsup2.html#T20" target="_self" title="RINFSUP2:th.20">Th20</a></span>;<br/><a NAME="E48:47_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A29</font></span>: 
<font color="Maroon" title="c11">rseq2</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rinfsup2.html#T15" target="_self" title="RINFSUP2:th.15">Th15</a></span>;<br/><a NAME="E49:47_1_1_2_1"/><span class="lab"><font color="Green" title="E64">A30</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="c8">rseq1</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#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">rseq2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_2_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">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="c8">rseq1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">rseq2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span><br/>

<a NAME="E1:47_1_1_2_1_3"/><span class="lab"><font color="Green" title="E65">A31</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">seq2</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></span>;<br/>
<a NAME="E2:47_1_1_2_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">seq1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:47_1_1_2_1_3"/>
<font color="Maroon" title="c8">rseq1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">rseq2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47"><span class="lab"><font color="Green" title="E44">A3</font></span></a>, <a class="txt" href="#E1:47_1_1_2_1_3"><span class="lab"><font color="Green" title="E65">A31</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="E50:47_1_1_2_1"/><span class="lab"><font color="Green" title="E65">A32</font></span>: 
<font color="Maroon" title="c1">seq1</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">k</font> is  <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_1"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="ref" href="rinfsup2.html#T20" target="_self" title="RINFSUP2:th.20">Th20</a></span>;<br/><a NAME="E51:47_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A33</font></span>: 
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c8">rseq1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rinfsup2.html#T15" target="_self" title="RINFSUP2:th.15">Th15</a></span>;<br/><a NAME="E52:47_1_1_2_1"/><span class="lab"><font color="Green" title="E67">A34</font></span>: 
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c11">rseq2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:47_1_1_2_1"><span class="lab"><font color="Green" title="E62">A28</font></span></a>, <a class="ref" href="rinfsup2.html#T15" target="_self" title="RINFSUP2:th.15">Th15</a></span>;<br/><a NAME="E53:47_1_1_2_1"/>
<font color="Maroon" title="c8">rseq1</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:47_1_1_2_1"><span class="lab"><font color="Green" title="E65">A32</font></span></a>, <a class="ref" href="rinfsup2.html#T15" target="_self" title="RINFSUP2:th.15">Th15</a></span>;<br/><span class="kw">hence </span><a NAME="E54:47_1_1_2_1"/>
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c1">seq1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c2">seq2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:47_1_1_2_1"><span class="lab"><font color="Green" title="E63">A29</font></span></a>, <a class="txt" href="#E51:47_1_1_2_1"><span class="lab"><font color="Green" title="E66">A33</font></span></a>, <a class="txt" href="#E52:47_1_1_2_1"><span class="lab"><font color="Green" title="E67">A34</font></span></a>, <a class="txt" href="#E49:47_1_1_2_1"><span class="lab"><font color="Green" title="E64">A30</font></span></a>, <a class="txt" href="#E31:47_1_1_2_1"><span class="lab"><font color="Green" title="E57">A21</font></span></a>, <a class="txt" href="#E11:47_1_1_2_1"><span class="lab"><font color="Green" title="E51">A13</font></span></a>, <a class="ref" href="seq_2.html#T32" title="SEQ_2:th.32">SEQ_2:32</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_2_2"><span class="kw">suppose </span></a><a NAME="E1:47_1_1_2_2"/><span class="lab"><font color="Green" title="E47">A35</font></span>: 
<font color="Maroon" title="c2">seq2</font> is  <a href="mesfunc5.html#V8" title="MESFUNC5:attr.8">convergent_to_+infty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c1">seq1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c2">seq2</font></span><br/><div class="add"><a NAME="E2:47_1_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">seq2</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#D11" title="MESFUNC5:def.11">MESFUNC5:def 11</a></span>;<br/><a NAME="E3:47_1_1_2_2"/><span class="kw">then </span>
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c2">seq2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_1_2_2"><span class="lab"><font color="Green" title="E47">A35</font></span></a>, <a class="ref" href="mesfunc5.html#D12" title="MESFUNC5:def.12">MESFUNC5:def 12</a></span>;<br/><span class="kw">hence </span><a NAME="E4:47_1_1_2_2"/>
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c1">seq1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c2">seq2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T3" title="XXREAL_0:th.3">XXREAL_0:3</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_2"><span class="kw">suppose </span></a><a NAME="E1:47_1_2"/><span class="lab"><font color="Green" title="E45">A36</font></span>: 
<font color="Maroon" title="c1">seq1</font> is  <a href="mesfunc5.html#V8" title="MESFUNC5:attr.8">convergent_to_+infty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c1">seq1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c2">seq2</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">g</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">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"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">g</font> implies  ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Maroon" title="c3">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">seq2</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">m</font> )</span><br/><span class="kw">assume </span><a NAME="E1:47_1_2_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Maroon" title="c3">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">seq2</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">m</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:47_1_2_1"/><span class="lab"><font color="Green" title="E46">A37</font></span>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c3">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">seq1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_2"><span class="lab"><font color="Green" title="E45">A36</font></span></a>, <a class="ref" href="mesfunc5.html#D9" title="MESFUNC5:def.9">MESFUNC5:def 9</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">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="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">m</font> implies <font color="Maroon" title="c3">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">seq2</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">m</font> )</span><br/><a NAME="E1:47_1_2_1_1"/>
<font color="Maroon" title="c5">m</font> is    <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="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><a NAME="E2:47_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A38</font></span>: 
<font color="Maroon" title="c1">seq1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">seq2</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47"><span class="lab"><font color="Green" title="E44">A3</font></span></a></span>;<br/><span class="kw">assume </span><a NAME="E3:47_1_2_1_1"/>
<font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">m</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="c3">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">seq2</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">m</font></span><br/><a NAME="E4:47_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">seq1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47_1_2_1"><span class="lab"><font color="Green" title="E46">A37</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:47_1_2_1_1"/>
<font color="Maroon" title="c3">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">seq2</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47_1_2_1_1"><span class="lab"><font color="Green" title="E47">A38</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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="E5:47_1_2_1"/>
 ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Maroon" title="c3">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">seq2</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">m</font>
 ; <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="E3:47_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A39</font></span>: 
<font color="Maroon" title="c2">seq2</font> is  <a href="mesfunc5.html#V8" title="MESFUNC5:attr.8">convergent_to_+infty</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#D9" title="MESFUNC5:def.9">MESFUNC5:def 9</a></span>;<br/><a NAME="E4:47_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">seq2</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#D11" title="MESFUNC5:def.11">MESFUNC5:def 11</a></span>;<br/><a NAME="E5:47_1_2"/><span class="kw">then </span>
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c2">seq2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47_1_2"><span class="lab"><font color="Green" title="E46">A39</font></span></a>, <a class="ref" href="mesfunc5.html#D12" title="MESFUNC5:def.12">MESFUNC5:def 12</a></span>;<br/><span class="kw">hence </span><a NAME="E6:47_1_2"/>
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c1">seq1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c2">seq2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T3" title="XXREAL_0:th.3">XXREAL_0:3</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_3"><span class="kw">suppose </span></a><a NAME="E1:47_1_3"/><span class="lab"><font color="Green" title="E45">A40</font></span>: 
<font color="Maroon" title="c1">seq1</font> is  <a href="mesfunc5.html#V9" title="MESFUNC5:attr.9">convergent_to_-infty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c1">seq1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c2">seq2</font></span><br/><div class="add"><a NAME="E2:47_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c1">seq1</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#D11" title="MESFUNC5:def.11">MESFUNC5:def 11</a></span>;<br/><a NAME="E3:47_1_3"/><span class="kw">then </span>
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c1">seq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_3"><span class="lab"><font color="Green" title="E45">A40</font></span></a>, <a class="ref" href="mesfunc5.html#D12" title="MESFUNC5:def.12">MESFUNC5:def 12</a></span>;<br/><span class="kw">hence </span><a NAME="E4:47_1_3"/>
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c1">seq1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c2">seq2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T5" title="XXREAL_0:th.5">XXREAL_0:5</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></div><span class="kw">end;</span></div>

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